Stat 231 Exam 2 Fall 2013 I have neither given nor received unauthorized assistance on this exam. Name Signed Date Name Printed 1
1. Some IE 361 students worked with a manufacturer on quantifying the capability of a machining process to produce metal parts meeting engineering requirements. A sample of n 15 parts had measured diameters with x 1.53633 in and s.00010 in. a) Give 95% two-sided confidence limits for the average measured diameter of these parts. (Plug in completely, but you need not simplify.) 13 pts b) The (mid-specification and) target for the diameter in question was 1.5365 in. With.05, is there sufficient evidence to force the conclusion that the current machining process is producing diameters with an average different from the target value? (Carefully show the whole 7-step format.) 2
6 pts c) If you had access to the students' data, what would you do with them before using a Stat 231 method to say where you expect (say) 95% of all measured diameters from this process to lie? Why? d) Give two-sided limits that you are "95% sure" enclose 99% of all measured diameters (assuming the process is producing normally distributed diameters). (Plug in completely, but you don't need to simplify.) e) Give an upper bound that you are 95% sure is larger than the standard deviation of measured diameters (assuming the process is producing normally distributed diameters). (Plug in completely, but you don't need to simplify.) 3
After adjusting the process, a sample of n2 200 parts produced a sample mean diameter x2 1.53652 in, sample standard deviation s2.00030 in, and 183 diameters meeting engineering specifications. f) Give two-sided 90% confidence limits for any change in measured mean part diameter produced by the adjusting. (Plug in completely, but you don't need to simplify.) g) Give two-sided 95% confidence limits for the fraction of diameters produced by the adjusted process that meet engineering specifications. (Plug in completely, but you don't need to simplify.) 6 pts h) Set up (and plug into completely) an equation for a sample size that you believe would in the future allow you to determine the process mean with a (90%) margin of error of.00001 in. (You need not solve for the sample size.) 4
2. The following is a problem about the concepts of hypothesis testing. It is not an application of the any of the standard inference methods outlined on the inference summary sheets. The variable X number of hits on certain high-traffic web site during a 10 second period is modeled as Poisson with mean. In what follows, use without needing to make any justification Poisson distribution is the fact (that is actually a consequence of the CLT) that for large a approximately normal. (Note also that you know the variance of a Poisson variable.) An engineer wants to be able to detect the possibility that 100. a) First suppose that the engineer will reject H 0 : 100 if X 110. i) Approximately what value of is this engineer using? ii) Approximately what is the probability of a Type II error if in fact 150? 6 pts b) Approximately what p -value would be associated with an observation X 120? 5
10 pts 3. A (uniform) random number generator produces values between 0 and 1 with mean.5 and standard deviation.2887. What is the approximate probability that the sample mean of the next 100 values it generates is between.45 and.55? 10 pts 4. Here are five 2-point True/False Questions. Write T or write F in the blank in front of each one. x is If a 90% prediction interval for new 31.2, 45.7, then this interval can also serve as a confidence interval for with a confidence level at least 90%. If a 90% confidence interval for is 31.2, 45.7, then a p -value for testing H 0 : 50.0 is larger than.10. A value of ˆp based on a sample size of n 100 produces a p -value of.10 for testing H 0 : p #. That same value of ˆp would then produce a smaller p -value for testing the same null hypothesis if it were based on n 400. The small n confidence limits for are fairly "robust" to moderate non-normality ("working" well even if the distribution sampled departs somewhat from a normal shape), but the confidence limits for are less "robust." A statistical tolerance interval for a part dimension is directly concerned with whether parts meet engineering tolerances on that dimension. 6