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Please check attaced documents then sent a bid !Class 6 - Assignment 1 Create frequency and summary tables for categorical data Background: The data provided are from a larger study on college students which also included students’ pulse rate before and after running, height, weight, and exercise level. Since we are only dealing with categorical variables here, I deleted all the other data. Still, the data regarding the smoking status can be seen as a sample taken from the population of college students, since the subjects were enrolled randomly. Raw data are in posted the Excel file: Class 6 Assignment raw data - smoking status among college students.xls Use Excel and minitab to carry out the counts of the raw data for each combination of category values. Combine the frequency counts into a 2-way frequency table. Calculate the respective proportions and create a 2-way summary table. Note: This task is very common and very basic, so you should know how to do this in Excel. You may not have minitab available all the time. Plus the minitab worksheet isn’t the best format to work with tables (see below). Part A. Using Excel I created a youtube tutorial to walk you through all the steps of this assignment. http://www.youtube.com/watch?v=RrXRoJDO90U Watch it until 30 min, 30 seconds. Skip the last 10 minutes! I screwed up a little with my wording for the row and column proportions. Better use the wording I put below. So here are the steps (as shown in the video as well): 1. Create a frequency table and a summary table for the raw data, in the same worksheet, next to the raw data (as shown in the video). 2. For the frequency table, use the Excel formula COUNTIFS to get the counts automatically inserted into your table fields. 3. Then calculate the marginal totals and the grand total. 4. For the summary table, use the frequencies to calculate the joint proportions. 5. Next, calculate the row proportions and the column proportions and list them in rows underneath the table (not within the table, so these proportions won’t get confused with the joint proportions of the table). Row proportions in this example: Proportion of all males that smoke / don’t smoke (=100\%) Proportion of all females that smoke / don’t smoke (=100\%) Column proportions in this example Proportion of all smokers that are male / female (=100\%) Proportion of all non-smokers that are male / female (=100\%) [You see, there are a lot of ways to calculate proportions. Be sure to keep them apart and when you see a number for a proportion, ask or verify what that proportion actually is.] When you are done, save the file twice (as I mention in the video): Once with the original title, plus your name, to upload as a homework. Then save it again, calling it “Template to create frequency and summary tables for categorical data.xls” or something like that... So now each time you have categorical data, you just paste them into the columns of this template and your tables are generated automatically. That may come inLecture 6 Statistics with categorical variables Summary statistics Binomial Distribution Sampling distribution of a proportion Errors & confidence intervals for a proportion Dr. Doerre Data Analyses and Statistical Concepts in Biotechnology FSU Math 924 03-04-2021 A quick refresher about categorical variables What are they? How are they defined? What are some of their properties? How can we organize and display them? What do their summary statistics look like? Examples of categorical variables Examples of “values” or outcomes for categorical variables Dr. Doerre Data Analyses and Statistical Concepts in Biotechnology FSU Math 924 Organizing two binary categorical variables in two-way tables (also called fourfold tables, cross-tabulations, or contingency tables)* Smoker Non-smoker Marginal row totals Male Σ R1 Female Σ R2 Marginal column totals Σ C1 Σ C2 Grand total* *Grand total: Either add up the columns or the rows! Either one will add up to the total number of subjects in the study. *The two-way/fourfold table is a special type of a cross-tabulation or contingency table, which can have many more categorical variables and not just binary variables . 2. Summarize frequency data by calculating proportions Frequency data 1. Enter counts from raw data or “case-level data“ (i.e. individual observations) Variable 2: smoking status 2 possible values Event counts (frequency of outcomes) Variable 1: Gender 2 possible values Dr. Doerre Data Analyses and Statistical Concepts in Biotechnology FSU Math 924 Smoker Non-smoker Marginal row totals Male Σ R1 Female Σ R2 Marginal column totals Σ C1 Σ C2 Grand total=1* 2. (cont’d): Divide each event count by grand total to obtain the respective proportion.* Optional: Multiply result by 100 to display values as percentages. Converting event counts to proportions * Note: The proportions in a 2-way table are always based on the total number of subjects (the grand total). Such proportions are also refered to as “joint distributions” *Grand total: Either add up the columns or the rows! Either one will add up to 1 (or 100, if you display the values in percent) Summary statistics for categorical variables Proportions of grand total* Variable 1: 2 possible values Variable 2: 2 possible values Dr. Doerre Data Analyses and Statistical Concepts in Biotechnology FSU Math 924 Anything else to add for the summary statistic? So what kind of statistical errors can occur in these numbers? No How about a sample mean? A sample standard deviation? There is none. The categories are clearly defined, and so are the values. There is no doubt in the counts (unless someone screwed up). None in tSheet1 MATH 924 SUMMARY up to Class 5 Type of variable Numerical Categorical How are values obtained? Measured Counted SUMMARY STATISTICS Measures of central tendency Sample mean population mean µ Median Measures of dispersion Sample standard deviation Population standard deviation Range = max - min 1st, 3rd quartile GRAPHICAL REPRESENTATIONS Dot plot (incl. frequency distributions) Box plot Histogram STATISTICAL ESTIMATION PROBABILITY DENSITY FUNCTIONS Normal distribution (Theoretical frequency distributions) Student t distributions SAMPLING DISTRIBUTION (of mean values) Condition for normality of sampling distribution n > 30 Mean of sampling distribution mean of mean values; = population mean Standard error (of the mean, SEM) Margin of error for a certain confidence level MOE = z x SEM Confidence interval (CI) C.I. = mean +/- MOE Conversion of a value x to a z-score z=(x-µ)/σ Use confidence intervals for: · one experimental mean value · comparing one experimental value to known value ·[comparing two experimental mean values] (possible, but we didnt cover this in class; instead, we use hypothesis testing) STATISTICAL DECISION (HYPOTHESIS TESTING) Purpose 1 Compare experimental value to known or claimed value (like an established population mean) Hypothesis test One sample t test, z test Null hypothesis Experimental value is equal to the known value. Purpose 2 Compare 2 experimental values Hypothesis test 2 sample t test Null hypothesis Two experimental mean values are equal. Purpose 3 Compare a series of pairs of experimental values Hypothesis test Paired t test Null hypothesis Pairs are equal, no differenceSheet1 Smokes Gender No M No M Yes M Yes M No M No M No M No M No M No M Yes M No M Yes M No M No M No M Yes M Yes M Yes M No M No M No M No M Yes M No F No F Yes F No F Yes F No F No F No F Yes F Yes F No F No M No M Yes M No M No M Yes M Yes M No M No M No M Yes M No M Yes M No M No M No M Yes M Yes M No M No M No M Yes M No M Yes M No M Yes M Yes M No M No M No M No M No M No F No F No F Yes F No F No F No F Yes F No F No F No F No F No F No F No F No F No F Yes F No F Yes F No F No F No F No FClass 6 - Assignment 2 Use Minitab to calculate the confidence interval for two proportions TASK: Create a confidence interval for the difference of the two proportions (male smokers minus female smokers) found in assignment 1. (Don’t worry, it is just a few clicks in minitab, but I have a lot of screenshots in this file) Background: In assignment 1 you determined the proportions of male and female college students that are smokers. You probably took a look at these results and checked whether the proportions of male and female smokers are indeed different. If not go back and check! Answer: For this sample of males and females there is a difference in smoking rates. The statistical question should now be obvious to you: Does the observed difference in smoking rates (proportions) in this sample also apply to the whole population of college students (maybe at least at that college)? You already know how to move from a sample mean to a population mean by creating a confidence interval for a sample mean. In the last class (#5) we went over the creation of a confidence interval for a proportion. Now you have two proportions and their difference and you want to know the confidence interval for the difference. No need to worry! Minitab does that for you with one click! You just need to interpret the result. Step 1: Define your question (most important step of all)! We could ask three questions about proportion differences here (because you calculated three types of proportions: 1. What is the difference of the joint proportion between males and females? This translates to: What is the difference in the proportion of female smokers and male smokers among all subjects in this sample? 2. What is the difference of the column proportions for smokers between males and females? This translates to: what is the difference of the proportion of males and females among all smokers? Here you would subtract the two column proportions in the smoker column. Note that the column proportions of male vs female smokers add up to 1 (or 100\%). 3. What is the difference of the row proportions of smoking rates between males and females? This translates to: Is there a difference in the proportion of males who smoke (out of all males) and the proportion of females who smoke (out of all females). Now I am telling you: Only one of these questions makes sense to ask, because only one of these questions is a “fair” question. All the others compare apples to oranges. Have you already figured out which one? Here’s a hint: the number of male and female subjects are not the same! Think about is a little more and try to come up with an answer or at least a guess before you read on. Because the number of male and female subjects are not the same, the joint proportion won’t be fair, because it compares the male and female smokers to all the subjects, and there are fewer females. So of course there would be fewer female smokers even if their smoking rate was the same