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Chapter 10

Chapter 10. Two-Sample Tests. 10-1: Comparing Two Independent Samples. Hypothesis Test 1: z Test for the difference in means. Hypothesis Test 2: Pooled-variance t Test for the difference in two means. Hypothesis Test 3: Separate-Variance t Test for Differences in two means.

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Chapter 10

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  1. Chapter 10 Two-Sample Tests

  2. 10-1: Comparing Two Independent Samples • Hypothesis Test 1: z Test for the difference in means. • Hypothesis Test 2: Pooled-variance t Test for the difference in two means. • Hypothesis Test 3: Separate-Variance t Test for Differences in two means. • Confidence Interval Estimate for the difference between the means of two independent groups.

  3. 10-1: Comparing the Means of Two Independent Populations • Two means—other tests are coming. • Two Independent samples. • Describe Hypothesis testing procedure. • Chapter 10 Summary chart, page 382.

  4. Configuring Hypotheses (!) • What’s the variable of interest? Hypothesizing about a difference. • 1-tail or 2-tail? • Key words • Status quo • What do you want to conclude? • Which test to use? • Do you know the variance? • Do you suspect that the variances of the 2 populations are the same?

  5. Rejection Region • Reject Null Hypothesis for calculated value of test statistic more extreme than critical value of test statistic. • 2-tailed tests have 2 rejection regions! • Reject Null Hypothesis for observed significance (p-value) less than alpha.

  6. Test Statistic • Test statistic = (point estimate – hypothesized value) / appropriate standard deviation • The appropriate standard deviation is more often called the “standard error.” Its formula will usually change depending on the test that you are using: • Z • Pooled-variance t • Different variance t

  7. Notes • Random sampling. • However you define populations 1 and 2, you must get the same result. • Text considers upper tail only. • Z-test, pooled-variance t test, and separate variance t test might give conflicting answers. • Separate-Variance t test has a simpler standard error but a more complicated degrees of freedom in critical value.

  8. More Notes • Populations are assumed to be normally distributed. • The pooled-variance t test is robust to departures from normality, provided the sample sizes are large. • Normality Comment, page 351. • Equation 10.3, page 352.

  9. 10.2 Comparing the Means of Two Related Populations • Paired or matched samples. Sometimes described as “dependent” test or sampling or as “repeated measurements.” • Table 10.3, page 359: “Difference” is the variable of interest.

  10. Hypotheses • Can be 1-tail or 2-tail. • H0: μD = 0 or 1-tail.

  11. Rejection Region or Decision Rule; Test Statistic • Reject H0 for value of test statistic more extreme than critical value. • Critical value is either a “z” or “t” value from the appropriate table, obtained in the usual way. • Degrees of freedom for “t” is = n –1 (# of pairs!). • Test statistic is either “z” or “t”. It is calculated in the usual way:

  12. Notes • Random samples. • Normally distributed population (tests are robust to this assumption). • The confidence interval exists. • See note on page 365. • We will return to this concept when we discuss “blocking.”

  13. 10.3 Comparing Two Population Proportions • Consider only the z test on page 368. • Proportions are calculated for ___ data. • The confidence interval exists (page 372).

  14. 10.4: F Test for the difference between two variances • Recall that we had 2 “t” tests for differences in means (in 10.1). • The difference between those 2 “t” tests is whether or not you can assume that the variance for population 1 is the same as the variance for population 2. • You can test for differences between the 2 variances.

  15. Hypothesis Test • Can be 1 tail or 2 tail. • Can be written for variance or standard deviation.

  16. Rejection Region Reject null hypothesis for calculated test statistic more extreme than a critical value. • Reject null hypothesis for p-value less than alpha. • The critical value is “F.” It has two d.f. calculations.

  17. More on Rejection Region • Critical values of F come from a table. • A 2-tail test requires 2 critical values: an upper and a lower (Figure 10.18, p 377) • The table will not give lower critical values directly. Use formula 10.12, page 377. • A 1-tail test should be written so that it is an upper-tail test.

  18. Examine Sample, Calculate Test Statistic • The test statistic is “F” • F = S12 / S22

  19. Decide and Conclude • At alpha = 0.05, there is sufficient information to say that the population variances are not equal. • At alpha = 0.05, there is insufficient information to say that the population variances are not equal. • At alpha = 0.05, there is sufficient information to say that the first population variance is larger than the second population variance.

  20. Notes • Test is most often used to decide which “t” test to use on the means. Use the 2-tail test. • When concerned about comparing variability across processes, you will most often use a 1-tail test. • This test is sensitive to departures from normality.

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