DEPARTMENT OF ENGINEERING MATHEMATICS APPLIED REGRESSION ANALYSIS
In this assignment use Minitab to calculate sample means and sample standard deviations then carry out the calculations of confidence intervals and tests of hypotheses by hand using the formulas given in class and the statistical tables provided. You should then show the Minitab output giving the same results.
1. A car manufacturer plans on using their current minivan engine in a new line of Sport Utility Vehicles they are developing. The mean fuel consumption rate is dependent on, amongst other things, the vehicle mass, rolling resistance, wind resistance, and driver agressiveness. In an effort to determine the mean highway fuel consumption rate for this vehicle, an engineer sends 25 of the vehicles out on separate highway test routes. The sample mean fuel consumption rate is determined to be 7.5 `/100 km with a standard deviation of 0.9`/100 km.
(a) Find a 95% confidence interval for the mean highway feul consumption rate for the vehicle.
(b) Suppose that the engineer wants to be 95% confident that the estimated mean highway fuel consumption rate is within 0.1 `/100 km of the true mean highway consumption rate. How many vehicles should be sent out?
(c) Suppose that the company wants to show that the new line of Sport Utility Vehicles has gas fuel consumption rate no more than 7 `/100 km. Test this hypothesis at the 5% level of significance and give the p–value.
2. A machine is producing cylindrical shafts. The specifications for the shafts call for a nominal diameter of 5 cm and the standard deviation of diameter is to be at most 0.1 cm. A random sample of 10 shafts have diameters as follows: 5.26, 5.08, 5.00, 4.84, 5.05, 4.95, 5.24, 5.06, 5.25, 5.01 (see the file diameter.txt).
(a) Compute a 95% confidence interval for the mean diameter of the shafts. Do you think the machine is producing shafts which meet the specifications as to nominal diameter ?
(b) Test the hypothesis, at the 5% level of significance, that the machine is producing shafts which meet the specifications as to nominal diameter. What is the p-value of the test?
(c) Compute a 95% confidence interval for the variance of the diameter of the shafts. Do you think the machine is producing shafts which meet the specifications as to standard deviation?
(d) Test the hypothesis, at the 5% level of significance, that the machine is producing shafts which meet the specifications as to standard deviation. What is the p–value of the test?
3. A chemical engineer is attempting to assess the concentration of lead remaining unabsorbed from a gas after passing it over a catalyst. This will be done by measuring the remaining lead content in the gas, in parts per million. Eight measurements of the lead content in the gas after passing it over the catalyst are stored in the file lead.txt.
(a) Assuming that the unabsorbed lead content is (at least approximately) normally distributed, construct a 95% confidence interval for the mean unabsorbed lead content.
(b) The engineer is hoping that the catalyst will reduce the mean unabsorbed lead content to 0.830 parts per million (which is what a competitor is claiming their catalyst achieves). Does it seem likely that the catalyst is achieving this goal? Explain your answer by refering to the confidence interval found above.
(c) In the situation described the engineer is interested in the lower limit of the lead content. Test the hypotheis: H0 : µ = 0.830 versus the one sided alternative H1 : µ > 0.830 using the 5% level of significance.
(d) How can you reconcile your answers to b) and c)?
4. Epidemiologists have theorized that the risk of coronary heart disease can be reduced by an increased consumption of fish. One study, begun in 1980, monitored the diet and health of a random sample of middle-aged men. The men were divided into groups according to the number of grams of fish consumed per day. Twenty years later, the level of HDL (good) cholesterol present in each was recorded. A subset of the results are summarized in the following table
No Fish Consumption 0 grams/day High Fish Consumption 45 grams/day
Sample Size 29 21
Sample Mean 1.10 1.58
Sample Stdev 0.66 0.75
(a) Find 95% confidence intervals for the mean of each group.
(b) Based on the confidence intervals in a) can we say that fish consumption changes the mean HDL cholesterol level?
(c) Use the 2–sample t test with equal variances to test the hypotheses H0 : µf ish−µnof ish = 0 versus H1 : µf ish − µnof ish 6= 0 at the 5% level of significance. What is the p–value of the test?
(d) How can you reconcile your answers to b) and c)?
(e) It would seem that a one sided test of hypotheses would be appropriate for this situation. Test the hypotheses H0 : µf ish − µnof ish = 0 versus H1 : µf ish − µnof ish > 0 at the 5% level of significance. What is the p–value of the test?
5. A new coal liquefaction process is being studied. It is claimed that the new process results in higher yield of distillate synthetic fuel than the current process. The observations, stored in the file fuel.txt, were obtained on the number of kilograms of distillate synthetic fuel produced per kilogram of hydrogen consumed in the process. (“Liquefaction Process Promised Better Efficiency”, Modern Power Systems, May 1983, p. 13.)
(a) Test whether the two random variables have the same standard deviation.
(b) Assuming that these two random variables have the same standard deviation, find a 95% confindence interval for the difference of mean distillate.
(c) Test the hypothesis that the new process results in higher yield at the 5% level of significance. What is the p-value of the test?
(d) Would you recommend the new process?
6. A taxi company is trying to decide whether the use of radial tires instead of regular belted tires improves fuel economy. Twelve cars were equipped with radial tires, driven over a prescribed test course and the fuel consumption was recorded for each car. Without changing drivers, the same cars were then equipped with belted tires, driven once again over the test course and the fuel consumption was recorded for each car. The fuel consumption in `/100 km are stored in the file tires.txt.
(a) Find 95% confidence intervals for the fuel consumption when using the radial and belted tires. Can we determine whether the radial tires result in lower fuel consumption based on these intervals?
(b) Find a 95% confidence interval of the difference of fuel consumptions. Can we determine whether the radial tires result in lower fuel consumption based on this interval?
(c) Carry out a test of hypotheses, at the 5% level of significance, to determine whether the radial tires result in lower fuel consumption. What is the p–value?