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The formula for comparing the means of two populations using pooled variance is
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Where s 1and s 2are the standard deviations of the two samples and n 1 and n 2are the sizes of the two samples. The formula for the pooled estimator of σ 2 is For this reason, the pooled variance method should be used with caution. Although using pooled variance in a t‐test is generally more likely to yield significant results than using separate variances, it is often hard to know whether the variances of the two populations are equal. If the two population distributions can be assumed to have the same variance-and, therefore, the same standard deviation- s 1and s 2 can be pooled together, each weighted by the number of cases in each sample. The fact that the interval contains 0 means that if you had performed a test of the hypothesis that the two population means are different (using the same significance level), you would not have been able to reject the null hypothesis of no difference. You can be 90 percent confident that Brand A cereal has between 2.81 fewer and 19.81 more raisins per box than Brand B.
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According to Table 3 in "Statistics Tables," the critical value for t. A 90 percent confidence interval is equivalent to an alpha level of 0.10, which is then halved to give 0.05. The degrees of freedom is the smaller of (6 – 1) and (9 – 1), or 5. The difference between and is 102.1 – 93.6 = 8.5. The degrees of freedom parameter for looking up the t‐value is the smaller of n 1 – 1 and n 2– 1.Įstimate a 90 percent confidence interval for the difference between the number of raisins per box in two brands of breakfast cereal. Where a and b are the limits of the confidence interval, and are the means of the two samples, is the value from the t‐table corresponding to half of the desired alpha level, s 1and s 2 are the standard deviations of the two samples, and n 1and n 2are the sizes of the two samples. This test has not provided statistically significant evidence that intensive tutoring is superior to paced tutoring. The computed t of 1.166 does not exceed the tabled value, so the null hypothesis cannot be rejected. 05,9in the t‐table (Table 3 in "Statistics Tables"), which gives a critical value of 1.833. Because this is a one‐tailed test, the alpha level (0.05) is not divided by two. The degrees of freedom parameter is the smaller of (12 – 1) and (10 – 1), or 9. Two randomly chosen groups are tutored separately and then administered proficiency tests. The number of degrees of freedom for the problem is the smaller of n 1– 1 and n 2– 1.Īn experiment is conducted to determine whether intensive tutoring (covering a great deal of material in a fixed amount of time) is more effective than paced tutoring (covering less material in the same amount of time). Where and are the means of the two samples, Δ is the hypothesized difference between the population means (0 if testing for equal means), s 1 and s 2are the standard deviations of the two samples, and n 1and n 2are the sizes of the two samples. Online Quizzes for CliffsNotes Statistics QuickReview, 2nd Edition.Quiz: Test for Comparing Two Proportions.Quiz: Test for a Single Population Proportion.Test for a Single Population Proportion.Quiz: Two-Sample t-test for Comparing Two Means.Two Sample t test for Comparing Two Means.Quiz: Two-Sample z-test for Comparing Two Means.Quiz: Introduction to Univariate Inferential Tests.Two-Sample z-test for Comparing Two Means.Quiz: Point Estimates and Confidence Intervals.Point Estimates and Confidence Intervals.Quiz: Normal Approximation to the Binomial.Quiz: Populations, Samples, Parameters, and Statistics.Populations, Samples, Parameters, and Statistics.
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Quiz: Introduction to Numerical Measures.