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CHAPTER 4 HYPOTHESIS TESTING WITH TWO SAMPLES Introduction There are many instances researchers wish to compare two sample means. Examples: The average lifetimes of two different brands of batteries might be compared to examine whether one brand has longer lifetime than the other. Two types of materials might be tested to see whether there is any difference in tensile strength. In the comparison of two means, the same basic steps for hypothesis testing shown in Chapter 3 are used. 4.1 Testing the Difference Between Two Means: Large Samples Lesson outcome: Test the difference between two large sample means, using the z-test. Testing the Difference Between Two Means: Large Samples Introduction Suppose Pn Azlyna wants to determine whether there is a difference in the average score of engineering students in IQ Test for University A and University B. In this case Pn Azlyna is not interested in the average score for both universities. She is interested in comparing the means of the two groups. The hypotheses: H 0 : 1 2 1 Mean score for University A H1 : 1 2 2 Mean score for University B Testing the Difference Between Two Means: Large Samples Assumptions for testing the difference between two means: The samples must be independent of each other (there can be no relationship between the subjects in each sample). The population from which the samples were obtained must be normally distributed or approximately normally distributed, and the population standard deviations of the variables must be known or the sample sizes must be ≥ 30. Testing the Difference Between Two Means: Large Samples The theory behind testing the difference between two means is based on selecting pairs of samples and comparing the means of the pairs. All possible pairs of samples are taken from populations. The means for each pair of samples are computed and then subtracted. After that, the differences are plotted. Testing the Difference Between Two Means: Large Samples Distribution of X1 X 2 0 If both populations have the same mean, then most of the differences will be zero or close to zero. There will be a few large differences due to chance alone, some positive and others negative. If the differences are plotted, the curve will be shaped like the normal distribution and have mean zero. Testing the Difference Between Two Means: Large Samples Formula for the z-test for comparing two means from independent populations ( X 1 X 2 ) ( 1 2 ) z 12 2 2 n1 n2 4.1 Based on the general format (observed value) (expected value) Test value standard error Testing the Difference Between Two Means: Large Samples If 1 and 2 are not known, use s1 and s2 2 2 2 2 But both sample sizes must be n1 ≥ 30 and n2 ≥ 30 ( X 1 X 2 ) ( 1 2 ) z 2 2 s s 1 2 n1 n2 4.2 Testing the Difference Between Two Means: Large Samples The tests can be one-tailed or two-tailed using the following hypotheses: H0 : μ1 = μ2 H :μ -μ =0 Two-tailed test OR 0 1 2 H1 : μ1 ≠ μ2 H1 : μ1 - μ2 ≠ 0 H0 : μ1 ≤ μ2 H0 : μ1 - μ2 ≤ 0 One-tailed test Right-tailed OR H1 : μ1 > μ2 H1 : μ1 - μ2 > 0 H0 : μ1 ≥ μ2 H0 : μ1 - μ2 ≥ 0 Left-tailed OR H1 : μ1 < μ2 H1 : μ1 - μ2 < 0 We have one tail or two tails ??? Testing the Difference Between Two Means: Large Samples Hypothesis testing situations in the comparison of means Sample 1 Sample 2 Sample 1 Sample 2 X1 X2 X1 X2 Population Population Population μ1 = μ2 μ1 μ2 Difference is not significant Difference is significant Do not reject H 0 : 1 2 Reject H 0 : 1 2 X 1 X 2 is not significant X 1 X 2 is significant Testing the Difference Between Two Means: Large Samples Example 4.1 A researcher hypothesizes that the average number of sports that private colleges offer for males is greater than the average number of sports offer for females. A sample of the number of sports offered by colleges is collected and the results are given below: For the males : n1 50 , X 1 8.6, s1 3.3 For the females : n2 50 , X 2 7.9, s2 3.3 At α = 0.10, is there enough evidence to support the claim. Testing the Difference Between Two Means: Large Samples Sometimes, we are interested in testing a specific difference in means other than zero. Example: For Example 4.1, the researcher might hypothesize that the average number of sports that private colleges offer for males is more than three compared to the average number of sports offer for females. In this case the hypotheses are: H 0 : 1 2 3 H1 : 1 2 3 Testing the Difference Between Two Means: Large Samples Confidence intervals for the difference between two means: large samples When one is hypothesizing a difference of 0, if the ☻ confidence interval contains 0 → H0 is not rejected. ☻ confidence interval does not contain 0 → H0 rejected. 12 2 2 ( X1 X 2 ) z / 2 1 2 n1 n2 12 2 2 ( X1 X 2 ) z / 2 n1 n2 4.3 Note: The relationship presented here is valid for two-tailed tests only. 4.2 Testing the Difference Between Two Variances Lesson outcome: Test the difference between two variances or standard deviations. Testing the Difference Between Two Variances Introduction In addition to comparing two means, a researcher may be interested in comparing two variances or standard deviations. Example: Is the variation in the temperatures for a certain month for two cities different? For the comparison of two variances or standard deviations, an F-test is used. The F-test should not be confused with the Chi- square test, which compares a single sample variance to a specific population variance, as shown in Chapter 3. Testing the Difference Between Two Variances Characteristics of the F Distribution The values of F cannot be negative, because variances are always positive or zero. The distribution is positively skewed. The mean value of F is approximately equal to 1. The F distribution is a family of curves based on the degrees of freedom of the variance of the numerator and the degrees of freedom of the variance of the denominator. Testing the Difference Between Two Variances Assumptions for testing the difference between two variances The populations from which the samples were obtained must be normally distributed. (The test should not be used when the distribution depart from normality.) The samples must be independent of each other, Testing the Difference Between Two Variances Formula for the F-test for testing the difference between two variances 2 s1 F 2 s2 4.4 s12 s2 2 F test has two terms for degrees of freedom (DOF): ☻DOF of the numerator, v1 n1 1 ☻DOF of the denominator, v2 n2 1 n1 is the sample size from which the larger variance was obtained Testing the Difference Between Two Variances Notes for the use of the F-test For a two-tailed test, the α value must be divided by 2 and the critical value placed on the right side of the F-curve. Note: When one conducting a two-tailed test, α is split and even though there are two values, only the right tail is used. This is because the F test value is always ≥ 1 If the standard deviations instead of the variances are given in the problem, they must be squared for the formula for the F-test. When the DOF cannot be found from TABLE 9, the closest value on the smaller side should be used. Note: This guideline keeps type I error ≤ α Testing the Difference Between Two Variances Example 4.2 A medical researcher wishes to see whether the variance of the heart rates (in beats per minute) of smokers is different from the variance of heart rates of people who do not smoke. Two samples are selected and data are shown below. Using α = 0.05, is there enough evidence to support the claim? Smokers Nonsmokers n1 26 n2 18 s12 36 s2 10 2 Testing the Difference Between Two Variances Note It is not necessary to place the larger variance in the numerator when one performing the F-test (specifically in left-tailed test) Critical values for left-tailed hypotheses tests can be found by interchanging the DOF and taking the reciprocal (salingan) of the value found in TABLE 9. Caution: When performing the F-test, the data can run contrary to the hypotheses on rare occasions. Example: If the hypotheses are H 0 : 12 2 2 but s12 s2 2 H1 : 12 2 2 then F-test should not be performed and one would not reject H0. Testing the Difference Between Two Variances Example 4.3 A mathematics professor claims that the variance on placement tests when the students use graphing calculators will be smaller than the variance on placement tests when the students use scientific calculator. A randomly selected group of 50 students who used graphing calculators had a variance of 32, and a randomly selected group of 40 students who used scientific calculators had a variance of 37. Is the professor correct, using α = 0.05? Testing the Difference Between Two Variances Example 4.4 Find the p-value interval for each F-test value. (a) F 2.97 , v1 9, v2 14, right-tailed test (b) F 3.32, v1 6, v2 12, two-tailed test (c) F 1.16, v1 6, v2 12, left-tailed test

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CONFIDENCE INTERVALS FOR VARIANCES AND STANDARD DEVIATIONS, Testing the Difference Between Two Means Small Dependent Samples, Data Description, ANALYSIS OF VARIANCE

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posted: | 4/14/2010 |

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