Chapter 9 Comparing Two Population Means 9 1

Chapter 9. Comparing Two Population Means 9. 1 9. 2 9. 3 9. 4 9. 5 NIPRL Introduction Analysis of Paired Samples Analysis of Independent Samples Summary Supplementary Problems 1

9. 1 Introduction 9. 1. 1 Two Sample Problems(1/7) • A set of data observations x 1, … , xn from a population A with a cumulative dist. FA(x), a set of data observations y 1, … , ym from another population B with a cumulative dist. FB(x). • How to compare the means of the two populations, (Fig. 9. 1) and ? • What if the variances are not the same between the two populations ? (Fig. 9. 2) NIPRL 2

9. 1. 1 Two Sample Problems (2/7) Fig. 9. 1 Comparison of the means of two prob. dists. Fig. 9. 2 Comparison of the variance of two prob. dists. NIPRL 3

9. 1. 1 Two Sample Problems (3/7) • Example 49. Acrophobia Treatments - In an experiment to investigate whether the new treatment is effective or not, a group of 30 patients suffering from acrophobia are randomly assigned to one of the two treatment methods. - 15 patients undergo the standard treatment, say A, and 15 patients undergo the proposed new treatment B. Fig. 9. 3 Treating acrophobia. NIPRL 4

9. 1. 1 Two Sample Problems (4/7) - observations x 1, … , x 15 ~ A (standard treatment), observations y 1, … , y 15 ~ B (new treatment). - For this example, a comparison of the population means and provides an indication of whether the new treatment is any better or any worse than the standard treatment. NIPRL , 5

9. 1. 1 Two Sample Problems (5/7) - It is good experimental practice to randomize the allocation of subjects or experimental objects between standard treatment and the new treatment, as shown in Figure 9. 4. - Randomization helps to eliminate any bias that may otherwise arise if certain kinds of subject are “favored” and given a particular treatment. • Some more words: placebo, blind experiment, double-blind experiment NIPRL Fig. 9. 4 Randomization of experimental subjects between two treatment 6

9. 1. 1 Two Sample Problems (6/7) • Example 44. Fabric Absorption Properties - If the rollers rotate at 24 revolutions per minute, how does changing the pressure from 10 pounds per square inch (type A) to 20 pounds per square inch (type B) influence the water pickup of the fabric? - data observations xi of the fabric water pickup with type A pressure and observations yi with type B pressure. - A comparison of the population means and shows how the average fabric water pickup is influenced by the change in pressure. NIPRL 7

9. 1. 1 Two Sample Problems (7/7) • Consider testing • What if a confidence interval of • contains zero ? Small p-values indicate that the null hypothesis is not a plausible statement, and there is sufficient evidence that the two population means are different. • How to find the p-value ? Just in the same way as for one-sample problems NIPRL 8

• 9. 1. 2 Paired Samples Versus Independent Samples (1/2) • Example 53. Heart Rate Reduction - A new drug for inducing a temporary reduction in a patient’s heart rate is to be compared with a standard drug. - Since the drug efficacy is expected to depend heavily on the particular patient involved, a paired experiment is run whereby each of 40 patients is administered one drug on one day and the other drug on the following day. - blocking: it is important to block out unwanted sources of variation that otherwise might cloud the comparisons of real interest NIPRL 9

9. 1. 2 Paired Samples Versus Independent Samples (2/2) • Data from paired samples are of the form (x 1, y 1), (x 2, y 2), …, (xn, yn) which arise from each of n experimental subjects being subjected to both “treatments” • The comparison between the two treatments is then based upon the pairwise differences zi = xi – yi , 1 ≤ i ≤ n Fig. 9. 9 Paired and independent samples NIPRL 10

9. 2 Analysis of Paired Samples 9. 2. 1 Methodology • Data observations (x 1, y 1), (x 2, y 2), …, (xn, yn) One sample technique can be applied to the data set zi = xi – yi , 1 ≤ i ≤ n, in order to make inferences about the unknown mean (average difference). • NIPRL 11

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9. 2. 2 Examples(1/2) • Example 53. Heat Rate Reduction - An initial investigation of the data observations zi reveals that 30 of 40 are negative. This suggests that - - NIPRL 13

9. 2. 2 Examples(2/2) - - - Consequently, the new drug provides a reduction in a patient’s heart rate of somewhere between 1% and 4. 25% more on average than the standard drug. NIPRL 14

9. 3 Analysis of Independent Samples NIPRL size Population A n Population B m mean standard deviation 15

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9. 3. 1 General Procedure (Smith-Satterthwaite test) • • NIPRL 17

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• • Fig. 9. 22 • p-value=0. 0027 NIPRL 21

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9. 3. 2 Pooled Variance Procedure • • NIPRL 23

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• • Fig. 9. 24 p-value=0. 0023 NIPRL 25

9. 3. 3. z-Procedure • • • - NIPRL 26

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9. 3. 4. Examples(1/2) • Example 49. Acrophobia Treatments - unpooled analysis (from Minitab) NIPRL Fig. 9. 25 Acrophobia treatment data set (improvement scores) 28

9. 3. 4. Examples(2/2) - Analysis using pooled variance * Almost same as in the unpooled case. NIPRL 29

9. 3. 5. Sample Size Calculations • Interest : determination of appropriate sample sizes n and m, or an assessment of the precision afforded by given sample sizes • • NIPRL 30

• Example 44. Fabric Absorption Properties NIPRL 31

– – to meet the specified goal, the experimenter can estimate that total sample sizes of n=m=95 will suffice. NIPRL 32

Summary problems (1) In a one-sample testing problem of means, the rejection region is in the same direction as the alternative hypothesis. (yes) (2) The p-value of a test can be computed without regard to the significance level. (yes) (3) The length of a t-interval is larger than that of a z-interval with the same confidence level. (no) (4) If we know the p-value of a two-sided testing problem of the mean, we can always see whether the mean is contained in a two-sided confidence interval. (yes) (5) Independent sample problems may be handled as paired sample problems. (no) NIPRL 33