11_2sam_hypotst2

11_2sam_hypotst2 - Two-sample hypothesis testing II 9.07...

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Two-sample hypothesis testing, II 9.07 3/16/2004
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Small sample tests for the difference between two independent means • For two-sample tests of the difference in mean, things get a little confusing, here, because there are several cases. • Case 1: The sample size is small, and the standard deviations of the populations are equal . • Case 2: The sample size is small, and the standard deviations of the populations are not equal .
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Inhomogeneity of variance • Last time, we talked about Case 1, which assumed that the variances for sample 1 and sample 2 were equal. • Sometimes, either theoretically, or from the data, it may be clear that this is not a good assumption. • Note: the equal-variance t-test is actually pretty robust to reasonable differences in the variances, if the sample sizes, n 1 and n 2 are (nearly) equal. When in doubt about the variances of your two samples, use samples of (nearly) the same size.
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Case 2: Variances not equal • Sometimes, however, it either isn’t possible to have an equal number in each sample, or the variances are very different. • In which case, we move on to Case 2, the t- test for difference in means when the variances are not equal.
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Case 2: Variances not equal • Basically, one can deal with unequal variances by making a correction in the value for degrees of freedom. • Equal variances: d.f. = n 1 + n 2 –2 • Unequal variances: 1 ) / ( 1 ) / ( ) / / ( d.f. 2 2 2 2 2 1 2 1 2 1 2 2 2 2 1 2 1 + + = n n n n n n σ
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Note on corrected degrees of freedom • There are several equations out there for correcting the number of degrees of freedom. • This equation is a bit on the conservative side – it will lead to an overestimate of the p-value. • An even easier conservative correction: d.f. = min(n 1 -1, n 2 -1) • You will NOT be required to memorize any of these equations for an exam. • Use the one on the previous slide for your homework.
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Case 2: Variances not equal • Once we make this correction, we proceed as with a usual t-test, using the equation for SE from last time, for unequal variances. • SE(difference in means) = sqrt( σ 1 2 /n 1 + σ 2 2 /n 2 ) •t obt = (observed – expected)/SE • Compare with t crit from a t-table, for d.f. degrees of freedom, from the previous slide.
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Example • The math test scores of 16 students from one high school showed a mean of 107, with a standard deviation of 10. • 11 students from another high school had a mean score of 98, and a standard deviation of 15. • Is there a significant difference between the scores for the two groups, at the α =0.05 level?
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Set up the null and alternative hypotheses, and find t crit •H 0 : µ 1 µ 2 = 0 a : µ 1 µ 2 0 •s 1 2 = 10 2 ; s 2 2 = 15 2 ; n 1 = 16; n 2 = 11 • d.f. = (100/16+225/11) 2 / [(100/16) 2 /15 + (225/11) 2 /10] 16 1 ) / ( 1 ) / ( ) / / ( d.f. 2 2 2 2 2 1 2 1 2 1 2 2 2 2 1 2 1 + + = n n n n n n σ
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Determining t crit • d.f. = 16, α =0.05 • Two-tailed test (H a : µ 1 µ 2 0) •t crit = 2.12
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Compute t obt , and compare to t crit •m 1 = 107; m 2 = 98, s 1 = 10; s 2 = 15; n 1 = 16; n 2 = 11 •t obt = (observed – expected)/SE • Observed difference in means = 107-98 = 9 •S E = s q r t ( σ 1 2 /n 1 + σ 2 2 /n 2 )
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This note was uploaded on 11/11/2011 for the course BIO 9.07 taught by Professor Ruthrosenholtz during the Spring '04 term at MIT.

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11_2sam_hypotst2 - Two-sample hypothesis testing II 9.07...

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