HW 6 (Due Nov. 14, 2017)

Transcription

1 HW 6 (Due Nov. 14, 2017) Name: HW 6.1. The thickness of a plastic film (in mils) on a substrate material is thought to be influenced by the temperature at which the coating is applied. In completely randomized experiment, 11 substrates are coated at 125 o F, resulting in a sample mean coating thickness of x 1 = and a sample standard deviation of s 1 = Another 13 substrates are coated at 150 o F for which x 2 = 99.7 and s 2 = 20.1 are observed. It was originally suspected that raising the process temperature would reduce mean coating thickness. Test H 0 : σ 1 = σ 2 against H a : σ 1 > σ 2 using α =

2 HW 6.2. The manager of a fleet of automobiles is testing two brands of radial tires and assigns one tire of each brand at random to the two rear wheels of eight cars and runs the cars until the tires wear out. The data (in kilometers) follow. Find a 99% confidence interval on the difference in mean life. Which brand would you prefer based on this calculation? Further test whether the mean difference is zero or not using α =

3 HW 6.3. The compressive strength of concrete is being studied, and four different mixing techniques are being investigated. The following data have been collected. Test the hypothesis that mixing techniques affect the strength of the concrete. Use α =

4 HW 6.4. An electronics engineer is interested in the effect on tube conductivity of five different types of coating for cathode ray tubes in a telecommunications system display device. The following conductivity data are obtained. Is there any difference in conductivity due to coating type? Use Use α =

5 HW 6.5. Article The Use of Residual Maximum Likelihood to Model Grain Quality Characteristics of Wheat with Variety, Climatic and Nitrogen Fertilizer Effects (the Journal of Agricultural Science, 1997, ) investigated means of wheat grain crude protein content (CP) and Hagberg falling number (HFN) surveyed in the United Kingdom. The analysis used a variety of nitrogen fertilizer applications (kg N/ha), temperature ( o C), and total monthly rainfall (mm). The following data below describe temperatures for wheat grown at Harper Adams Agricultural College between 1982 and The temperatures measured in June were obtained as follows: 15.2, 14.2, 14.0, 12.2, 14.4, 12.5, 14.3, 14.2, 13.5, 11.8, 15.2 Assume the true distribution is normal. Answer the following questions. (DO NOT forget the interpretation). (a) Assume that the standard deviation σ is known to be σ = 0.5. Construct a 95% confidence interval on the mean temperature. Further find a 95% confidence lower and upper bound on the mean temperature. (b) Assume that the standard deviation σ is known to be σ = 0.5. Suppose you wanted to be 95% confident that the error in estimating the mean temperature is less than 0.05 degrees Celsius. What sample size should be used? (c) Now assume that the standard deviation σ is not known. Construct a 95% confidence interval on the mean temperature. (d) Assume that the standard deviation σ is not known. Construct a 95% confidence interval on σ 2. 5

6 HW 6.6. In semiconductor manufacturing, wet chemical etching is often used to remove silicon from the backs of wafers prior to metallization. The etch rate is an important characteristic in this process and known to follow a normal distribution. Two different etching solutions have been compared using two random samples. The observed etch rates are as follows (in mils per minute): Solution 1: 9.9, 9.4, 9.3, 9.6, 10.2, 10.6, 10.3, 10.0, 10.3, 10.1 Solution 2: 10.2, 10.6, 10.7, 10.4, 10.5, 10, 10.2, 10.7, 10.4, 10.3, 10.5 Assuming normality holds for both distributions. Let σ 1 and σ 2 denote the standard deviations of the etch rates using solution 1 and solution 2, respectively. Let µ 1 and µ 2 denote the means of the etch rates using solution 1 and solution 2, respectively. (a) Assuming that σ 1 = σ 2 = 0.3, construct a 95% confidence interval on µ 1 µ 2. (b) Assuming that σ 1 = σ 2 but both of them are unknown, construct a 95% confidence interval on µ 1 µ 2. (c) Assuming that both σ 1 and σ 2 are unknown, construct a 95% confidence interval on µ 1 µ 2. (d) Assuming that both σ 1 and σ 2 are unknown, construct a 95% confidence interval on σ 1 /σ 2. 6

7 HW 6.7. In Connecticut, a random sample of n = 200 legally-registered automobiles was recently taken. Of the 200, only 124 passed the state s emission test for pollution. Denote p 1 as the population proportion of automobiles that meet the state emissions standards in Connecticut. (a) Construct a 95% confidence interval for p 1. (b) An environmental engineer would like to design a larger study to estimate p 1. She would like to write a 95 percent confidence interval that will have margin of error equal to How many cars will she need to sample? In South Carolina, a random sample of n = 250 legally-registered automobiles was recently taken. Of the 250, only 193 passed the state s emission test for pollution. Denote p 2 as the population proportion of automobiles that meet the state s emissions standards in South Carolina. (c) Construct a 95% confidence interval for p 1 p 2. (c) Find a 95% confidence upper bound for p 1 p 2. 7

8 HW 6.8. The manager of a large taxi company in Los Angeles (with 1000s of cars) is trying to decide whether using radial tires (instead of using belted tires) improves his fleet s fuel economy on average. He randomly samples n = 12 cars equipped with radial tires and has them driven over a test course. Without changing drivers, the same cars are then equipped with belted tires and are driven through the same test course. The gasoline consumption (recorded in km per liter) was recorded for each car and tire type: Car Radial Belted (a) Explain why this is a matched pairs study. Are the two samples independent or dependent? Why? (b) Construct a 95% confidence intervals for the population mean difference gasoline consumption between radial (Group 1) and belted (Group 2) tires. Interpret the result for the manager. Which tires do you think is more fuel efficient? 8