Applied Regression Analysis, Assignment 1 1

DEPARTMENT OF ENGINEERING MATHEMATICS

APPLIED REGRESSION ANALYSIS

ENGM 6671

ASSIGNMENT # 1

1. A car manufacturer plans on using their current minivan engine in a new line of Sport UtilityVehicles they are developing. The mean fuel consumption rate is dependent on, amongst otherthings, the vehicle mass, rolling resistance, wind resistance, and driver agressiveness. In aneffort to determine the mean highway fuel consumption rate for this vehicle, an engineer sends25 of the vehicles out on separate highway test routes. The sample mean fuel consumptionrate is determined to be 7.8 /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 highwayfuel consumption rate is within 0.1 /100 km of the true mean highway consumption rate.How many vehicles should be sent out?

2. A study of the electromechanical protection devices used in electrical power systems showedthat of 193 devices that failed when tested, 75 were due to mechanical parts failures. (Reliabil-ity of Protection Equipment in Operation, H. Hubensteiner, Brown Boveri Review, February,1983, pp. 111-114.)

(a) Find a point estimate of p, the proportion of failures that are due to mechanical failures.

(b) Find a 95% confidence interval for p.

(c) How large a sample is required to estimate p to within .03 with 95% confidence?

3. A machine is producing cylindrical shafts. The specifications for the shafts call for a nominaldiameter of 5 cm and the standard deviation of diameter is to be at most 0.1 cm. A randomsample of 10 shafts have diameters as follows: 5.263, 5.079, 5.003, 4.811, 5.048, 4.945, 5.244,5.055, 5.253, 5.011 (see the file diameter.txt).

(a) Compute a 95% confidence interval for the mean diameter of the shafts. Do you thinkthe machine is producing shafts which meet the specifications as to nominal diameter ?

(b) Test the hypothesis that the machine is producing shafts which meet the specificationsas 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. Doyou think the machine is producing shafts which meet the specifications as to standarddeviation?

(d) Test the hypothesis that the machine is producing shafts which meet the specificationsas to standard deviation. What is the pvalue of the test?

4. A chemical engineer is attempting to assess the concentration of lead remaining unabsorbedfrom a gas after passing it over a catalyst. This will be done by measuring the remaining leadcontent in the gas, in parts per million. Eight measurements of the lead content in the gasafter passing it over the catalyst are stored in the file lead.txt.

(a) Assuming that the unabsorbed lead content is (at least approximately) normally dis-tributed, 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 to0.830 parts per million (which is what the competitor is claiming their catalyst achieves).Does it seem likely that the catalyst is achieving this goal? Explain your answer byrefering to the confidence interval found above.

Applied Regression Analysis, Assignment 1 2

(c) In the situation described the engineer is interested in the lower limit of the lead content.Test the hypotheis: H0 : = 0.830 versus the one sided alternative H1 : > 0.830 usingthe 5% level of significance.

(d) How can you reconcile your answers to b) and c)?

5. Epidemiologists have theorized that the risk of coronary heart disease can be reduced by anincreased consumption of fish. One study, begun in 1980, monitored the diet and healthof a random sample of middle-aged men. The men were divided into groups according tothe 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 followingtable

No Fish Consumption High Fish Consumption0 grams/day 45 grams/day

Sample Size 29 21Sample Mean 1.10 1.58Sample Stdev 0.66 0.75

(a) Find 95% confidence intervals for the mean and standard deviation of each group.

(b) Based on the confidence intervals in a) can we say that fish consumption changes themean HDL cholesterol level?

(c) Use the 2sample t test with equal variances to test the hypothesesH0 : fishnofish = 0versus H1 : fish nofish 6= 0. What is the pvalue 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 : fish nofish = 0 versus H1 : fish nofish > 0. What isthe pvalue of the test?

6. A new coal liquefaction process is being studied. It is claimed that the new process results inhigher yield of distillate synthetic fuel than the current process. The observations, stored inthe file fuel.txt, were obtained on the number of kilograms of distillate synthetic fuel producedper kilogram of hydrogen consumed in the process. (Liquefaction Process Promised BetterEfficiency, Modern Power Systems, May 1983, p. 13.)

(a) Assuming that these two random variables have the same standard deviation, find thepooled standard deviation for the two data sets.

(b) Find a 95% confindence interval for the difference of mean distillate.

(c) Test the hypothesis that the new process results in higher yield. What is the p-value ofthe test?

(d) Would you recommend the new process?

7. A study was conducted to decide whether a new statistical package has lower cost than theone currently in use. To do so, 15 data sets are used. Each is analyzed by each package andthe cost of the analysis is recorded. The observations are stored in the file cost.txt.

(a) Find 95% confidence intervals for the costs when using the new and old packages. Canwe determine whether the new package has lower cost than the old one based on theseintervals?

(b) Find a 95% confidence interval of the difference of costs. Can we determine whether thenew package has lower cost than the old one based on this interval?

(c) Carry out a test of hypotheses to determine whether the new package has lower cost thanthe old one. What is the pvalue?