10.9 Assignment 10

Purposes

The following questions assess your knowledge of properties of the distribution of the sample proportion [latex]\widehat{p}[/latex] and the distribution of difference between two sample proportions [latex]{\widehat{p}}_{1} - {\widehat{p}}_{2}[/latex], conducting a one-proportion [latex]z[/latex] test and obtaining a one-proportion [latex]z[/latex] interval, performing a two-proportion [latex]z[/latex] test and determining a two-proportion [latex]z[/latex] interval, and using R commander to conduct a one-proportion and a two-proportion [latex]z[/latex] test.

Resources

M10_BloodPressure_Proportion_Q5.xlsx

Instructions

Part A

Complete the following:

1. Think of a scenario (you can make up one) you can observe in your daily life and answer the following questions about the basic notation and terminology for proportions.
   1. What is a population proportion? (2 marks)
   2. What symbol is used for a population proportion? (1 mark)
   3. What is a sample proportion? (2 marks)
   4. What symbol is used for a sample proportion? (1 mark)
   5. Explain why the sample proportion is a special case of the sample mean. (4 marks)
2. This exercise involves using an unrealistically small population to provide a concrete illustration for the exact distribution of a sample proportion. A population consists of three men and two women. The men’s first names are Jose, Pete, and Carlo; the women’s first names are Gail and Frances. Suppose that the specified attribute is “female.”
   1. Determine the population proportion, [latex]p[/latex]. (2 marks)
   2. The first column of the following table provides the possible samples of size 2, where each person is represented by the first letter of their first name; the second column gives the number of successes—the number of females obtained—for each sample; and the third column shows the sample proportion. Complete the table. (4 marks)
      

      Sample	Number of females [latex]x[/latex]	Sample proportion [latex]\hat{p}[/latex]
      J, G	1	0.5
      J, P	0	0.0
      J, C	0	0.0
      J, F	1	0.5
      G, P
      G, C
      G, F
      P, C
      P, F
      C, F

   3. Use the third column of the table to obtain the mean of the variable. (2 marks)
   4. Compare your answers from parts (a) and (c). What property about the distribution of the sample mean does it verify? (3 marks)
3. In a poll of 1,961 randomly selected U.S. adults, 1,137 said that they do not believe that abstinence programs are effective in reducing or preventing AIDS.
   1. At the 2% significance level, do the data provide sufficient evidence to conclude that a majority of all U.S. adults feel that way? (8 marks)
   2. Obtain a confidence interval for the percentage of U.S. adults who believe that abstinence programs effectively reduce or prevent AIDS, corresponding to the hypothesis test in part (a). (4 marks)
   3. Interpret the confidence interval obtained in part (b). Does this interval support the conclusion of the hypothesis test in part (a)? Justify your answer. (4 marks: 2+2)
4. In a clinical trial, 56 patients were randomly assigned to use the Bug Buster kit and 70 were assigned to use the standard treatment. Thirty-two patients in the Bug Buster kit group were cured, whereas nine of those in the standard treatment group were cured.
   1. At the 5% significance level, do these data provide sufficient evidence to conclude that a difference exists in the cure rates of the two types of treatment? (8 marks)
   2. Determine a confidence interval for the difference in cure rates for the two types of treatment corresponding to the hypothesis test in part (a). (4 marks)
   3. Interpret the confidence interval obtained in part (b). Does this interval support the conclusion of the hypothesis test in part (a)? Justify your answer. (4 marks: 2+2)
5. The following table summarizes the age ([latex]\leq 50[/latex] or above 50) and blood pressure (BP; normal or high blood pressure) status of 380 Canadian adults in 2017.
   

   	Age 50 or Below	Age Above 50	Total
   High BP	74	74	148
   Normal BP	139	93	232
   Total	213	167	380

   1. It was reported that the percentage of Canadian adults with high blood pressure was 30% in 2007. Test at the 1% significance level whether the percentage of Canadian adults with high blood pressure in 2017 differs from that in 2007. Report the P-value of the test. (8 marks)
   2. Obtain a 99% confidence interval for the proportion of Canadian adults having high blood pressure. (4 marks)
   3. Interpret the confidence interval obtained in part (b). Does this interval support the conclusion of the hypothesis test in part (a)? Justify your answer. (4 marks: 2+2)
   4. Test at the 5% significance level whether the proportion of high blood pressure is higher among Canadian adults above age 50 than among those age 50 or below. (8 marks)
   5. Obtain a confidence interval for the difference between the proportions of high blood pressure among Canadian adults above age 50 and among those age 50 or below corresponding to the hypothesis test in part (d). (5 marks)
   6. Interpret the confidence interval obtained in part (e). Does this interval support the conclusion of the hypothesis test in part (d)? Justify your answer. (4 marks: 2+2)

Part B

Finish the following questions using R and R commander. Make sure that you copy and paste the computer outputs as required and write down your answers in statements.

Refer to Question 5 in Part A. The data are provided in the data file M10_Bloodpressure_Proportion_Q5.xlsx. Import the data into R commander.

1. Re-conduct the test in Question 5 (a) using R commander. Make sure to include all the six components of a hypothesis test. Copy and paste the computer output first and then compare the answer with the one you obtained by hand in Question 5 (a). (5 marks)
2. Obtain a confidence interval corresponding to the hypothesis test in part (b). Compare the one you obtained by hand in Question 5 (b). (2 marks)
3. Re-conduct the test in Question 5 (d) using R commander. Compare the answer with the one you obtained by hand in Question 5 part (d). (5 marks)
4. Obtain a confidence interval corresponding to the hypothesis test in part (c). Compare the one you obtained by hand in Question 5 part (e). (2 marks)