8.9 Assignment 8

Purposes

This assignment has two parts. The first part assesses your knowledge of conducting a one-sample [latex]z[/latex] test and a one-sample [latex]t[/latex] test, explaining and calculating the P-value of a hypothesis test, stating the Type I and Type II errors of a hypothesis test, and explaining the relationship between the results of a confidence interval and a hypothesis test. The second part assesses your skills in using R commander to conduct a one-sample [latex]t[/latex] test for the population mean [latex]\mu[/latex].

Resources

M08_Age_Millionaire_Q4.xlsx

M08_BloodPressure_Diabete_Q6.xlsx

M08_Hour_Q7.xlsx

Instructions

Part A

Complete the following:

1. The following statement appeared on a box of Tide laundry detergent: “Individual packages of Tide may weigh slightly more or less than the marked weight due to normal variations incurred with high-speed packaging machines, but each day’s production of Tide will average slightly above the marked weight.”
   1. Explain in statistical terms what the statement means. (2 marks)
   2. Suppose that the marked weight is 1 liter. State in words the null and alternative hypotheses for the hypothesis test. (2 marks)
   3. State the Type I and Type II errors in the context of this application. (4 marks)
   4. Propose procedures to collect data to test the statement. (4 marks)
2. For a fixed sample size, what happens to the probability of a Type II error if the significance level is decreased from 0.05 to 0.01? (2 marks)
3. For a one-sample [latex]t[/latex] test with [latex]\small{df = 40}[/latex], find the P-values of the following hypothesis tests. (9 marks: 3+3+3)
   1. [latex]\small{H_{0}: \mu = 4 \text{ years versus }\ H_{a}: \mu \neq 4\text{ years}}[/latex], with the observed test statistic [latex]\small{t_{o} = 1.5}[/latex].
   2. [latex]\small{H_{0}: \mu \geq 4 \text{ years versus }\ H_{a}: \mu < 4 \text{ years}}[/latex], with the observed test statistic [latex]\small{t_{o} = - 2.5}[/latex].
   3. [latex]\small{H_{0}: \mu \leq 4 \text{ years versus }\ H_{a}: \mu > 4\text{ years}}[/latex], with the observed test statistic [latex]\small{t_{o} = 3.5}[/latex].
4. The following table gives the age (in years) of 36 randomly selected U.S. millionaires. The sample mean [latex]\small{\bar{x} = 58.53}[/latex] years. Assume that the standard deviation of ages of all U.S. millionaires is 13.0 years. (See data on file: M08_Age_Millionaire_Q4.xlsx)
   

   31	45	79	64	48	38	39	68	52
   59	68	79	42	79	53	74	66	66
   71	61	52	47	39	54	67	55	71
   77	64	60	75	42	69	48	57	48

   1. Test at the 10% significance level whether the average year of all U.S. millionaires is above 55 years. (8 marks)
   2. What is the P-value of the hypothesis test in part (a)? (2 marks)
   3. Obtain a confidence interval corresponding to the hypothesis test in part (a). (4 marks)
   4. Does the interval in part (c) support the result in part (a)? Explain your answer. (3 marks)
5. The mean retail price of agriculture books in 2005 was $57.61. This year’s retail mean price for 28 randomly selected agriculture books was $54.97. Assume that the population standard deviation of prices for this year’s agriculture books is $8.45.
   1. At the 5% significance level, do the data provide sufficient evidence to conclude that this year’s mean retail price of agriculture books has changed from the 2005 mean? (8 marks)
   2. What is the P-value of the test in part (a)? (2 marks)
   3. Obtain a confidence interval corresponding to the test in part (a). (4 marks)
   4. Does the interval in part (c) support the result in part (a)? Explain your answer. (3 marks)
6. Past studies showed that maternal diabetes results in obesity, blood pressure, and glucose tolerance complications in the offspring. Following are the arterial blood pressures, in millimetres of mercury (mm Hg), for a random sample of 16 children of diabetic mothers. The sample mean is [latex]\bar{x} = 85.99[/latex] mm Hg and the sample standard deviation is [latex]s = 8.08[/latex] mm Hg. (See data on file: M08_BloodPressure_Diabete_Q6.xlsx)
   

   81.6	84.1	87.6	82.8	82.0	88.9	86.7	96.4
   84.6	101.9	90.8	94.0	69.4	78.9	75.2	91.0

   1. Test at the 5% significance level whether the mean arterial blood pressure of all children of diabetic mothers is above 85 mm Hg. (8 marks)
   2. What is the P-value of the test in part (a)? (2 marks)
   3. Obtain a confidence interval corresponding to the test in part (a). (4 marks)
   4. Does the interval in part (c) support the result in part (a)? Explain your answer. (3 marks)
7. Previous studies showed that the average person watched 4.55 hours of television daily in 2005. A random sample of 20 people gave the following number of hours of television watched per day for last year. The sample mean is [latex]\bar{x} =[/latex]4.76 hours and the sample standard deviation is [latex]s = 2.30[/latex] hours. (See data on file: M08_Hour_Q7.xlsx)
   

   1.0	4.6	5.4	3.7	5.2
   1.7	6.1	1.9	7.6	9.1
   6.9	5.5	9.0	3.9	2.5
   2.4	4.7	4.1	3.7	6.2

   1. Test at the 1% significance level, do the data provide sufficient evidence to conclude that the amount of television watched per day last year by the average person differed from that in 2005? (8 marks)
   2. What is the P-value of the test in part (a)? (2 marks)
   3. Obtain a confidence interval corresponding to the test in part (a). (4 marks)
   4. Does the interval in part (c) support the result in part (a)? Explain your answer. (3 marks)

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.

1. Refer to the data in Question 6 in Part A.
   1. Use the proper graphical tools in R and R commander to assess whether applying the one-sample [latex]t[/latex] test procedure is reasonable. Make sure to write down the assumptions of the procedure and address whether every assumption is satisfied. (5 marks)
   2. Conduct a one-sample [latex]t[/latex] test at the 5% significance level using R commander to test whether the mean arterial blood pressure of all children of diabetic mothers is above 85 mm Hg. Make sure to include all the six components of a hypothesis test. Keep the computer outputs in an appendix. Compare the answer with the one you obtained by hand in Question 6 parts (a) and (b). (5 marks)
   3. Obtain a confidence interval corresponding to the hypothesis test in part (b). Compare the one you obtained by hand in Question 6 part (c). (2 marks)
2. Refer to the data in Question 7 in Part A.
   1. Use the proper graphical tools in R and R commander to assess whether applying the one-sample [latex]t[/latex] test procedure is reasonable. Make sure to write down the assumptions of the procedure and address whether every assumption is satisfied. (5 marks)
   2. Conduct a one-sample [latex]t[/latex] test at the 1% significance level using R commander to test whether the amount of television watched per day last year by the average person differed from that in 2005. Make sure to include all the six components of a hypothesis test. Paste the computer output into the space below first, then compare the answer with the one you obtained by hand in Question 7 parts (a) and (b). (5 marks)
   3. Obtain a confidence interval corresponding to the hypothesis test in part (b). Compare the one you obtained by hand in Question 7 part (c). (2 marks)