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## Generalizing Tests of Significance

Recall that confidence intervals follow a pattern

 (8.1)

where ESTIMATE can either be a , , , or depending on the type of data and problem at hand.

Test statistics also follow a general pattern

The ESTIMATE's that may be used in tests are listed in Table  8.1 along with their SE's and the correct curve from which to read P-values. In effect, we may replace in any of the previous tests by or or . As long as we remember to use the corresponding SE and distribution curve, the test procedure is the same as before.

Table 8.1: Summary of Test Statistics

 ESTIMATE SE OF ESTIMATE P-value CURVE t (df=n-1) where if , z if , t (df=n1+n2-2 ) z where , z

We describe once more the procedure for a lower-tailed test of hypotheses; but this time we use the generic term ESTIMATE instead of the sample mean .

1. Hypotheses: H0: Parameter Hypothesized Value versus H1: Parameter < Hypothesized Value
2. Test Statistic = .
3. P-value: Presuming H0, the likelihood of chance producing a test statistic as low as or lower than observed in Step 2 is the area to its left under the P-value curve.
4. Conclusion: If P-value the observed difference in Step 2 is statistically significant. Therefore, Reject H0 and conclude that the sample provides enough evidence to prove H1.

For upper-tailed tests, we reject H0 if we observe values of the estimate larger than the hypothesized value. Therefore the P-value is the area to the right of the observed test statistic in Step 2.

For two-tailed tests, the P-value is the area both left and right of the observed test statistic in Step 2.

Next: Examples Up: Test of Significance involving Previous: The t-test is not

2003-09-08