Fluent Essays

This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 Practice Problems Section 4F (#1-10) Use each of the following categorical association 𝜒𝜒2-test statistics and the corresponding critical values to fill out the table. 𝜒𝜒2-test stat Sentence to explain 𝜒𝜒2-test statistic. Critical Value Does the 𝜒𝜒2-test statistic fall in a tail determined by the critical value? (Yes or No) Does sample data significantly disagree with 𝐻𝐻0? 1. +1.573 +4.117 2. +6.226 +5.118 3. +2.144 +4.121 4. +3.415 +5.091 5. +13.718 +7.189 6. +0.972 +4.812 7. +31.652 +12.557 8. +11.185 +5.181 9. +25.443 +7.008 10. +1.133 +8.336 (#11-20) Use each of the following P-values and corresponding significance levels to fill out the table. P-value Proportion P- value \% Sentence to explain the P-value Significance Level \% Significance level Proportion If 𝐻𝐻0 is true, could the sample data occur by random chance or is it unlikely? Reject 𝐻𝐻0 or Fail to reject 𝐻𝐻0? 11. 0.263 10\% 12. 0.0042 1\% 13. 5.22 × 10−4 5\% 14. 0.0639 1\% 15. 0 5\% 16. 0.539 10\% 17. 0.0419 5\% 18. 0.0027 10\% 19. 7.73 × 10−8 1\% 20. 0.674 5\% 21. If we have two raw categorical data sets, what must we click on in Statcato to perform a categorical association test? 22. If we have summary counts organized in a contingency table, what must we click on in Statcato to perform a categorical association test? 23. What are the assumptions for a categorical association test if the data was collected from on random sample? 24. What are the assumptions for a categorical association test if the data was collected from multiple random samples? 25. How are the expected counts calculated in a categorical association test? 26. If the expected counts from the null hypothesis are significantly different from the observed sample counts, describe the effect on the Chi-Squared test statistic. 27. If the expected counts from the null hypothesis are close to the observed sample counts, describe the effect on the Chi-Squared test statistic. https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 (#28-31) Directions: For each of the following problems, use the Statcato printout provided to answer the following questions. a) Write the null and alternative hypothesis. Make sure to label which one is the claim. b) Check the assumptions for the categorical association test. c) What is the Chi-squared test statistic? Write a sentence to explain the test statistic. d) Does the test statistic fall in the tail determined by the critical value? e) Does the sample data significantly disagree with the null hypothesis? Explain your answer. f) Are the observed counts significantly different from the expected counts? Explain your answer. g) What is the P-value? Write a sentence to explain the P-value. h) Compare the P-value to the significance level. Should we reject the null hypothesis or fail to reject the null hypothesis? Explain your answer. i) If the null hypothesis was true, could the sample data or more extreme have occurred by sampling variability or is it unlikely to be sampling variability? Explain your answer. j) Write a conclusion for the test addressing evidence and the claim. Explain your conclusion in non-technical language. k) Are the categories related or not? Explain your answer. 28. A random sample of male college students were asked their major. Later, a random sample of female college students were asked their major. The goal of the study was to show that gender is not related to major. Use a 5\% significance level and the Statcato printout below to answer the questions given above. https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 29. A random sample of adults were asked their blood type and Rh status. (Blood tests were provided for those that did not know their blood type and Rh status.) The goal of the study was to show that blood type is related to Rh status (dependent). Use a 10\% significance level and the Statcato printout below to answer the questions given above. 30. A hospital wanted to determine if the age of a patient is not related to what part of the hospital they were in. They took a random sample of patients that have visited their hospital and determined both their age and the part of the hospital. The ages were broken up into age groups. Use a 1\% significance level and the Statcato printout below to answer the questions given above. https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 31. A random sample of American adults was taken and their health and education status obtained. Test to test the claim that health and education are related. Use a 5\% significance level and the Statcato printout below to answer the questions given above. (#32-35) Directions: Use StatKey at www.lock5stat.com to simulate the following chi-squared categorical association tests. Go to the “More Advanced Randomization Tests” menu at the bottom of the StatKey page. Click on the button that says, “𝜒𝜒2 Test for Association”. Click on “Edit Data” and type in the contingency table provided. Click on “Generate 1000 Samples” a few times to create the simulated sampling distribution and answer the following questions. a) Write the null and alternative hypothesis. Make sure to label which one is the claim. b) Check the assumptions for the categorical association test. Assume the data was collected randomly. Under “Original Sample”, click on “Show Details” to see the expected counts. c) Use the formula df = (r – 1)(c – 1) to calculate the degrees of freedom. “r” is the number of rows and “c” is the number of columns not counting the totals. d) What is the Chi-squared test statistic? Write a sentence to explain the test statistic. e) Put the significance level proportion in the right tail proportion to calculate the critical value. What is the critical value? (Answers will vary slightly.) Does the original sample 𝜒𝜒2 test statistic fall in the tail determined by the critical value? f) Does the sample data significantly disagree with the null hypothesis? Explain your answer. g) Are the observed counts significantly different from the expected counts? Explain your answer. h) Put the original sample test 𝜒𝜒2 test statistic in the bottom box in the simulation to calculate the P-value. What is the P-value? (Answers will vary slightly.) Write a sentence to explain the P-value. https://creativecommons.org/licenses/by/4.0/ http://www.lock5stat.com/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 i) Compare the P-value to the significance level. Should we reject the null hypothesis or fail to reject the null hypothesis? Explain your answer. j) If the null hypothesis was true, could the sample data or more extreme have occurred by sampling variability or is it unlikely to be sampling variability? Explain your answer. k) Write a conclusion for the test addressing evidence and the claim. Explain your conclusion in non-technical language. l) Are the categories related or not? Explain your answer. 32. We want to know if the state a home is built in is related to the size of the home. A random sample of homes in the U.S was taken. Click on “Edit Data” in StatKey and type in the following contingency table. Do not forget to include a space after the commas. Use a 5\% significance level and randomized simulation to test the claim that the state is not related to size of the home. [blank], CA, NJ, NY, PA Large, 7, 6, 7, 3 Small, 23, 24, 23, 27 33. Open the “Car Data” at www.matt-teachout.org. Copy and paste the “Country” and “Cylinders” columns next to each other in a new Excel spreadsheet. Then copy the two columns together. Click on “Edit Data” in StatKey and paste the two columns into StatKey. Use a 1\% significance level to test the claim that the country a car is made in is related to the cylinders. Answer the questions above. 34. We want to show that gender is related to getting an award. A random sample of people that won famous awards in the Olympic, Academia, and Nobel was taken and their gender was noted. Click on “Edit Data” in StatKey and type in the following contingency table. Do not forget to include a space after the commas. Use a 10\% significance level and randomized simulation to test the claim that awards are related to gender. [blank], Olympic, Academy, Nobel Male, 109, 11, 73 Female, 73, 20, 76 35. Open the “Math 140 Fall 2015 Survey Data” at www.matt-teachout.org. Copy and paste the “Tattoo” and “Favorite Social Media” columns next to each other in a new Excel spreadsheet. Then copy the two columns together. Click on “Edit Data” in StatKey and paste the two columns into StatKey. Use a 5\% significance level to test the claim that having a tattoo or not is not related to social media. Answer the questions above. -------------------------------------------------------------------------------------------------------------------------------------------------------- https://creativecommons.org/licenses/by/4.0/ http://www.matt-teachout.org/ http://www.matt-teachout.org/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 Practice Problems Section 4G 1. How can tell which variable should be the explanatory variable and which variable should be the response variable? 2. How can we use the correlation coefficient (r) to determine if there is strong positive correlation? How can we use the correlation coefficient (r) to determine if there is strong negative correlation? How can we use the correlation coefficient (r) to determine if there is no correlation? 3. What is the definition of the coefficient of determination (𝑟𝑟2)? 4. What are the two definitions for the standard deviation of the residual errors (𝑠𝑠𝑒𝑒)? 5. What is the definition of the slope of the regression line? 6. What is the definition of the y-intercept of the regression line? 7. What is extrapolation? Why should we avoid extrapolation? (#8-16) Directions: Go to www.matt-teachout.org, click on “Statistics” and then “Data Sets”. Open the indicated data. Copy and paste the two indicated columns of quantitative data next to each other on a new Excel spreadsheet. Then copy the two columns together. Now go to www.lock5stat.com and click on StatKey. Under the “Descriptive Statistics and Graphs” menu click on “Two Quantitative Variables”. Click on “Edit Data” and paste the two columns together into StatKey. Then answer indicated questions. 8. Open the cigarette data. Let the explanatory variable (X) represent the amount of nicotine (milligrams) and the response variable (Y) represent the amount of tar (milligrams). a) Look at the scatterplot and the correlation coefficient (r). Describe the strength and direction of the linear relationship. b) Square the correlation coefficient in StatKey to calculate 𝑟𝑟2. This is also called the coefficient of determination. Write a sentence to explain 𝑟𝑟2. c) Find the slope of the regression line. Write a sentence to explain the slope. d) Find the y-intercept. Write a sentence to explain the y-intercept. Does the y-intercept make sense in the context of this data? e) The standard deviation of the residual errors was 1.3 mg. Explain the two meanings of this statistic. f) Use the regression line formula to predict the amount of tar if a cigarette contains 1.2 mg of nicotine. How much error could there be in this prediction. 9. Open the cigarette data. Let the explanatory variable (X) represent the amount of nicotine (mg) and the response variable (Y) represent the amount of carbon monoxide in parts per million (PPM). a) Look at the scatterplot and the correlation coefficient (r). Describe the strength and direction of the linear relationship. b) Square the correlation coefficient in StatKey to calculate 𝑟𝑟2. This is also called the coefficient of determination. Write a sentence to explain 𝑟𝑟2. c) Find the slope of the regression line. Write a sentence to explain the slope. d) Find the y-intercept. Write a sentence to explain the y-intercept. Does the y-intercept make sense in the context of this data? e) The standard deviation of the residual errors was 2.3 PPM. Explain the two meanings of this statistic. f) Use the regression line formula to predict the amount of carbon monoxide if a cigarette contains 1.2 mg of nicotine. How much error could there be in this prediction. https://creativecommons.org/licenses/by/4.0/ http://www.matt-teachout.org/ http://www.lock5stat.com/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 10. Open the health data. Let the explanatory variable (X) represent the systolic blood pressure (mm of Hg) and the response variable (Y) represent the diastolic blood pressure (mm of Hg). Use the combined columns with 80 randomly selected adults. Do not separate by gender. a) Look at the scatterplot and the correlation coefficient (r). Describe the strength and direction of the linear relationship. b) Square the correlation coefficient in StatKey to calculate 𝑟𝑟2. This is also called the coefficient of determination. Write a sentence to explain 𝑟𝑟2. c) Find the slope of the regression line. Write a sentence to explain the slope. d) Find the y-intercept. Write a sentence to explain the y-intercept. Does the y-intercept make sense in the context of this data? e) The standard deviation of the residual errors was 7.4579 mm of Hg. Explain the two meanings of this statistic. f) Use the regression line formula to predict the diastolic blood pressure of a person who has a systolic blood pressure of 130. How much error might there be in that prediction? 11. Open the health data. Let the explanatory variable (X) represent the waist size in centimeters and the response variable (Y) represent the weight in pounds. Use the combined columns with 80 randomly selected adults. Do not separate by gender. a) Look at the scatterplot and the correlation coefficient (r). Describe the strength and direction of the linear relationship. b) Square the correlation coefficient in StatKey to calculate 𝑟𝑟2. This is also called the coefficient of determination. Write a sentence to explain 𝑟𝑟2. c) Find the slope of the regression line. Write a sentence to explain the slope. d) Find the y-intercept. Write a sentence to explain the y-intercept. Does the y-intercept make sense in the context of this data? e) The standard deviation of the residual errors was 14.6809 pounds. Explain the two meanings of this statistic. f) Use the regression line formula to predict the weight of a person who has a waist size of 100 cm. How much error might there be in that prediction? 12. Open the health data. Let the explanatory variable (X) represent the age in years and the response variable (Y) represent the cholesterol in milligrams per deciliter (mg/dL). Use the combined columns with 80 randomly selected adults. Do not separate by gender. a) Look at the scatterplot and the correlation coefficient (r). Describe the strength and direction of the linear relationship. b) Square the correlation coefficient in StatKey to calculate 𝑟𝑟2. This is also called the coefficient of determination. Write a sentence to explain 𝑟𝑟2. c) Find the slope of the regression line. Write a sentence to explain the slope. d) Find the y-intercept. Write a sentence to explain the y-intercept. Does the y-intercept make sense in the context of this data? e) The standard deviation of the residual errors was 255.3625 mg/dL. Explain the two meanings of this statistic. f) Use the regression line formula to predict the cholesterol of a person who is 40 years old. How much error might there be in that prediction? 13. Open the bear data. Let the explanatory variable represent the age of the bear in months and the response variable represent the length of the bear in inches. a) Look at the scatterplot and the correlation coefficient (r). Describe the strength and direction of the linear relationship. b) Square the correlation coefficient in StatKey to calculate 𝑟𝑟2. This is also called the coefficient of determination. Write a sentence to explain 𝑟𝑟2. c) Find the slope of the regression line. Write a sentence to explain the slope. https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 d) Find the y-intercept. Write a sentence to explain the y-intercept. Does the y-intercept make sense in the context of this data? e) The standard deviation of the residual errors was 7.51 inches. Explain the two meanings of this statistic. f) Use the regression line formula to predict the length of a bear that is 24 months old. How much error might there be in that prediction? 14. Open the bear data. Let the explanatory variable represent the neck circumference of the bear and the response variable represent the weight of the bear in pounds. a) Look at the scatterplot and the correlation coefficient (r). Describe the strength and direction of the linear relationship. b) Square the correlation coefficient in StatKey to calculate 𝑟𝑟2. This is also called the coefficient of determination. Write a sentence to explain 𝑟𝑟2. c) Find the slope of the regression line. Write a sentence to explain the slope. d) Find the y-intercept. Write a sentence to explain the y-intercept. Does the y-intercept make sense in the context of this data? e) The standard deviation of the residual errors was 43.9 pounds. Explain the two meanings of this statistic. f) Use the regression line formula to predict the weight of a bear that has a neck circumference of 24 inches. How much error might there be in that prediction? 15. Open the car data. Let the explanatory variable (X) represent the weight of the car in tons and the response variable (Y) represent the gas mileage in miles per gallon. a) Look at the scatterplot and the correlation coefficient (r). Describe the strength and direction of the linear relationship. b) Square the correlation coefficient in StatKey to calculate 𝑟𝑟2. This is also called the coefficient of determination. Write a sentence to explain 𝑟𝑟2. c) Find the slope of the regression line. Write a sentence to explain the slope. d) Find the y-intercept. Write a sentence to explain the y-intercept. Does the y-intercept make sense in the context of this data? e) The standard deviation of the residual errors was 2.8516 mpg. Explain the two meanings of this statistic. f) Use the regression line formula to predict the mpg for a car that weighs 3 tons. How much error might there be in that prediction? 16. Open the car data. Displacement is the amount of liquid in cubic centimeters forced out by the piston. Let the explanatory variable (X) represent the horsepower of the car and the response variable (Y) represent the displacement of the car (cc). a) Look at the scatterplot and the correlation coefficient (r). Describe the strength and direction of the linear relationship. b) Square the correlation coefficient in StatKey to calculate 𝑟𝑟2. This is also called the coefficient of determination. Write a sentence to explain 𝑟𝑟2. c) Find the slope of the regression line. Write a sentence to explain the slope. d) Find the y-intercept. Write a sentence to explain the y-intercept. Does the y-intercept make sense in the context of this data? e) The standard deviation of the residual errors was 44.138 cubic centimeters. Explain the two meanings of this statistic. f) Use the regression line formula to predict the number of cc’s of displacement for a car with 120 horsepower. How much error might there be in that prediction? -------------------------------------------------------------------------------------------------------------------------------------------------------- https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 Problems Section 4D (#1-10) Use each of the following Goodness of Fit 𝜒𝜒2-test statistics and the corresponding critical values to fill out the table. 𝜒𝜒2-test stat Sentence to explain 𝜒𝜒2-test statistic. Critical Value Does the 𝜒𝜒2-test statistic fall in a tail determined by the critical value? (Yes or No) Does sample data significantly disagree with 𝐻𝐻0? 1. +28.573 +9.117 2. +1.226 +7.113 3. +2.137 +5.521 4. +14.415 +6.114 5. +3.718 +7.182 6. +0.891 +3.994 7. +51.652 +14.881 8. +1.185 +4.181 9. +2.442 +8.619 10. +14.133 +10.336 (#11-20) Use each of the following P-values and corresponding significance levels to fill out the table. P-value Proportion P- value \% Sentence to explain the P-value Significance Level \% Significance level Proportion If 𝐻𝐻0 is true, could the sample data occur by random chance or is it unlikely? Reject 𝐻𝐻0 or Fail to reject 𝐻𝐻0? 11. 0.0006 10\% 12. 0.042 1\% 13. 9.16 × 10−7 5\% 14. 0.739 1\% 15. 0.0035 5\% 16. 0 10\% 17. 0.419 5\% 18. 0.0274 10\% 19. 3.77 × 10−5 1\% 20. 0.067 5\% 21. How is the degrees of freedom calculated in a Goodness of Fit test? 22. The 𝝌𝝌𝟐𝟐-test statistic compares the observed sample counts to the expected counts from 𝑯𝑯𝟎𝟎. Explain how the expected counts are calculated. 23. Explain how the 𝝌𝝌𝟐𝟐-test statistic is calculated from the observed and expected counts. 24. If the observed sample counts were significantly different from the expected counts, would the 𝜒𝜒2-test statistic be large or small? Explain why. 25. If the observed sample counts were close to the expected counts, would the 𝜒𝜒2-test statistic be large or small? Explain why. https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 (#26-29) Directions: Use StatKey at www.lock5stat.com to simulate the following Chi-squared Goodness of Fit tests. Go to “more advanced randomization tests” at the bottom of the StatKey page. Click on the button that says “𝜒𝜒2 Goodness of Fit”. Under “Edit Data”, type in the given sample data. Create a randomized simulation of the null hypothesis to answer the following questions. a) Write the null and alternative hypothesis. Include relationship implications. b) What is the degrees of freedom? c) What is the Chi-squared test statistic? Write a sentence to explain the test statistic. d) Adjust the right tail of your simulation to reflect the significance level. Did the Chi-squared test statistic fall in the tail? e) Does the sample data significantly disagrees with the null hypothesis? Explain your answer. f) Are the observed counts in the sample data significantly different from the expected counts from the null hypothesis? Explain your answer. g) Put the Chi-squared test statistic into the bottom box in the right tail of your simulation in order to calculate the P- value. What was the P-value? (Answers will vary.) Write a sentence to explain the P-value. h) Use the P-value and significance level to determine if the sample data could have occurred by random chance (sampling variability) or is it unlikely to random chance? Explain your answer. i) Should we reject the null hypothesis or fail to reject the null hypothesis? Explain your answer. j) Write a conclusion for the hypothesis test. Explain your conclusion in plain language. k) Is the population proportion related to the categorical variable or not? Explain your answer. 26. It is a big job to write and grade the AP-statistics exam for high school students each year. It is a difficult multiple-choice exam. All questions have five possible answers A-E. Use a 5\% significance level and the following sample data to test the claim that percent of A answers is the same as the percent of B answers which is the same as C, D and E. This would indicate that the letter of the answer is not related to the percentage of times it happens. You can assume that the sample data meets the assumptions. Type the following sample data under the “Edit Data” menu of StatKey. Choice, Count A, 85 B, 90 C, 79 D, 78 E, 68 https://creativecommons.org/licenses/by/4.0/ http://www.lock5stat.com/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 27. We collected data from all of the math 140 statistics students in the fall 2015 semester. A person that works at COC thinks that 80\% of COC students drive alone, 10\% carpool, 5\% are dropped off by someone, 2\% walk, 1\% bike, and 2\% use public transportation. Use a 5\% significance level and the following sample data to test the claim that these percentages are wrong. You can assume that the data meets the assumptions for inference. Type in the proportions under “Null Hypothesis” in StatKey. Under “Edit Data” type in the following sample data from the fall 2015 survey data. Choice, Count Bicycle, 1 Carpool, 30 Drive Alone, 267 Dropped Off, 18 Public Transportation, 6 Walk, 10 28. We collected data from all of the math 140 statistics students in the fall 2015 semester. Use a randomized simulation in StatKey, a 5\% significance level, and the following sample data to test the claim that the population percentages for the different political parties are different. This would indicate that the political party is related to the population percentages. You can assume that the data meets the assumptions for inference. Under “Edit Data”, type in the following sample data from the fall 2015 survey data. Choice, Count Democratic, 110 Republican, 63 Independent, 65 Other, 90 29. Juries are required to meet the racial demographic of the county they represent. Here is the racial demographic for Alameda county: 54\% Caucasian, 18\% African American, 12\% Hispanic American, 15\% Asian American, and 1\% other. We are worried that the juries in Alameda County may not be representing these percentages. Use randomized simulation, a 1\% significance level, and the following observed sample counts to test the claim that the juries do not represent the demographic of the county. Under the “Edit Data” menu in StatKey, type in the following sample counts. Jury Sample Data Observed Counts Race, Count Caucasian, 780 African American, 117 Hispanic American, 114 Asian American, 384 Other, 58 https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 Under the “Null Hypothesis” menu, type in the following. (#30-32) Directions: Use the following Statcato printouts to answer the following questions. a) Write the null and alternative hypothesis. Include relationship implications. b) Check the assumptions for a Goodness of Fit test. c) What is the Chi-squared test statistic? Write a sentence to explain the test statistic. d) Did the Chi-squared test statistic fall in the tail determined by the critical value? e) Does the sample data significantly disagrees with the null hypothesis? Explain your answer. f) Are the observed counts in the sample data significantly different from the expected counts from the null hypothesis? Explain your answer. g) What was the P-value? Write a sentence to explain the P-value. h) Use the P-value and significance level to determine if the sample data could have occurred by random chance (sampling variability) or is it unlikely to random chance? Explain your answer. i) Should we reject the null hypothesis or fail to reject the null hypothesis? Explain your answer. j) Write a conclusion for the hypothesis test. Explain your conclusion in plain language. k) Is the population proportion related to the categorical variable or not? Explain your answer. https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 30. An online sports magazine wrote an article about the favorite sports in America. It said that 43\% of Americans prefer Football, 23\% of Americans prefer Baseball, 20\% of Americans prefer Basketball, 8\% of Americans prefer Hockey, and 6\% of Americans prefer Soccer. When 130 randomly selected adults were asked their favorite sport, we found the following: 44 said Football, 26 said Baseball, 29 said Basketball, 13 said Hockey, and 18 said Soccer. Use a 5\% significance level to test the claim that the proportions match the distribution claimed in the magazine article. 31. Thousands of people die from car accidents across the U.S. every year, but is the day of the week related to the probability of having a fatal car accident? To test this claim, use a 1\% significance level and a Goodness of Fit test to determine if the probabilities of a fatal car accident are significantly different. The following random sample data summary gives the observed number of the number of deaths from car accidents in the U.S. for each day of a randomly selected week. The total number of deaths for the week was 805. https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 32. The National Highway Traffic Safety Administration (NHTSA) publishes reports about motorcycle fatalities and helmet use. The following distribution shows the proportion of fatalities by location of injury for motorcycle accidents. The random sample data below shows the distribution of 2068 randomly selected fatalities from riders that were not wearing a helmet. Use a 0.01 significance level to test the claim that the distribution for the sample does not match the proportions given by the NHTSA. Where is the largest discrepancy between the observed and expected value? What does this tell us about the importance of wearing helmets? -------------------------------------------------------------------------------------------------------------------------------------------------------- https://creativecommons.org/licenses/by/4.0/ 22. The ,𝝌-𝟐.-test statistic compares the observed sample counts to the expected counts from ,𝑯-𝟎.. Explain how the expected counts are calculated. 23. Explain how the ,𝝌-𝟐.-test statistic is calculated from the observed and expected counts. This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 Problems Section 4C (#1-10) Use each of the following two-population proportion Z-test statistics and the corresponding critical values to fill out the table. Z-test stat Sentence to explain Z-test statistic. Critical Value Does the Z-test statistic fall in a tail determined by a critical value? (Yes or No) Does sample data significantly disagree with 𝐻𝐻0? 1. −1.835 ±1.645 2. +0.974 +2.576 3. −1.226 −1.96 4. −3.177 ±1.96 5. +2.244 +1.645 6. +1.448 ±2.576 7. −0.883 −2.576 8. +1.117 +1.96 9. +2.139 ±2.576 10. −0.199 −1.645 (#11-20) Use each of the following P-values and corresponding significance levels to fill out the table. P-value Proportion P- value \% Sentence to explain the P-value Significance Level \% Significance level Proportion If 𝐻𝐻0 is true, could the sample data occur by random chance or is it unlikely? Reject 𝐻𝐻0 or Fail to reject 𝐻𝐻0? 11. 0.728 10\% 12. 0.0421 1\% 13. 2.11 × 10−4 5\% 14. 0.0033 1\% 15. 0.176 5\% 16. 0 10\% 17. 0.0628 5\% 18. 0.277 10\% 19. 3.04 × 10−6 1\% 20. 0 5\% 21. Explain the difference between random samples and random assignment. 22. List the assumptions that we need to check for a two-population proportion hypothesis test. 23. List the assumptions that we need to check for a two-population proportion hypothesis test that is using experimental design. 24. Explain how to use a two-population proportion hypothesis test to show that two categorical variables are related. 25. Explain how to use a two-population proportion hypothesis test to show there is a cause and effect between two categorical variables. https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 (#26-30) Directions: Use the following Statcato printouts to answer the following questions. a) Write the null and alternative hypothesis. Include relationship implications. Is this a left-tailed, right-tailed, or two-tailed test? b) Check all of the assumptions for a two-population proportion Z-test. Explain your answers. Does the problem meets all the assumptions? d) Write a sentence to explain the Z-test statistic in context. e) Use the test statistics and the critical value to determine if the sample data significantly disagrees with the null hypothesis. Explain your answer. f) Write a sentence to explain the P-value. g) Use the P-value and significance level to determine if the sample data could have occurred by random chance (sampling variability) or is it unlikely to random chance? Explain your answer. h) Should we reject the null hypothesis or fail to reject the null hypothesis? Explain your answer. i) Write a conclusion for the hypothesis test. Explain your conclusion in plain language. j) Is the population proportion related to the categorical variable or not? Explain your answer. 26. The United States has the highest teen pregnancy rate in the industrialized world. In 2008, a random sample of 1014 teenage girls found that 326 of them were pregnant before the age of 20. In 2012, a random sample of 1025 teenage girls was taken and 334 were found to be pregnant before the age of 20. Let population proportion 1 represent 2008 and population proportion 2 represent 2012. Use a 10\% significance level and the following Statcato printout to test the claim that the population percentage of teen pregnancies in the U.S. is lower in 2008 than it is in 2012. This claim would indicate that the population percentage of U.S. teen pregnancies is related to the year. https://creativecommons.org/licenses/by/4.0/ http://www.livestrong.com/article/12457-teen-pregnancy/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 27. While many Americans favor the legalization of marijuana, opponents of legalization argue that marijuana may be a gateway drug. They believe that if a person uses marijuana, then they are more likely to use other more dangerous illegal drugs. Use the table of random sample data given below and a 5\% significance level to test the claim that marijuana users have a higher percentage of other drug use than non-marijuana users. This claim also would indicate that using Marijuana is related to using other drugs. 28. Use a 1\% significance level and the following Statcato printout to test this claim that gender is not related to abstaining from drinking alcohol. If this is the case, then the percentage of men and women that do not drink alcohol should be the same. We took a random sample of 190 men and found that 66 of them never drink alcohol. We took a random sample of 250 women and found that that 137 of them never drink alcohol. We designated the proportion of men that never drink alcohol as population 1 and the women as population 2. https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 29. A health magazine claims that marriage status is one of the most telling factors for a person’s happiness. Use a 10\% significance level and the Statcato printout below to test the claim that the percent of married people that are unhappy is lower than the percent of single or divorced people that are unhappy. If this is the case, then perhaps being married, single or divorced is related to being unhappy. The following sample data was collected randomly. Population 1 represented married adults and population 2 represented single or divorced adults. 30. A tattoo magazine claimed that the percent of men that have at least one tattoo is greater than the percent of women with at least one tattoo. If this were true, then gender would be related to having a tattoo. Use a 5\% significance level and the following Statcato printout to test this claim. A random sample of 857 men found that 146 of them had at least one tattoo. A random sample of 794 women found that 137 of them had at least one tattoo. Population 1 was the proportion of men with at least one tattoo and population 2 was the proportion of women with at least one tattoo. https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 (#31-34) Directions: go to www.lock5stat.com and click on StatKey. Then under the “Randomization Hypothesis Test” menu click on “Test for Difference in Proportions”. Create a randomized simulation of the null hypothesis to answer the following questions. a) Write the null and alternative hypothesis. Include relationship implications. Is this a left-tailed, right-tailed, or two- tailed test? b) What is the difference between the sample proportions? Adjust the tails of your simulation to reflect the significance level. Did your sample proportion difference fall in the tail? c) Does the sample data significantly disagree with the null hypothesis? Explain your answer. d) Put the sample proportion difference into the bottom box in the appropriate tail of your simulation in order to calculate the P-value. What was the P-value? (Answers will vary.) Write a sentence to explain the P-value. e) Use the P-value and significance level to determine if the sample data could have occurred by random chance (sampling variability) or is it unlikely to random chance? Explain your answer. f) Should we reject the null hypothesis or fail to reject the null hypothesis? Explain your answer. g) Write a conclusion for the hypothesis test. Explain your conclusion in plain language. h) Is the population proportion related to the categorical variable or not? Explain your answer. i) Use the following formula to calculate the Z-test statistic. Write a sentence to explain the Z-test statistic in context. (Answers will vary.) 𝑍𝑍 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑡𝑡 = 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑃𝑃𝑃𝑃𝑃𝑃𝑆𝑆𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐷𝐷𝑃𝑃𝐷𝐷𝐷𝐷𝑆𝑆𝑃𝑃𝑆𝑆𝑃𝑃𝐷𝐷𝑆𝑆 𝑆𝑆𝑃𝑃𝑆𝑆𝑃𝑃𝑆𝑆𝑆𝑆𝑃𝑃𝑆𝑆 𝐸𝐸𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 31. A body mass index of 20-25 indicates that a person is of normal weight for their height and body type. A random sample of 760 women found that 198 of the women had a normal BMI. A random sample of 745 men found that 273 of them had a normal BMI. A fitness magazine claims that the percent of women with a normal BMI is lower than the percent of men with a normal BMI. This would imply that gender is related to having a normal BMI. Let population 1 be the proportion of women with a normal BMI and population 2 be the proportion of men with a normal BMI. Use a 10\% significance level and a randomized simulation in StatKey. 32. A new medicine has been developed that treats high cholesterol. An experiment was conducted and adults were randomly assigned into two groups. The groups had similar gender, ages, exercise patterns and diet. Of the 420 adults in the placebo group, 38 of them showed a decrease in cholesterol. Of the 410 adults in the treatment group, 49 of them showed a decrease in cholesterol. The FDA claims that the medicine is not effective in lowering cholesterol since the proportion for the placebo group and the treatment groups are about the same. Use a randomized simulation in StatKey, and a 1\% significance level to test this claim. 33. A study was done to see if there is a relationship between smoking and being able to get pregnant. Two random samples of women trying to get pregnant were compared. A random sample of 135 women that smoke (population 1) found that 38 were able to get pregnant in the allotted amount of time. A random sample of 543 women that do not smoke (population 2) found that 206 were able to get pregnant in the allotted amount of time. Test the claim that the population percent of smoking women that were able to get pregnant is lower than the population percent of non-smoking women. This claim also implies that smoking is related to getting pregnant. Use a randomized simulation in StatKey, and a 5\% significance level to test this claim. https://creativecommons.org/licenses/by/4.0/ http://www.lock5stat.com/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 34. A study was done to see if there is a relationship between the age of a person (teen or adult) and using text messages to communicate. A random sample of 800 teens (population 1) found that 696 of them use text messages regularly to communicate. A random sample of 2252 adults (population 2) found that 1621 of them use text messages regularly to communicate. Test the claim that population percentages are equal for the two groups implying that age is not related to using text messages. Use a randomized simulation in StatKey, and a 10\% significance level to test this claim. ------------------------------------------------------------------------------------------------------------------------------------------------------- https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 Practice Problems Section 4A (#1-10) Use each of the following two-population mean T-test statistics and the corresponding critical values to fill out the table. Type of Test T-test stat Sentence to explain T-test statistic. Critical Value Does the T-test statistic fall in a tail determined by a critical value? (Yes or No) Are the sample means from the two groups significantly different or not? Explain. Does sample data significantly disagree with 𝐻𝐻0? Explain. 1. Right Tailed +1.383 +2.447 2. Left Tailed −2.851 −1.773 3. Two Tailed −1.501 ±2.006 4. Right Tailed +3.561 +1.692 5. Two Tailed +0.887 ±1.943 6. Left Tailed −1.003 −2.759 7. Two Tailed −4.416 ±1.994 8. Right Tailed +0.275 +1.839 9. Left Tailed −1.461 −1.674 10. Two Tailed +2.330 ±2.138 (#11-20) Use each of the following P-values and corresponding significance levels to fill out the table. P-value Proportion P- value \% Sentence to explain the P-value Significance Level \% Significance level Proportion If 𝐻𝐻0 is true, could the sample data occur by sampling variability or is it unlikely? Reject 𝐻𝐻0 or Fail to reject 𝐻𝐻0? 11. 0.0007 10\% 12. 0.421 1\% 13. 8.71 × 10−5 5\% 14. 0.339 1\% 15. 0.076 5\% 16. 0 10\% 17. 0.528 5\% 18. 0.0277 10\% 19. 3.04 × 10−6 1\% 20. 0.178 5\% 21. Explain the difference between matched pair data and independent groups. 22. Explain the difference between random samples and random assignment. 23. List the assumptions that we need to check for a two-population mean hypothesis test from independent groups. 24. List the assumptions that we need to check for a two-population mean hypothesis test from matched pairs. https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 25. List the assumptions that we need to check for a two-population mean hypothesis test that is using experimental design. 26. Explain how to use a two-population mean hypothesis test to show that categorical and quantitative data are related. 27. Explain how to use a two-population mean hypothesis test to show there is a cause and effect between categorical and quantitative data. (#28-33) Directions: a) Determine if the following two-population mean tests are matched pair or independent groups b) Write the null and alternative hypothesis. Include relationship implications. c) Check all of the assumptions for a two-population mean T-test. Explain your answers. Does the problem meets all the assumptions? d) Write a sentence to explain the T-test statistic. e) Use the test statistics and the critical value to determine if the sample data significantly disagrees with the null hypothesis. Explain your answer. f) Write a sentence to explain the P-value. g) Use the P-value and significance level to determine if the sample data could have occurred by random chance (sampling variability) or is it unlikely to random chance? Explain your answer. h) Should we reject the null hypothesis or fail to reject the null hypothesis? Explain your answer. i) Write a conclusion for the hypothesis test. Explain your conclusion in plain language. j) Is the categorical variable related to the quantitative variable? Explain your answer. 28. The ACT exam is used by many colleges to test the readiness of high school students for college. Many high school students are now taking ACT prep classes. A local high school offers an ACT prep class, but wants to know if it really helps. Twenty students were randomly selected. They took the ACT exam before and after taking the ACT prep class. For each student the difference between the after and before scores were measured (d = after – before). Population 1 was the after prep class scores and population 2 was the before prep class scores. The mean of the differences was 1.5 ACT points with a standard deviation of 2.3 ACT points. A histogram of the differences yielded a bell shaped normal distribution. Use a 5\% significance level to test the claim that the after prep class scores are higher than the before prep class scores. What does this data indicate about the relationship between taking a prep class or not and ACT scores. https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 29. A random sample of 20 male German Shepherds found that their average weight was 112 pounds with a standard deviation of 28 pounds. A random sample of 14 male Dobermans found that their average weight is 107 pounds with a standard deviation of 24 pounds. Assume that weights are normally distributed. Use the Statcato printout below and a 5\% significance level to test the claim that the population mean average weight of male German Shepherds (population 1) is more than the population mean average weight of male Doberman Pinchers (population 2). What does this data indicate about the relationship between the weight and the type of dog? 30. Cotinine is an alkaloid found in tobacco and is used as a biomarker for exposure to cigarette smoke. It is especially useful in examining a person’s exposure to second hand smoke. A random sample of 32 non-smoking American adults was collected. These adults were not smokers and did not live with any smokers. The average cotinine level for this sample was 7.2 ng/mL with a standard deviation of 5.8 ng/mL. A second random sample of 35 non-smoking American adults was then collected. These adults did not smoke themselves, but did live with one or more smokers. The mean average cotinine level for this sample was 28.5 ng/mL and had a standard deviation of 11.4 ng/mL. Use a 1\% significance level to test the claim that people that do not live with smokers have a lower cotinine level than those people that do live with smokers. What does this data indicate about the relationship between cotinine levels and living with a smoker or not. 31. We want to see if the country a car is made in is related to its gas mileage in miles per gallon. Specifically we wanted to see if cars made in the U.S. have a lower population mean average mpg than those made outside the U.S. We used the random car data at www.matt-teachout.org and a 5\% significance level to create the following graphs and statistics with Statcato. Check the assumptions and perform the hypothesis test. https://creativecommons.org/licenses/by/4.0/ http://www.matt-teachout.org/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 32. We want to test the claim that the diastolic blood pressure of a person is less than the systolic blood pressure of a person. We used the random health data at www.matt-teachout.org, Statcato, and a 1\% significance level to create the following graphs and statistics. Check the assumptions and perform the hypothesis test. Notice that since the diastolic and systolic blood pressures came from the same randomly selected adults, we used a matched pair calculation. https://creativecommons.org/licenses/by/4.0/ http://www.matt-teachout.org/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 33. We want to see if the country a car is made in is related to the horsepower of the car. Specifically we wanted to see if cars made in the U.S. have a higher population mean average horsepower than those made outside the U.S. We used the random car data at www.matt-teachout.org and a 10\% significance level to create the following graphs and statistics with Statcato. Check the assumptions and perform the hypothesis test. https://creativecommons.org/licenses/by/4.0/ http://www.matt-teachout.org/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 (#34-37) Directions: For each problem, answer the following questions. a) Determine if the following two-population mean tests are matched pair or independent groups b) Write the null and alternative hypothesis. Include relationship implications. c) Use randomized simulation to calculate the P-value. Write a sentence to explain the P-value. d) Use the P-value and significance level to determine if the sample data could have occurred by random chance (sampling variability) or is it unlikely to random chance? Explain your answer. e) Use the sample mean difference and the standard error in the simulation to calculate the T-test statistic. 𝑇𝑇 = 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑀𝑀𝑆𝑆𝑆𝑆𝑀𝑀 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑆𝑆𝐷𝐷𝑆𝑆𝑀𝑀𝐷𝐷𝑆𝑆 𝑆𝑆𝑆𝑆𝑆𝑆𝑀𝑀𝑆𝑆𝑆𝑆𝐷𝐷𝑆𝑆 𝐸𝐸𝐷𝐷𝐷𝐷𝐸𝐸𝐷𝐷 e) Should we reject the null hypothesis or fail to reject the null hypothesis? Explain your answer. f) Write a conclusion for the hypothesis test. Explain your conclusion in plain language. g) Is the categorical variable related to the quantitative variable? Explain your answer. 34. Go to StatKey at www.lock5stat.com. Under the “Randomization Hypothesis Tests” menu, click on “Test for Difference for Difference in Means”. Under the data sets menu on the top left, click on “Commute Time (Atlanta vs St Louis)”. This took a random sample of people from Atlanta (population 1) and a random sample of people from St. Louis (population 2). Use randomized simulation and a 5\% significance level to test the claim that the mean average commute time for people in Atlanta is greater than the mean average commute time for people from St. Louis. What does this data indicate about the relationship between the city and the commute time? 35. Go to StatKey at www.lock5stat.com. Under the “Randomization Hypothesis Tests” menu, click on “Test for Single Mean”. Under the data sets menu on the top left, click on “Pulse Rate Difference (Quiz – Lecture)”. An experiment was done on college students to determine if heart rate is related to taking a quiz or not. The heart rates of students were measured on a day they were taking a quiz (population 1) and again on a day when there was just lecture (population 2). The same students were measured twice. Use randomized simulation and a 1\% significance level to test the claim that the mean average heart rate difference between the quiz and lecture days is greater than zero. This will indicate that the heart rates on quiz days tend to be higher than lecture days. What does this data indicate about the relationship between the heart rate and taking a quiz or not? 36. Go to StatKey at www.lock5stat.com. Under the “Randomization Hypothesis Tests” menu, click on “Test for Difference in Means”. Under the data sets menu on the top left, click on “Exercise Hours (Male vs Female)”. This took a random sample of male adults (population 1) and a random sample of female adults (population 2). Use randomized simulation and a 10\% significance level to test the claim that the mean average amount of time that males and females exercise is the same. What does this data indicate about the relationship between exercise hours and gender? 37. Use StatKey and the random health data to test the claim that the population mean average pulse rate for women is higher than for men. Go to www.matt-teachout.org and click on the statistics tab and then the data sets tab. Open the health data. Copy and paste the gender data and pulse data columns next to each other in a fresh excel spreadsheet. Now copy the two columns. Go to StatKey at www.lock5stat.com. Under the “Randomization Hypothesis Tests” menu, click on “Test for Difference for Difference in Means”. Under “Edit Data”, paste the gender and pulse rate columns into StatKey. Click on “Generate 1000 Samples” a few times. Click on “Right Tail” and put in the sample mean difference of 6.9 beats per minute in the bottom box in order to estimate the P-value. Now answer the questions above. -------------------------------------------------------------------------------------------------------------------------------------------------------- https://creativecommons.org/licenses/by/4.0/ http://www.lock5stat.com/ http://www.lock5stat.com/ http://www.lock5stat.com/ http://www.matt-teachout.org/ http://www.lock5stat.com/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 Practice Problems Section 4H (#1-10) Use either the correlation coefficient or the T-test statistic and the corresponding critical values to fill out the table. T-test statistic or Correlation Coefficient (r) Sentence to explain T- test statistic or Correlation Coefficient (r) Critical Value (T or r) Does the T-test statistic or r- value fall in a tail determined by a critical value? (Yes or No) Does sample data significantly disagree with 𝐻𝐻0? 1. T = −2.441 ±1.775 2. r = 0.183 0.316 3. T = +1.166 +2.003 4. r = −0.799 ±0.286 5. T = +3.118 +2.714 6. r = 0.921 0.339 7. T = −0.852 ±2.322 8. r = −0.026 −0.279 9. T = +1.339 ±1.997 10. r = 0.483 +0.303 (#11-20) Use each of the following P-values and corresponding significance levels to fill out the table. P-value Proportion P- value \% Sentence to explain the P-value Significance Level \% Significance level Proportion If 𝐻𝐻0 is true, could the sample data occur by random chance or is it unlikely? Reject 𝐻𝐻0 or Fail to reject 𝐻𝐻0? 11. 0.521 10\% 12. 0.0426 1\% 13. 3.41 × 10−5 5\% 14. 0.0033 1\% 15. 0.768 5\% 16. 0 10\% 17. 0.0428 5\% 18. 0.277 10\% 19. 6.04 × 10−6 1\% 20. 0.0178 5\% 21. List the assumptions that we need to check when performing a correlation hypothesis test. 22. How can we use the scatterplot and the correlation coefficient (r) to determine if the sample data follows a linear pattern? 23. Points in the scatterplot that are far from the regression line are considered outliers, but it is difficult to know if the outliers are influential or not. How can we use the scatterplot and the correlation coefficient (r) to determine if potential outliers are influential or not? 24. Explain the two assumptions that we check by using the histogram of the residuals. 25. Explain how to determine if the residual plot is evenly spread out or not. https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 (#26-29) Directions: For each of the following problems, use the Statcato printouts provided to answer the following questions. a) Write the null and alternative hypothesis for the correlation test. Address the quantitative relationship and label which is the claim. b) Write a sentence to explain the strength and direction based on the correlation coefficient (r). c) Write a sentence to explain the sample slope (𝑏𝑏1). d) Check all of the assumptions for the correlation test. Explain your answers. e) Write a sentence to explain the T-test statistic. f) Compare the T-test statistic to the critical value. Does the test statistic fall in a tail determined by the critical value? g) Does the sample data significantly disagree with the null hypothesis? Explain your answer. h) Is the sample slope significantly different from zero? Explain your answer. i) Write a sentence explaining the P-value. j) Compare the P-value to the significance level. Could the sample data or more extreme occur by sampling variability if the null hypothesis was true or is it unlikely? Explain your answer. k) Should we reject the null hypothesis or fail to reject the null hypothesis? Explain your answer. l) Write a conclusion for the test addressing evidence and the claim. 26. Use a 5\% significance level and the Statcato printout below to test the claim that there is a linear relationship between the height (X) of a man and his weight (Y). This printout came from the random health data at www.matt- teachout.org. https://creativecommons.org/licenses/by/4.0/ http://www.matt-teachout.org/ http://www.matt-teachout.org/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 27. Use a 5\% significance level and the Statcato printout below to test the claim that there is NO linear relationship between the systolic blood pressure (X) of a woman and her diastolic blood pressure (Y). This printout came from the random health data at www.matt-teachout.org. https://creativecommons.org/licenses/by/4.0/ http://www.matt-teachout.org/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 28. Use a 5\% significance level and the Statcato printout below to test the claim that there is a relationship between the head width (X) of a bear and its chest size (Y). This printout came from the random bear data at www.matt- teachout.org. https://creativecommons.org/licenses/by/4.0/ http://www.matt-teachout.org/ http://www.matt-teachout.org/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 29. Use a 5\% significance level and the Statcato printout below to test the claim that there is NO relationship between the neck circumference (X) of a bear and its weight (Y). This printout came from the random bear data at www.matt-teachout.org. https://creativecommons.org/licenses/by/4.0/ http://www.matt-teachout.org/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 (#30-32) Directions: Go to www.lock5stat.com and click on the “StatKey” button. Under “Randomization Hypothesis Tests”, click the one that says, “Test for Slope, Correlation”. Click “Generate 1000 Samples” a few times. Remember there are two ways of getting the P-value. If the top of the graph says “Randomization Dot plot of Correlation”, then the null hypothesis is ρ = 0. Remember rho looks like a “ ρ ” but it is not a “P”. If the top of the graph says “Randomization Dot plot of Slope”, then the null hypothesis is 1β = 0. Remember when StatKey simulates correlation we will be comparing the original r-value to all the simulated r-values in the simulation. When StatKey simulates the slope, we will be comparing the original sample slope to the simulated slopes. You will get about the same P-value from either of these. Assume the assumptions are met. Use the simulation in StatKey to answer the following questions. a) Write the null and alternative hypothesis for the correlation test. Address the quantitative relationship and label which is the claim. Is this a right-tailed, left-tailed, or two-tailed test? b) Write a sentence to explain the strength and direction based on the “original sample” correlation coefficient (r). c) Does the original sample correlation coefficient fall in a tail of the correlation simulation? d) Write a sentence to explain the original sample slope (𝑏𝑏1). e) Does the original sample slope fall in the tail of the slope simulation? f) Is the sample slope significantly different from zero? Explain your answer. g) Does the sample data significantly disagree with the null hypothesis? Explain your answer. h) Put the original sample slope into the slope simulation to calculate the P-value. What is your estimated P-value? (Answers will vary.) i) Write a sentence explaining the P-value. j) Compare the P-value to the significance level. Could the sample data or more extreme occur by sampling variability if the null hypothesis was true or is it unlikely? Explain your answer. k) Should we reject the null hypothesis or fail to reject the null hypothesis? Explain your answer. l) Write a conclusion for the test addressing evidence and the claim. m) Use the original sample slope, the estimated standard error in the simulation, and the following formula to calculate the T-test statistic. (Answers will vary.) Write a sentence to explain the T-test statistic. T-test statistic = (𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆−0) 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝐸𝐸𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 https://creativecommons.org/licenses/by/4.0/ http://www.lock5stat.com/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 30. Open the “Car Data” in Excel from www.matt-teachout.org. Copy and paste the miles per gallon (mpg) and horsepower into two columns in new excel spreadsheet. The mpg should be on the left and the horsepower should be on the right. The mpg will be the explanatory variable (X) and the horsepower will be the response variable (Y). Now go to www.lock5stat.coma and click on StatKey. Under “Randomization Hypothesis Tests” click on “Test for Slope, Correlation”. Under “Edit Data” paste the two columns into StatKey. Now click “Generate 1000 Samples” a few times. Use the randomized simulation in StatKey and a 1\% significance level to test the claim that there is a negative (inverse) relationship between mpg and horsepower. 31. Open the “Car Data” in Excel from www.matt-teachout.org. Copy and paste the horsepower and weight into two columns in new excel spreadsheet. The horsepower should be on the left and the weight should be on the right. The horsepower will be the explanatory variable (X) and the weight in tons will be the response variable (Y). Now go to www.lock5stat.coma and click on StatKey. Under “Randomization Hypothesis Tests” click on “Test for Slope, Correlation”. Under “Edit Data” paste the two columns into StatKey. Now click “Generate 1000 Samples” a few times. Use the randomized simulation in StatKey and a 10\% significance level to test the claim that there is a positive (direct) relationship between the horsepower and weight of a car. 32. Open the “Health Data” in Excel from www.matt-teachout.org. Copy and paste the age of women and the height of women into two columns in new excel spreadsheet. The age of women should be on the left and the height of women should be on the right. The age of women in years will be the explanatory variable (X) and the height of women in inches will be the response variable (Y). Now go to www.lock5stat.coma and click on StatKey. Under “Randomization Hypothesis Tests” click on “Test for Slope, Correlation”. Under “Edit Data” paste the two columns into StatKey. Now click “Generate 1000 Samples” a few times. Use the randomized simulation in StatKey and a 5\% significance level to test the claim that there is NO relationship between the age and height of women. -------------------------------------------------------------------------------------------------------------------------------------------------------- https://creativecommons.org/licenses/by/4.0/ http://www.matt-teachout.org/ http://www.lock5stat.coma/ http://www.matt-teachout.org/ http://www.lock5stat.coma/ http://www.matt-teachout.org/ http://www.lock5stat.coma/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 Problems Section 4B (#1-10) Use each of the following ANOVA F-test statistics and the corresponding critical values to fill out the table. F-test stat Sentence to explain F-test statistic. Critical Value Does the F-test statistic fall in a tail determined by the critical value? (Yes or No) Does sample data significantly disagree with 𝐻𝐻0? 1. +5.573 +2.886 2. +1.192 +3.113 3. +0.664 +2.949 4. +4.415 +3.125 5. +3.718 +4.117 6. +0.991 +2.009 7. +2.652 +1.875 8. +1.585 +3.225 9. +2.447 +2.798 10. +8.133 +2.891 (#11-20) Use each of the following P-values and corresponding significance levels to fill out the table. P-value Proportion P- value \% Sentence to explain the P-value Significance Level \% Significance level Proportion If 𝐻𝐻0 is true, could the sample data occur by random chance or is it unlikely? Reject 𝐻𝐻0 or Fail to reject 𝐻𝐻0? 11. 0.186 10\% 12. 0.0042 1\% 13. 2.59 × 10−4 5\% 14. 0.006 1\% 15. 0.353 5\% 16. 0 10\% 17. 0.041 5\% 18. 0.274 10\% 19. 1.04 × 10−8 1\% 20. 0.067 5\% 21. The F-test statistic compares the variance between the groups to the variance within the groups. Explain how the variance between the groups is calculated and what it tells us. Explain how the variance within the groups is calculated and what it tells us. How can we use the variance between and the variance within to calculate the F-test statistic? 22. If the variance between the groups were significantly larger than the variance within, would the F-test statistic be large or small? Explain why. 23. If the variance between the groups were about the same as the variance within, would the F-test statistic be large or small? Explain why. 24. The ANOVA printout involves the degrees of freedom within the groups, the degrees of freedom between the groups and the total degrees of freedom. How are the different degrees of freedom calculated? https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 (#25-28) Directions: Use the following Statcato statistics, graphs and ANOVA printout to test the population claims. For each of the following problems answer the following. a) Give the null and alternative hypothesis. b) Check the assumptions for a One-Way ANOVA test. c) Write a sentence to explain the F test statistic. d) Use the F test statistic and Critical Value to determine if the sample data significantly disagrees with the null hypothesis. Explain your answer. e) Use the P-value and Significance Level to answer the following: Could the sample data or more extreme have occurred because of sampling variability or is it unlikely that the sample data occurred because of sampling variability? Explain your answer. f) Should we reject the null hypothesis or fail to reject the null hypothesis? Explain your answer. g) Write a conclusion for the hypothesis test addressing evidence and the claim. h) What is the variance between the groups? What is the variance within the groups? Was the variance between significantly higher than the variance within? Explain how you know. i) Was the categorical and quantitative variables related or not. Explain your answer. 25. A random sample of black bears were weighed at various times of the year. Some of the bears were weighed in the spring, some in the summer and some in the fall. The bears were tagged so that the same bear was not measured more than once. Use a 1\% significance level and the following Statcato statistics, graphs and ANOVA printout to test the population claim that the time of year (season) is related to the weight of the bears. https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 26. A census of Math 075 pre-stat students was taken in the fall 2015 semester. The students were separated into three sleep groups: low amount of sleep, moderate amount of sleep, high amount of sleep. They were also asked how many total units they have completed at the college. Though the data was not random, you can assume it was representative of Math 075 students at COC. Use a 10\% significance level and the following Statcato statistics, graphs and ANOVA printout to test the claim that sleep is not related the total number of units completed. https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 27. A census of Math 075 pre-stat students was taken in the fall 2015 semester. The students were separated into four political parties: democratic, republican, independent party, and other political party. They were also asked number of alcoholic beverages they consume per week. Though the data was not random, you can assume it was representative of Math 075 students at COC. Use a 5\% significance level and the following Statcato statistics, graphs and ANOVA printout to test the claim that political party is not related to the number of alcoholic beverages. https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 28. A census of Math 075 pre-stat students was taken in the fall 2015 semester. The students were asked what their favorite social media is: Facebook, Instagram, Snapchat, or Twitter. They were also asked number minutes per day spent on social media. Though the data was not random, you can assume it was representative of Math 075 students at COC. Use a 5\% significance level and the following Statcato statistics, graphs and ANOVA printout to test the claim that the type of social media is related to the number of minutes per day spent on social media. https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 https://creativecommons.org/licenses/by/4.0/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 (#29-33) Directions: Go to www.lock5stat.com and click on the StatKey button. Under the “More advanced randomization tests” menu click on “ANOVA for Difference in Means”. For each of the following problems, use a randomized simulation to answer the following. Assume the data met the assumptions for an ANOVA hypothesis test. For each problem, answer the following questions. a) Give the null and alternative hypothesis. b) The F-test statistic is given under “Original Sample”. Write a sentence to explain the F test statistic. c) Simulate the null hypothesis and put the significance level in the right tail to calculate the critical value. What was the critical value? (Answers will vary.) d) Use the F test statistic and Critical Value to determine if the sample data significantly disagrees with the null hypothesis. Explain your answer. f) Put in the test statistic into the right tail to calculate the P-value. What was the P-value? (Answers will vary.) g) Use the P-value and Significance Level to answer the following: Could the sample data or more extreme have occurred because of sampling variability or is it unlikely that the sample data occurred because of sampling variability? Explain your answer. f) Should we reject the null hypothesis or fail to reject the null hypothesis? Explain your answer. g) Write a conclusion for the hypothesis test addressing evidence and the claim. h) What is the variance between the groups? What is the variance within the groups? Was the variance between significantly higher than the variance within? Explain how you know. i) Was the categorical and quantitative variables related or not. Explain your answer. 29. Use the random car data and a 1\% significance level to test the claim that the country a car is from is related to its gas mileage. Go to www.matt-teachout.org and open the random car data. Copy and paste the country and the miles per gallon columns next to each other in a new excel spreadsheet. The country should be on the left and the miles per gallon should be on the right. Then copy both columns together. Go to www.lock5stat.com and click on the StatKey button. Under the “More advanced randomization tests” menu click on “ANOVA for Difference in Means”. Click on the “Edit Data” button and paste the country and mpg columns into StatKey. Click on “Generate 1000 Samples” a few times and then “Right-Tail”. Put in the original sample F-test statistic in the bottom box to estimate the P-value. Complete the questions above. https://creativecommons.org/licenses/by/4.0/ http://www.lock5stat.com/ http://www.matt-teachout.org/ http://www.lock5stat.com/ This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18 30. Under the “ANOVA for Difference in Means” menu in StatKey, click on the button at the top left of the page and click on “Sandwich Ants”. We are studying the number of ants that are drawn to different kinds of food. In this data, we are looking at the mean average number of ants that come to three different types of sandwiches left out to spoil. Use a 5\% significance level to test the claim that the number of ants is not related to the type of sandwich. 31. Use the random car data and a 10\% significance level to test the claim that the country a car is from is not related to its horsepower. Go to www.matt-teachout.org and open the random car data. Copy and paste the country and the horsepower columns next to each other in a new excel spreadsheet. The country should be on the left and the horsepower should be on the right. Then copy both columns together. Go to www.lock5stat.com and click on the StatKey button. Under the “More advanced randomization tests” menu click on “ANOVA for Difference in Means”. Click on the “Edit Data” button and paste the country and horsepower columns into StatKey. Click on “Generate 1000 Samples” a few times and then “Right-Tail”. Put in the original sample F-test statistic in the bottom box to estimate the P-value. Complete the questions above. 32. Under the “ANOVA for Difference in Means” menu in StatKey, click on the pulse rate and award data. This data looks at the average pulse rates of those people that have won Olympic, Academy and Nobel awards. Use a 1\% significance level to test the claim that the population mean average pulse rate is related to the type of award the person won. 33. Under the “ANOVA for Difference in Means” menu in StatKey, click on the Homes for Sale (price by state) data. This data looks at the average selling price of homes in four different states. Use a 10\% significance level to test the claim that the population mean average home price is related to the state the home is sold in. -------------------------------------------------------------------------------------------------------------------------------------------------------- https://creativecommons.org/licenses/by/4.0/ http://www.matt-teachout.org/ http://www.lock5stat.com/ Problems Section 4B 21. The F-test statistic compares the variance between the groups to the variance within the groups. Explain how the variance between the groups is calculated and what it tells us. Explain how the variance within the groups is calculated and what i... (#25-28) Directions: Use the following Statcato statistics, graphs and ANOVA printout to test the population claims. For each of the following problems answer the following. a) Give the null and alternative hypothesis. b) Check the assumptions for a One-Way ANOVA test. c) Write a sentence to explain the F test statistic. d) Use the F test statistic and Critical Value to determine if the sample data significantly disagrees with the null hypothesis. Explain your answer. e) Use the P-value and Significance Level to answer the following: Could the sample data or more extreme have occurred because of sampling variability or is it unlikely that the sample data occurred because of sampling variability? Expl... f) Should we reject the null hypothesis or fail to reject the null hypothesis? Explain your answer. g) Write a conclusion for the hypothesis test addressing evidence and the claim. h) What is the variance between the groups? What is the variance within the groups? Was the variance between significantly higher than the variance within? Explain how you know. i) Was the categorical and quantitative variables related or not. Explain your answer. a) Give the null and alternative hypothesis. b) The F-test statistic is given under “Original Sample”. Write a sentence to explain the F test statistic. c) Simulate the null hypothesis and put the significance level in the right tail to calculate the critical value. What was the critical value? (Answers will vary.) d) Use the F test statistic and Critical Value to determine if the sample data significantly disagrees with the null hypothesis. Explain your answer. f) Put in the test statistic into the right tail to calculate the P-value. What was the P-value? (Answers will vary.) g) Use the P-value and Significance Level to answer the following: Could the sample data or more extreme have occurred because of sampling variability or is it unlikely that the sample data occurred because of sampling variability? Ex... f) Should we reject the null hypothesis or fail to reject the null hypothesis? Explain your answer. g) Write a conclusion for the hypothesis test addressing evidence and the claim. h) What is the variance between the groups? What is the variance within the groups? Was the variance between significantly higher than the variance within? Explain how you know.