7.5 Assignment 7

Purposes

This assignment has two parts. The first part assesses your knowledge of distinguishing a statistic and a parameter, an estimator and a point estimate, obtaining and interpreting a one-sample [latex]Z[/latex] interval and a one-sample [latex]t[/latex] interval and calculating the required sample size given the margin of error and confidence level. The second part assesses your skills in using R commander to obtain a one-sample confidence interval for the population mean [latex]\mu[/latex].

Resources

M07_Age_Millionaire_Q9.xlsx

M07_BloodPressure_Diabete_Q10.xlsx

Instructions

Part A

Complete the following:

1. The value of a statistic used to estimate a parameter is called a ________ of the parameter. (2 marks)
2. Explain the relationship between the population mean [latex]\mu[/latex], the sample mean [latex]\bar{X}[/latex], and a value of the sample mean [latex]\bar{x}[/latex]. Which is the population parameter, which is a statistic, which is a point estimate, and which is an estimator? (4 marks)
3. What is a confidence interval estimate of a parameter? Why is such an estimate superior to a point estimate? (3 marks)
4. Explain the similarities and differences between a standard normal distribution and a [latex]t[/latex] distribution. (3 marks)
5. Must the variable under consideration be normally distributed for you to use the [latex]z[/latex]-interval procedure or [latex]t[/latex]-interval procedure? Explain your answer. (3 marks)
6. Given that [latex]n[/latex] = 36, use the t-score Table (Table IV; you can find a copy online or in the Blackboard course) to find
   1. [latex]t_{0.025} =  [/latex]____________ (1 mark)
   2. [latex]t_{0.005} =  [/latex]____________ (1 mark)
   3. [latex]P(t \geq 2.030) =  [/latex]_____________, that is the area under the [latex]t[/latex] density curve to the right of 2.030. (2 marks)
   4. [latex]P(t \leq - 2.030) =  [/latex]_____________, that is the area under the [latex]t[/latex] density curve to the left of -2.030. (2 marks)
   5. [latex]P(t \geq 2.6) =  [/latex]_____________, that is the area under the [latex]t[/latex] density curve to the right of 2.6. (2 marks)
7. If you obtained one thousand 95% confidence intervals for a population mean, [latex]\mu[/latex], roughly how many of the intervals would actually contain [latex]\mu[/latex]? (2 marks)
8. A confidence interval for a population mean has a margin of error of 10.7.
   1. Obtain the length of the confidence interval. (2 marks)
   2. If the mean of the sample is 75.2, determine the confidence interval. (2 marks)
9. The following table gives the age (in years) of 36 randomly selected U.S. millionaires. The sample mean [latex]\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: M07_Age_Millionaire_Q9.xlsx.)
   

   31	45	79	64	48	38	39	68	52
   59	68	79	42	49	53	74	66	66
   71	61	52	47	39	54	67	55	71
   77	64	60	75	42	69	48	57	48

   1. Obtain a 95% confidence interval for [latex]\mu[/latex], the mean age of all U.S. millionaires. (4 marks)
   2. Interpret the confidence interval obtained in part (a). (2 marks)
   3. According to the confidence interval obtained in part (b), could you claim that the average age of all U.S. millionaires is above 55 years? Explain your answer. (3 marks)
   4. Determine the number of millionaires who should be picked to guarantee that the error of [latex]\bar{x}[/latex] in estimating [latex]\mu[/latex] is at most 0.5 years with 98% confidence. (4 marks)
10. 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: M07_BloodPressure_Diabete_Q10.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. Obtain a 95% confidence interval for the mean arterial blood pressure of all children of diabetic mothers. (4 marks)
    2. Interpret the confidence interval obtained in part (a). (2 marks)
    3. Obtain a 90% confidence interval for the mean arterial blood pressure of all children of diabetic mothers. (4 marks)
    4. Compare the confidence intervals obtained in parts (a) and (c). Which interval is wider? Write a sentence to summarize the relationship between the length of a confidence interval and the confidence level. (4 marks)

Part B

Finish the following questions using R and R commander. Make sure that you copy and paste the computer outputs into the space below each question, and write down your answers in statements.

1. For Question 9 in Part A (See data on file: M07_Age_Millionaire_Q9.xlsx), use the proper graphical tools in R and R commander to assess whether it is reasonable to apply the one-sample [latex]z[/latex] interval procedure. Make sure to write down the assumptions of the procedure and address whether the assumptions are satisfied. (5 marks)
2. Refer to the data in Question 10 in Part A; also see data on file: M07_BloodPressure_Diabete_Q10.xlsx.
   1. Use the proper graphical tools in R and R commander to assess whether applying the one-sample [latex]t[/latex] interval procedure is reasonable. Make sure to write down the assumptions of the procedure and address whether the assumptions are satisfied. (5 marks)
   2. Obtain a 95% confidence interval for the mean arterial blood pressure of all children of diabetic mothers. Compare the answer you obtained by hand. (2 marks)
   3. Obtain a 90% confidence interval for the mean arterial blood pressure of all children of diabetic mothers. Compare the answer you obtained by hand. (2 marks)