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Name: _____________________________ Collaborators (if any): ________________________ Student number: _____________ Quiz Section: ______ Problem Set 7 Please submit a hardcopy of your solutions to this problem set to your instructor (Will Brown) at the beginning of lecture on Monday 24 February 2020. The late work policy will be applied to any submission received after this deadline. You are encouraged to work on this problem set with your classmates, but if you do, please be sure to note the names of any classmates you work with. Show your work so that we may award partial credit in case you have made mathematical errors but demonstrate that you understand how to solve the problems presented to you. Please feel free to use a calculator but refrain from using any other electronic device to achieve these calculations (for example, calculator apps on your cell phone or computer). Inferences about single proportions and comparisons of two proportions The contingency table below compares the number of individuals in CSSS/SOC/STAT 221 who stated that they drink primarily with their right hand, versus those who do not (i.e. those who drink either primarily with their left hand or who use both). The full sample represented in this contingency table combines two samples collected at different times: Summer 2019 and Fall 2019 students. Academic quarter SU2019 FA2019 Row sums Right drinking hand Right 25 93 118 Left/Both 13 95 108 Col sums 38 188 226 We have no reason to believe that drinking hand should correspond with the timing of enrollment, yet it appears as if the these two variables may be associated: more than half of the Summer 2019 students favor their right hand when drinking, while barely less than half of the Fall 2019 students do. If this apparent pattern were real, it would be surprising, and it might lead to interesting and insightful follow-up research. However, before we invest our time and resources attempting to understand the reasons behind this difference, the first thing we should do is evaluate whether this pattern is real or merely the consequence of random sampling error. Let’s call the Summer 2019 students “Group 1” (indexed with a subscript 1) and the Fall 2019 students “Group 2” (indexed with a subscript 2). We will use the letter as a placeholder for the group. Let’s also consider the right-hand drinkers our “successes.” (This is not a value judgment; I fall in the “failure” category, myself). In the following sections, you will apply different methods of statistical inference to evaluate whether the perceived difference between these two samples might actually be real. Part 1: Confidence intervals for two sample proportions Calculate 95% confidence interval estimates for each group’s underlying probability of right-handed drinking: and . To do this, identify each group’s number of successes, number of trials, and sample proportions. For each group, you will also need to apply the relevant success-failure condition. If you satisfy your success-failure condition, you then need to calculate a standard error, then apply the confidence interval equation to calculate the lower and upper boundaries of each confidence interval, using a Z-score that is appropriate for a 95% confidence interval. Note: for this part of the problem set, you need to analyze each sample separately; you do not need to combine information between groups here. Group 1 Group 2 Given these calculations, do the two groups’ confidence intervals overlap? (To answer this question, you might consider visualizing each group’s confidence interval by drawing side-by-side pictures of the two intervals.) What does your decision suggest about whether drinking hand is associated with quarter of enrollment? ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Part 2: a goodness of fit test for the Summer 2019 sample Comparing confidence intervals like you did above may be suggestive, but it doesn’t provide a conclusive or valid way to decide whether or not the two groups have a common value for their underlying parameters. A significance test is better suited to this goal. Let’s treat the Fall 2019 sample proportion as if it were true of the 2019 UW population as a whole. Based on this, we will treat the Fall 2019 sample proportion as a null value for a hypothesis about the Summer 2019 students, i.e. . By implication, the alternative hypothesis is that . Some of your calculations from Part 1 will help you test this null hypothesis. However, you will also need to apply a success-failure condition that is suitable for null hypothesis significance testing. If this condition is met, then you will also need to calculate a standard error that is suitable for this hypothesis test (; a standard error for the sample proportion assuming the hypothetical value of ). Then, you will need to standardize the sample proportion (i.e., calculate a Z-score for it, ) based on your null value and your standard error. Finally, you will need to identify the critical values , using a 0.05 level of significance, and compare to these scores. Group 1 Given and , do you reject, or fail to reject, your null hypothesis? Why? Based on this decision, what can you conclude about the relationship between drinking hand and quarter of enrollment? ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Part 3: A test for independence To complete Part 2, you assumed that the pattern observed for the Fall 2019 sample was true of the UW 2019 student population as a whole, but this assumption isn’t a sound one. Instead, to be truly thorough in your analysis, you should apply a statistical test that treats both sample proportions as sample proportions. A two-sample test for independence is more ideal in this situation. To perform this test, you can reuse the sample sizes, numbers of successes, and proportions you identified in Part 1 above. Using these, you will then need to do the following: calculate the difference between sample proportions (); calculate the pooled sample proportion; evaluate success-failure conditions for both samples based on this pooled sample proportion; calculate a standard error for the difference in sample proportions (); standardize (i.e. calculate based on and ); identify the critical values using a 0.05 level of significance (these will be the same critical values you used in Part 2 above); and compare to these critical values. Group 1 Group 2 Given and , do you reject, or fail to reject, your null hypothesis? Why? Based on this decision, what can you conclude about the relationship between drinking hand and quarter of enrollment? ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Part 4: Making decisions based on statistical inferences has real-world consequences Now imagine that you oversee budgeting allocations for a research funding agency. Your role comes with a responsibility to make sure that funding is allocated in a responsible way, to research endeavors that are likely to yield new insights about the world. This responsibility includes evaluating whether research proposal associated with funding requests are based on rigorous background research. Imagine that someone has submitted a request for funding to your agency based on the above data, claiming that the handedness of college students varies between academic quarters, proposing a reason for this pattern based on their favorite psychological theory. Given your above analyses (particularly Part 3), do you think it would be responsible to reward the requested research funding to this researcher? Why or why not? ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 4