9.6 Assignment 9
Purposes
This assignment has two parts. The first part assesses your knowledge of properties of the distribution of the difference between two sample means [latex]{\bar{X}}_{1} - {\bar{X}}_{2}[/latex], conducting a two-sample [latex]t[/latex] test, and performing a paired [latex]t[/latex] test. The second part assesses your skills in using R commander to conduct a two-sample and a paired [latex]t[/latex] test in comparing two population means [latex]\mu_{1}[/latex] and [latex]\mu_{2}[/latex].
Resources
Instructions
Part A
Complete the following:
- Consider the quantities [latex]\mu_{1}[/latex], [latex]{\bar{X}}_{1}[/latex], [latex]{\bar{x}}_{1}[/latex] [latex]s_{1}[/latex], [latex]\mu_{2}[/latex], [latex]{\bar{X}}_{2}[/latex], [latex]{\bar{x}}_{2}[/latex], and [latex]s_{2}[/latex].
- Which quantities represent parameters and which represent statistics? (4 marks)
- Which quantities are fixed numbers and which are variables? (4 marks)
- A variable of two populations has a mean of 8 and a standard deviation of 6 for one of the populations and a mean of 7 and a standard deviation of 5 for the other population.
- For independent samples of sizes 3 and 6, respectively, find the mean and standard deviation of [latex]{\bar{X}}_{1} - {\bar{X}}_{2}[/latex]. (3 marks)
- Must the variable under consideration be normally distributed on each of the two populations for you to answer part (a)? Explain your answer. (2 marks)
- Can you conclude that the variable [latex]{\bar{X}}_{1} - {\bar{X}}_{2}[/latex] is normally distributed? Explain your answer. (3 marks)
- A variable of two populations has a mean of 40 and a standard deviation of 12 for one of the populations and a mean of 40 and a standard deviation of 6 for the other population. Moreover, the variable is normally distributed on each of the two populations.
- For independent samples of sizes 9 and 4, respectively, determine the mean and standard deviation of [latex]{\bar{X}}_{1} - {\bar{X}}_{2}[/latex]. (3 marks)
- Can you conclude that the variable [latex]{\bar{X}}_{1} - {\bar{X}}_{2}[/latex] is normally distributed? Explain your answer. (3 marks)
- Determine the percentage of all pairs of independent samples of sizes 9 and 4, respectively, from the two populations with the property that the difference [latex]{\bar{X}}_{1} - {\bar{X}}_{2}[/latex] between the sample means is between -10 and 10. (5 marks)
- A study examined the sense of direction of 30 male and 30 female students. After being taken to an unfamiliar wooded park, the students were given some spatial orientation tests, including pointing to the south, which tested their absolute frame of reference. The students pointed by moving a pointer attached to a [latex]360^{o}[/latex] protractor. Following are the absolute pointing errors, in degrees, of the participants.
Male Female 13 130 39 33 10 14 8 20 3 138 13 68 18 3 11 122 78 69 111 3 38 23 60 5 9 128 31 18 35 111 59 5 86 22 70 109 36 27 32 35 58 3 167 15 30 12 27 8 3 80 8 20 67 26 19 91 68 66 176 15 Male Female Mean [latex]{\bar{x}}_{1} = 37.6[/latex] [latex]{\bar{x}}_{2} = 55.8[/latex] SD [latex]s_{1} = 38.49[/latex] [latex]s_{2} = 48.26[/latex] - Given the statistical summaries, test at the 1% significance level whether males have a better sense of direction than females on average. (8 marks)
- What is the P-value of the hypothesis test in part (a)? (2 marks)
- Obtain a confidence interval for the difference between the mean absolute pointing errors for males and females corresponding to the hypothesis test in part (a). (4 marks)
- Interpret the confidence interval obtained in part (c). Does this interval support the conclusion of the hypothesis test in part (a)? Justify your answer. (4 marks: 2+2)
- Frustrated passengers, congested streets, time schedules, and air and noise pollution are just some of the physical and social pressures that lead many urban bus drivers to retire prematurely with disabilities such as coronary heart disease and stomach disorders. An intervention program was implemented to improve the work conditions of the city’s bus drivers. The following table reported the heart rates, in beats per minute, of the drivers who drove on the improved routes (intervention) and those who drove on the regular routes (control).
[latex]{\bar{x}}_{1} = 67.90[/latex], [latex]s_{1} = 4.36[/latex], [latex]n_{1} = 10[/latex] [latex]{\bar{x}}_{2} = 67.35[/latex], [latex]s_{2} = 9.66[/latex], [latex]n_{2} = 31[/latex] Intervention ([latex]\mu_{1}[/latex]) Control ([latex]\mu_{2}[/latex]) 68 66 72 62 69 63 68 71 64 76 74 52 67 63 77 57 80 77 58 72 54 73 54 55 82 63 60 68 64 66 75 72 55 71 84 63 79 59 74 58 82 - Is applying the pooled two-sample [latex]t[/latex] test reasonable? Justify your answer. (2 marks)
- At the 5% significance level, do the data provide sufficient evidence that the intervention program reduces the mean heart rate of urban bus drivers? (8 marks)
- Obtain a confidence interval for the difference between the mean heart rates of urban bus drivers in the two environments corresponding to the hypothesis test in part (a). (4 marks)
- Interpret the confidence interval obtained in part (c). Does this interval support the conclusion of the hypothesis test in part (a)? Justify your answer. (4 marks: 2+2)
- Eleven tires were each measured for treadwear by two methods, one based on weight and the other on groove wear. The following are the data, in thousands of miles.
[latex]{\bar{x}}_{1} = 23.71[/latex], [latex]s_{1} = 7.19[/latex] [latex]{\bar{x}}_{2} = 19.95[/latex], [latex]s_{2} = 5.77[/latex] [latex]\bar{d} = 3.75[/latex], [latex]{\ \ \ s}_{d} = 3.22[/latex] Weight method ([latex]\mathbf{\mu}_{\mathbf{1}}[/latex]) Groove method ([latex]\mathbf{\mu}_{\mathbf{2}}[/latex]) Difference ([latex]\mathbf{\mu}_{\mathbf{1}}\mathbf{-}\mathbf{\mu}_{\mathbf{2}}[/latex]) 30.5 28.7 1.8 30.9 25.9 5 31.9 23.3 8.6 30.4 23.1 7.3 27.3 23.7 3.6 20.4 20.9 -0.5 24.5 16.1 8.4 20.9 19.9 1 18.9 15.2 3.7 13.7 11.5 2.2 11.4 11.2 0.2 - Are the data two independent samples or a simple paired sample? (2 marks)
- At the 5% significance level, do the data provide sufficient evidence to conclude that, on average, the two measurement methods give different results? (8 marks)
- What is the P-value of the hypothesis test in part (b)? (2 marks)
- Obtain a confidence interval for the mean difference in measurement by the weight and groove methods corresponding to the hypothesis test in part (b). (4 marks)
- Interpret the confidence interval obtained in part (d). Does this interval support the conclusion of the hypothesis test in part (b)? Justify your answer. (4 marks: 2+2)
- The manufacturer of an Engine Energizer System (EES) claims that it improves gas mileage and reduces emissions in automobiles by using magnetic-free energy to increase the amount of oxygen in the fuel for greater combustion efficiency. Following are test results, performed under international and U.S. government agency standards, on a random sample of 14 vehicles. The data (also see file M09_Gas_Paired.txt or M09_Gas_Q7.xlsx) give the carbon monoxide (CO) levels, in parts per million, of each vehicle tested, both before installation of EES and after installation.
Before After 1.60 0.15 0.30 0.20 3.80 2.80 6.20 3.60 3.60 1.00 1.50 0.50 2.00 1.60 2.60 1.60 0.15 0.06 0.06 0.16 0.60 0.35 0.03 0.01 0.10 0.00 0.19 0.00 - Test at the 1% significance level whether, on average, EES reduces CO emissions. (11 marks)
- Obtain a 99% confidence interval for the difference between the mean CO emissions before and after installation of EES corresponding to the test in part (a). (4 marks)
- 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)
Part B
Finish the following questions using R and R commander. Make sure that you copy and paste the computer outputs and write down your answers in statements.
- Refer to Question 4 in Part A. The data are provided in the data file M09_Direction_Q4.xlsx. Import the data into R commander.
- Use the proper graphical tools in R and R commander to assess whether it is reasonable to apply the procedure you chose in Question 4 of Part A. Make sure to write down the assumptions of the procedure and address whether every assumption is satisfied. (5 marks)
- Re-conduct the test in Question 4 (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 on the output with the one you obtained by hand in Question 4 (a) and (b). (5 marks)
Note: R commander uses Female-Male by default; please pay attention to this when you set up the hypotheses. - Obtain a confidence interval corresponding to the hypothesis test in part (b). Compare it to the one you obtained by hand in Question 4 (c). (2 marks)
- Refer to Question 5 in Part A. The data are provided in the data files M09_Driver_Q5_twocolumn.xlsx and M09_Driver_Q5.xlsx.
- Use the proper graphical tools in R and R commander to assess whether applying the procedure you chose in Question 5 (b) is reasonable. Make sure to write down the assumptions of the procedure and address whether every assumption is satisfied. (Hint: use the data file M09_Driver_Q5_twocolumn.xlsx to draw the normal probability plots). (5 marks)
- Import the data file M09_Driver_Q5.xlsx into R commander. Re-conduct the test in Question 5 (b) using R commander. Make sure to include all the six components of a hypothesis test. Copy and paste the computer output first, then compare the answer with the one you obtained by hand in Question 5 (b). (5 marks)
- Obtain a confidence interval corresponding to the hypothesis test in part (b). Compare it to the one you obtained by hand in Question 5 (c). (2 marks)
- Refer to Question 6 in Part A. The data are provided in the data file M09_Treadwear_Q6.xlsx.
- Use the proper graphical tools in R and R commander to assess whether applying the procedure you chose in Question 6 (b) is reasonable. Make sure to write down the assumptions of the procedure and address whether every assumption is satisfied. (5 marks)
- Re-conduct the test in Question 6 (b) using R commander. Make sure to include all the six components of a hypothesis test. Copy and paste the computer output first, then compare the answer with the one you obtained by hand in Question 6 (b) and (c). (5 marks)
- Obtain a confidence interval corresponding to the hypothesis test in part (b). Compare the one you obtained by hand in Question 6 (d). (2 marks)
- Refer to Question 6. The data are provided in the data file M09_Gas_Q7.xlsx.
- Use the proper graphical tools in R and R commander to assess whether applying the procedure you chose in Question 7 (a) is reasonable. Make sure to write down the assumptions of the procedure and address whether every assumption is satisfied. (5 marks)
- Re-conduct the test in Question 7 (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 7 (a). (5 marks)
- Obtain a confidence interval corresponding to the hypothesis test in part (b). Compare it to the one you obtained by hand in Question 7 (b). (2 marks)
