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Chapter 11 Outline Park University - EC315 - Chapter 11 Lecture Notes Page 1 Chapter 11 Lecture Notes Introduction In this chapter we continue our study of hypothesis testing. Recall that in Chapter 10 we considered hypothesis tests in which we compared the results of a single sample statistic to a population parameter. In this chapter , we expand the concept of hypothesis testing to two samples. We select random samples from two independent populations and conduct a hypothesis test to determine whether the population means are equal. We might want to test to see if there is a difference in the mean number of defects produced on the 7:00 AM to 3:00 PM shift and the 3:00 PM to 11:00 PM shift at the DaimlerChrysler Jeep Liberty plant in Toledo, Ohio. We also conduct a hypothesis tests to determine if two sample proportions come from populations which are equal. For example, we may want to determine if the proportion of Jumpin’ Java customers who purchase frozen coffee drinks is the same for New England stores versus stores in the southeast. Two-Sample Tests of Hypothesis: Independent Samples As noted above, we expand the concept of hypothesis testing to two samples. When there are two populations, we can compare two sample means to determine if they came from populations with the same or equal means. For example, a purchasing agent is considering two brands of tires for use on the company's fleet of cars. A sample of 60 Rossford tires indicates the mean useful life to be 65,000 miles. A sample of 50 Maumee tires reveals the useful life to be 68,000 miles. Could the difference between the two sample means be due to chance? The assumption is that for both populations (Rossford and Maumee) the standard deviations are known. The test statistic follows the standard normal distribution and its value is computed from text formula [11-2]: XX12− Test Statistic for No Difference z= [11− 2] σσ22 Between Two Sample Means 12+ nn12 Where: X1 and X 2 refer to the two sample means. 2 2 σ1 and σ2 refer to the two sample variances. n1 and n2, refer to the two sample sizes. The following are assumptions necessary for this two-sample test of means: 1. The two populations must be unrelated; that is, independent. 2. The standard deviations for both populations must be known. Two-Sample Tests about Proportions We are often interested in whether two sample proportions came from populations that are equal. For example, we want to compare the proportion of rural voters planning to vote for the incumbent governor with the proportion of urban voters. The test statistic is formula [11-3]: © McGraw Hill – May not be duplicated without written permission Park University - EC315 - Chapter 11 Lecture Notes Page 2 pp− Two-Sample Test of Proportions z = 12 [11− 3] pppp()11−−() cccc+ nn12 Where: p1 is the proportion in the first sample possessing the trait. p2 is the proportion in the second sample possessing the trait. n1 is the number of observations in the first sample. n2 is the number of observations in the second sample. pc is the pooled proportion possessing the trait in the combined samples. It is called the pooled estimate of the population proportion and is found by formula [11-4] XX12+ Pooled Proportion pc = [11− 4] nn12+ Where: X1 is the number possessing the trait in the first sample. X2 is the number possessing the trait in the second sample. Comparing Population Means with Unknown Population Standard Deviations (the Pooled t-test) We now consider the case in which the population standard deviations are unknown. The following assumptions are required: 1. The sampled populations follow the normal distribution. 2. The two samples are from independent populations. 3. The standard deviations of the two populations are unknown but equal. If these assumptions are met, the t distribution can be used for the test statistic for a test of hypothesis for the difference between two population means. The t statistic for the two sample cases is similar to that employed for the z statistic, with one additional calculation. The two sample variances must be “pooled” to form a single estimate of the unknown population variance. This is accomplished by using text formula [11-5]: 22 2 (nsns112−+−11) ( ) 2 Pooled Variance sp = [11− 5] nn12+−2 Where: 2 sp is the pooled estimate of the population variance. 2 s1 is the variance of the first sample. 2 s2 is the variance of the second sample. n1 is the number of observations in the first sample. n2 is the number of observations in the second sample. The value of t is then computed using text formula [11-6]. © McGraw Hill – May not be duplicated without written permission Park University - EC315 - Chapter 11 Lecture Notes Page 3 XX− t = 12 [11− 6] Two-Sample Test of Means—Unknown σ1 and σ2 2 ⎛⎞11 sp ⎜⎟+ ⎝⎠nn12 Where: X1 is the mean of the first sample. X 2 is the mean of the second sample. n1 is the number of observations in the first sample. n2 is the number of observations in the second sample. 2 sp is the polled estimate of the population variance. The number of degrees of freedom for a two-sample test is the total number of items sampled minus the number of samples. It is found by: (n1 + n2 − 2). Two-Sample Tests of Hypothesis: Dependent Samples Another hypothesis testing situation occurs when we are concerned with the difference in paired or related observations. These are situations in which the samples are not independent. Typically, it is a before-and-after situation, where we want to measure the difference. To illustrate, suppose we administer a reading test to a sample of ten students. We have them take a course in speed reading and then we test them again. Thus the test focuses on the reading improvements of each of the ten students. The distribution of the population of differences is assumed to be approximately normal. The test statistic is t, and text formula [11-7] is used. d Paired t test t = []11− 7 snd Where: d is the mean of the difference between the paired or related observations. sd is the standard deviation of the differences between the paired or related observations. n is the number of paired observations. For a paired difference test there are (n − 1) degrees of freedom. The standard deviation of the differences sd is computed using the familiar formula for the Σ−()dd2 standard deviation except that d is substituted for X. The text formula is: s = d n −1 Comparing Dependent and Independent Samples When working with paired data, we need to distinguish between dependent samples and independent samples. Dependent samples: Two samples that are related to each other. © McGraw Hill – May not be duplicated without written permission Park University - EC315 - Chapter 11 Lecture Notes Page 4 There are two types of dependent samples: 1. Samples characterized by a measurement, an intervention of some type, and then another measurement. This is often referred to as a “before” and “after” study. For example: A group of teenagers are enrolled in a weight reduction program. They are weighed, go through a diet and exercise program, and then they are weighed again. The two weights are paired weights and are considered to be dependent samples. The paired samples are dependent because the same individual is a member of both samples. 2. Samples characterized by matching or pairing observations. For example: The transportation manager wants to study the amount of “tire wear” on two brands of tire. One tire of each brand is placed on 15 company trucks and the wear is measured after 20,000 miles. The manager would have 30 observations with 15 pairs of data. The paired samples are dependent because the pairs came off the same truck. When the samples chosen at random are in no way related to each other they are considered independent samples. Independent samples: Two samples that are unrelated to each other. Independent samples are essentially samples taken from entirely different populations. Keep in mind, however that the populations need to share some similar characteristics. For example: The human resource director might want to test the “Microsoft Word” skills of two sets of graduates from two different secretarial business programs. © McGraw Hill – May not be duplicated without written permission .
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