ICA11-Fall2023 (1)

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North Carolina State University *

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311

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Statistics

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Apr 3, 2024

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ST 311 Week 11 In-Class Activity 1 | P a g e Two-Sample Inference Problem 1: Does resting heart rate differ between men and women? A random sample of 30 American adults was taken, consisting of 18 women and 12 men. Their sample mean heart rates were 70.56 and 72.33 beats per minute respectively, with sample standard deviations of 8.47 and 5.00 beats per minute. It is known that resting heart rates among men and women tend to follow a normal distribution. a. Which confidence interval formula do we use? A difference of two proportions, or two means? b. Describe the parameters relevant to the problem, including the variables of interest and populations of interest. c. Check that the assumptions necessary for the calculation of this confidence interval are satisfied. d. Calculate a 99% confidence interval for the difference. Assume the correct critical value is 2.5 and use this value in your calculation. Write a sentence summarizing your confidence interval in the context of the problem.

ST 311 Week 11 In-Class Activity 2 | P a g e Problem 2: It is believed that the average grade on an English essay in a particular school system for females is higher than for males. After verifying that the necessary assumptions are satisfied, we find that a 95% confidence interval for the difference in average grade on the English essay between females and males in this school system to be (4.11, 7.89). Based on this confidence interval, what is the conclusion to the following hypothesis tests? Define 𝜇 1 to be the average grade on the English essay of all females in the school system, and 𝜇 2 the average grade on the English essay of all males in the school system. a. 𝐻 0 : 𝜇 1 − 𝜇 2 = 0 vs. 𝐻 𝑎 : 𝜇 1 −𝜇 2 ≠ 0 at an 𝛼 = 0.05 level b. 𝐻 0 : 𝜇 1 − 𝜇 2 = 0 vs. 𝐻 𝑎 : 𝜇 1 −𝜇 2 > 0 at an 𝛼 = 0.05 level c. 𝐻 0 : 𝜇 1 − 𝜇 2 = 0 vs. 𝐻 𝑎 : 𝜇 1 −𝜇 2 ≠ 0 at an 𝛼 = 0.1 level Problem 3: A Kaiser Family Foundation poll for US adults in 2019 asked self-identified Democrats, Independents, and Republicans whether or not they supported a generic “National Health Plan”. This was based on a random sample of all Americans. SUPPORT DON’T SUPPORT TOTAL DEMOCRATS 274 73 347 INDEPENDENTS 339 278 617 We wish to find an appropriate 95% confidence interval for the difference between the proportion of Democrats and the proportion of Independents supporting the National Health Plan. a. Check that the assumptions necessary for calculation of the relevant confidence interval are verified.

ST 311 Week 11 In-Class Activity 3 | P a g e b. The confidence interval for 𝑝 ??? − 𝑝 𝑖?? is computed to be (0.1769, 0.3035) . Write a sentence interpreting your confidence interval in the context of the problem. c. Based on the calculated interval, does is appear reasonable to claim that there is no difference in the proportion of Democrats and Independents supporting the National Health Plan, and the difference we observe is just due to sampling variability? d. A 95% confidence interval for the difference between the proportion of Republicans and Independents supporting the National Health Plan is calculated from the above data as (-0.370, - 0.245). (All assumptions hold.) Which of the following statements about this confidence interval are true? i. If a random sample of Republicans and a random sample of Independents are polled, the difference between the proportion of those sampled Republicans and the sampled Independents supporting the National Health Plan will be between -0.370 and -0.245 with probability 0.95. ii. There must be a calculation error, since proportions can’t be negative. iii. This confidence interval indicates that it is reasonable to conclude that more Independents than Republicans support the National Health Plan. iv. This confidence interval indicates that it is reasonable to conclude that a greater proportion of Republicans than Independents support the National Health Plan.

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ST 311 Week 11 In-Class Activity 4 | P a g e Problem 4: To test the difference in the effectiveness of two different medicines treating high blood pressure, 200 patients in an ICU having a specific ailment are randomly selected. Doctors had assigned 80 of these patients to be treated with medication A, while the rest are treated with medication B. a. What hypotheses would we test? b. We observe a statically significant difference in the blood pressure between the two medications. What conclusion does that suggest about our hypotheses? c. Does this suggest one medication is more effective than the other?

ICA11-Fall2023 (1)

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