Fluent Essays

2 Solutions to Practice 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? 4. -3.177 The sample proportion for group 1 is 3.177 standard errors below the sample proportion for group 2. ±1.96 No The critical values are ± 1.96. So, the Z-test statistic does not fall in either tail. Since it does not fall in either tail, the sample data does not significantly disagree with the null hypothesis (𝐻𝐻0). 8. +1.117 The sample proportion for group 1 is 1.117 standard errors above the sample proportion for group 2. +1.96 Yes The critical values are +1.96. So, the Z-test statistic falls in the right tail. Since it falls in one tail, then the sample data significantly disagrees with the null hypothesis. Table. (#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? 12. 0.0421 4.21\% The P-value is high above zero (0.0421), indicating a possibility of the sample data occurring because of random chance. 1\% 0.01 The sample data could occur by random chance Fail to reject the null hypothesis 16. 0 0\% The P-value is low (exactly at zero), then it is unlikely to be random chance 10\% 0.10 The sample data is unlikely to have occurred by random chance Reject the null hypothesis 20. 0 0\% The P-value is low (exactly at zero), then it is unlikely to be random chance 5\% 0.05 The sample data is unlikely to have occurred by random chance Reject the null hypothesis Question 22 The assumptions that we need to check for a two-population proportion hypothesis test · The sampling method for each population is simple random sampling. · The samples are independent. · Each sample includes at least 10 successes and 10 failures. · Each population is at least 20 times as big as its sample. Question 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 were 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. Solution : population percentage of teen pregnancies in the U.S. in 2008 : population percentage of teen pregnancies in the U.S. in 2012 : ≠ (the population percentage of U.S. teen pregnancies is not related to the year.) : = (the population percentage of U.S. teen pregnancies is related to the year.) (claim) Assumptions Both samples were collected randomly, Data values within and between the samples should be independent of each other. At least 10 success and at least 10 failures. Z-test statistic = -0.210 The sample proportion for group 1 was 0.210 standard errors bellow the sample proportion for group 2 The sample proportions are significantly different since the test statistic fell in the tail determined by the critical value P-value = 0.4168 or 41.68\% If the null hypothesis is true, then there is about a 41.68\% probability of getting the sample data or more extreme because of sampling variability. The P-value is close to zero, so it is very unlikely that the sample data occurred because of sampling variability. Since the P-value is greater than the significance level, we fail to reject the null hypothesis. There is significant evidence to support the claim that (the population percentage of U.S. teen pregnancies is related to the year. Question 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. Solution : Population proportion of married people that are unhappy Population proportion of non-married people that are unhappy : ≠ ( (Being married is not related to being unhappy) : = (Being married is related to being unhappy) (claim) Assumptions Both samples were collected randomly, Data values within and between the samples are independent of each other. At least 10 success and at least 10 failures Z-test statistic = −2.325 The sample proportion for group 1 was 2.325 standard errors bellow the sample proportion for group 2 The sample proportions are significantly different since the test statistic fell in the tail determined by the critical value P-value = 0.0100 = 1.0\% If the null hypothesis is true, then there is about a 1.0\% probability of getting the sample data or more extreme because of sampling variability. The P-value is less than the significance level, so it is unlikely that the sample data occurred because of sampling variability. Since the P-value is less than the significance level, we will reject the null hypothesis. There is significant evidence to support the claim that married people have a lower percentage of unhappiness than non-married people. This also indicates that being married or not is related to being unhappy. Question 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. Solution : Population proportion of women with a normal BMI : Population proportion of men with a normal BMI : ≠ (Gender is NOT related to having a normal BMI) : = (Gender is related to having a normal BMI) (claim) The sample proportion difference is -0.109 The original sample difference of -0.109 does fall in the tail determined by the significance level and the simulation. This implies that the sample proportion for women was significantly lower than the sample proportion for men. P-value = 0 = 0\% If the null hypothesis is true, there is a 0\% probability of getting this sample data or more extreme because of sampling variability. It is very unlikely that this sample data occurred because of sampling variability. Since the P-value is less than the significance level, we will reject the null hypothesis. There is significant evidence to support the claim that percentage of women with normal BMI is lower than the percentage of men. This also implies that having a normal BMI is related to gender. Question 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. Solution : Population proportion of teens who use text message : Population proportion of adults who use text message : ≠ (age is related to using text messages) : = (age is not related to using text messages) (claim) The sample proportion difference is 0.150 The original sample difference of − 0.098 does fall in the tail determined by the significance level and the simulation. This implies that the sample proportion for smokers was significantly lower than the sample proportion for non-smokers. P-value = 0 = 0\% If the null hypothesis is true, there is a 0\% probability of getting this sample data or more extreme because of sampling variability It is very unlikely that this sample data occurred because of sampling variability. Since the P-value is less than the significance level, we will reject the null hypothesis. There is significant evidence to support the claim that age is not related to using text messages 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/