Class_13 - The t-test Independent and Paired(Dependent...

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The t-test: Independent and The t-test: Independent and Paired (Dependent) Samples Paired (Dependent) Samples
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Recapitulation Recapitulation 1. Still dealing with random samples . 2. However, they are partitioned into two subsamples . 3. Interest is in whether the means of some variable differ significantly between the two subsamples. 4. For large samples, the sampling distribution of mean differences is normally shaped . 5. For smaller samples, the sampling distribution takes the shape of one of the Student’s t distributions .
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6. In either case, the significance tests involve a test of the null hypothesis (H 0 ) that in general (i.e., in the universe) the means do NOT differ. Symbolically, H 0 : μ 2 - μ 1 = 0.00 7. The alternate hypothesis (H 1 ) can either be nondirectional H 1 : μ 2 - μ 1 0.0 or directional , either H 1 : μ 2 - μ 1 > 0.0 or H 1 : μ 2 - μ 1 < 0.0
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Reject a null hypothesis (H 0 ) when either: 1. the value of the statistical test ( χ 2 , z, t , or F) exceeds the critical value at the chosen α -level; or, 2. the p-value for the statistical test is smaller than the chosen value of α .
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An Example of a T-Test PPD 404 The TTEST Procedure Statistics Lower CL Upper CL Lower CL Upper CL Variable Class N Mean Mean Mean Std Dev Std Dev Std Dev Std Err MANUFPCT Newer 38 22.928 26.526 30.125 8.9262 10.949 14.165 1.7761 MANUFPCT Older 25 20.1 23.92 27.74 7.2268 9.2553 12.875 1.8511 MANUFPCT Diff (1-2) -2.706 2.6063 7.9183 8.7659 10.316 12.536 2.6565 T-Tests Variable Method Variances DF t Value Pr > |t| MANUFPCT Pooled Equal 61 0.98 0.3304 MANUFPCT Satterthwaite Unequal 57.1 1.02 0.3139 Equality of Variances Variable Method Num DF Den DF F Value Pr > F MANUFPCT Folded F 37 24 1.40 0.3897
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An Example of a T-Test PPD 404 The TTEST Procedure Statistics Lower CL Upper CL Lower CL Upper CL Variable Class N Mean Mean Mean Std Dev Std Dev Std Dev Std Err MANUFPCT Newer 38 22.928 26.526 30.125 8.9262 10.949 14.165 1.7761 MANUFPCT Older 25 20.1 23.92 27.74 7.2268 9.2553 12.875 1.8511 MANUFPCT Diff (1-2) -2.706 2.6063 7.9183 8.7659 10.316 12.536 2.6565 T-Tests Variable Method Variances DF t Value Pr > |t| MANUFPCT Pooled Equal 61 0.98 0.3304 MANUFPCT Satterthwaite Unequal 57.1 1.02 0.3139 Equality of Variances (null hypoth. Variances are equal) Variable Method Num DF Den DF F Value Pr > F MANUFPCT Folded F 37 24 1.40 0.0389
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The tests that we have performed thus all assumed that the “subuniverse” variances were equal. Remember, parameters (e.g., universe variances) are almost always unknown . Therefore, we let subsample variances be proxies for the unknown universe variances. If the two variances in the universe are greatly different —as indicated by statistically significant differences between the subsample variances —, then a different method must be used to estimate the standard error of the difference.
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A hypothesis test exists to decide whether the two variances are significantly different. In this F' test , a ratio
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