Statistical Processes #2 1. The yields from an ethanol-water distillation column have a standard deviation of 0.01. Process specifications call for a target yield of 0.93. Management wishes to detect any decrease in the true mean yield. a. A random sample of 8 recent batches produced the following yields. Conduct a hypothesis test to determine whether the true mean yield has decreased. Use an =0.01. 0.90 0.90 0.93 0.87 0.95 0.93 0.86 0.92 b. Based on the sample of 8 observations in part a, construct a 99% confidence interval for the true mean yield of the process. c. Discuss the relationship between your response to part a and part b. d. What is , the probability of not rejecting the null hypothesis of part a, when in fact the true yield of the process is 0.925? What is (1-), the probability of rejecting the null hypothesis of part a, when in fact the true yield of the process is 0.925? 2. A company manufactures steel bolts that continuously feed an assembly line downstream. Historically, the thickness of these bolts follows a normal distribution with a standard deviation of 1.6 mm. A recent random sample of 10 observations yielded these values of thickness, in mm: 9.7 9.4 9.9 9.9 10.3 10.1 10.1 9.7 10.5 10.3 a. Construct a 95% confidence interval for the true mean thickness. b. Find the number of observations, n, required to estimate the true mean thickness to within 0.2 mm using a 95% confidence interval. 3. Consider the particle size of a particular product. The product specifications require that the variance of the particle sizes be 14.40. Production personnel measured the particle size for a sample of 25 observations, with the following results: 99.6 102.8 101.5 103.2 102.3 92.1 100.9 96.7 97.5 93.8 103.8 100.5 96.8 98.3 102.7 95.3 102.7 97.8 105.8 94.9 101.6 96.9 104.7 100.6 94.9 a. Conduct a hypothesis test to test the claim that the variance of the particle sizes is 14.40. Use an = 0.05. b. Construct a 95% confidence interval for the true variance of the particle sizes. c. What assumptions did you have to make in order to perform the above analyses? Please evaluate how well the data meet the assumptions. 4. An automobile manufacturer gives a 5 year/60,000 mile warranty on its drive train. Historically, 7% of this manufacturer's cars have required service under this warranty. Recently, a design team proposed an improvement that should extend the drive train's life. A random sample of 200 cars underwent 60,000 miles of road testing, in which tests the drive train of 12 cars failed. a. Conduct the most appropriate hypothesis test using a 0.05 significance level. b. Construct a 95% confidence interval for the true proportion of automobiles with drive trains that fail. 5. A company compared the performance of two temperature instruments: one coupled to a process computer and the other used for visual control. Ideally, these two instruments should agree. The following data are the two temperatures over a five day period: Day Temp 1 Temp 2 1 84.6 85.2 2 84.5 85.1 3 84.4 84.9 4 84.6 85.3 5 84.3 85.0 a. Conduct the most appropriate hypothesis test using a 0.05 significance level to determine if there is a statistically significant difference between the reading of both instruments. b. Construct a 95% confidence interval for the true mean difference in temperature readings. 6. The October 24, 1996 issue of The Wall Street Journal gave the following results for a poll: Clinton Dole Perot Undecided 52% 35% 6% 7% According to the article the above results were based on a nationwide survey of 1008 households. Determine a 95% confidence interval for each of the above results