L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 1 MER301:Engineering Reliability LECTURE 9: Chapter 4: Decision Making for a Single Sample, part 2

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 2 Summary  Hypothesis Testing Procedure  Inference on the Mean,Known Variance (z-test) Hypothesis Test Criteria P-value Choice of sample size Confidence interval

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 3 Populations and Parameters, Samples and Statistics….  A Population has a Distribution that is characterized by Parameters and that give the Mean and Variance, respectively.  The intent of drawing a Sample is to make estimates of either the Population Mean or the Variance, or Both The Sample Mean is a Statistic used to estimate the value of the Population Mean The Sample Variance is a Statistic that may be used to estimate the Population Variance  A larger number of samples gives a more precise estimate

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 4 Summary of Hypothesis Testing  Comments on Hypothesis Testing Null Hypothesis is what is tested Rejection of the Null Hypothesis always leads to accepting the Alternative Hypothesis Test Statistic is computed from Sample data Critical region is the range of values for the test statistic where we reject the Null Hypothesis in favor of the Alternative Hypothesis Rejecting when it is true is a Type I error Failing to reject when it is false is a Type II error

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 5 Hypothesis Testing Procedure See inside front and back flaps of text…

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 6 Example 9.1  A computer system currently has 10 terminals and uses a single printer. The average turnaround time for the system is 15 minutes.  10 new terminals and a second printer are added to the system.  We want to determine whether or not the mean turnaround time is affected.  Describe the hypothesis.

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 7 Example 9.2  The acceptable level for exposure to microwave radiation in the US is taken as 10 microwatts per square centimeter. It is feared that a large television transmitter may be pushing the the level of microwave radiation above the acceptable level.  Write the appropriate hypothesis test.

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 8 Example 9.3  Design engineers are working on a low- effort steering system that can be used in vans modified to fit the needs of disabled drivers. The old-type steering system required a force of 54 ounces to turn the van’s 15in diameter steering wheel. The new design is intended to reduce the average force required to turn the wheel.  State the appropriate hypothesis.

L Berkley Davis Copyright 2009 Inference on the Mean, Variance Known (z-test)

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 10 Inference on the Mean, Variance Known  This case typically arises when samples are being drawn from a population with known mean and standard deviation but subject to variation in processes, for example as in manufacturing.  For such cases, samples are drawn to test whether the process is producing parts with the required quality.  Random Samples of size n are drawn from the population to give a test statistic

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 11 Inference on the Mean, Variance Known  The test may be to establish that a process remains centered (two sided) or that the process does not drift beyond a critical upper or lower bound (one sided). 4-10

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 12 Inference on the Mean, Variance Known  For a two sided test to see if a process is centered, the hypothesis is Reject H 0 if the observed value of the test statistic z 0 is either: z 0 > z  /2 or z 0 < -z  /2 Fail to reject H 0 if -z  /2 < z 0 < z  /2

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 13 Hypothesis Testing Summary Inference on the Mean, Variance Known

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 14 Example 9.4  Continuing with the Example 9.1  30 samples of turnaround time are taken with the following results Sample Average = 14.0 (Population)Standard deviation = 3  Can we reject the null hypothesis? Set the probability of making a Type 1 error at

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 15 P-Value

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 16 What does P-value tell you?  It is customary to call the test statistic and the data significant when the null hypothesis is rejected…..therefore, the p-value is the smallest at which the data are significant.  Another way to think of the p-value is the probability that is true and the sample results (the value of the test statistic) were obtained by pure chance…

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 17 Two tailed P-Values

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 18 Example 9.5  What is the P value for the results of Examples 9.1/9.4?

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 19 Type II Error 4-11

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 20 Example 9.6  For the previous Example 9.4, what was the probability of failing to reject the null hypothesis when it is false? Assume the true mean is equal to the Sample Mean=14  Compute the power of this statistical test.

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 21 Impact of Sample Size on Type II Error, Two Sided Test

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 22 Impact of Sample Size on Type II Error, One Sided Test

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 23 Example 9.7  Consider Example 9.1/9.4/9.5/9.6… again.  The engineer wishes to design a test so that if the true mean turnaround time differs from 15 minutes by at least 0.9 minutes, the test will detect this with probability of 0.9. The Population Standard Deviation is 3min.  What number of samples is required?

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 24 Distribution of the Mean 4-13

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 25 Confidence Intervals on the Mean 4-12

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 26 Confidence Intervals on Mean Known Variance

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 27 One-Sided Confidence Interval on Mean with Variance Known

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 28 Example 9.8  The lifetime of a mechanical relay in a heating system is assumed to be a normal random variable with variance 6.4 days 2.  Five items are tested and fail at 104.1, 86.2, 94.1, 112.7, and 98.8 days  What is the 95% confidence interval on the mean.

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Estimating Sample Size for a Given Error

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Estimating Sample Size for a Given Error  In general, a sample size n necessary to ensure a confidence interval of length L is given by  The smaller the desired L, the larger n must be…  n increases as the square of More population variability requires a larger sample size  n is an increasing function of confidence interval since as decreases increases

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 10.1  Extensive monitoring of a computer time sharing system has suggested that response time to a particular edit command is normally distributed with standard deviation 25 msec.  A new operating system has been installed and it is desired to estimate the true average response time µ for the new environment.  Assume that the response times are still normally distributed with σ=25 msec.  What sample size is necessary to ensure that the resulting 95% confidence interval has a length of, at most, 10 msec.

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 9 32 Summary  Hypothesis Testing Procedure  Inference on the Mean,Known Variance Criteria P-value Choice of sample size Confidence interval