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Generalizing Tests of Significance

Recall that confidence intervals follow a pattern

 \begin{displaymath}
\mbox{ESTIMATE } \pm \mbox{ CRITICAL VALUE * (SE of ESTIMATE)}
\end{displaymath} (8.1)

where ESTIMATE can either be a $\overline{X}$, $\overline{X}_1-\overline{X}_2$, $\hat{p}$, or $\hat{p_1}-\hat{p_2}$ depending on the type of data and problem at hand.

Test statistics also follow a general pattern 

  % latex2html id marker 6194
\fbox{ \parbox{5.5in}{
\vspace*{1ex}
\begin{equation...
...- HYPOTHESIZED VALUE}}
{\mbox{SE of ESTIMATE}}.
\end{equation}\vspace*{1ex}
} }

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 $\overline{X}$ in any of the previous tests by $\overline{X}_1-\overline{X}_2$ or $\hat{p}$ or $\hat{p_1}-\hat{p_2}$. 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
$\overline{X}$ $S/\sqrt {n}$ t (df=n-1)
$\overline{X}_1-\overline{X}_2$ $\sqrt{ \mbox{SE}_1^2 + \mbox{SE}_2^2}$ where  
  if $\mbox{SE}_1=S_1/\sqrt{n_1}$, $\mbox{SE}_2=S_2/\sqrt{n_2}$ z
  if $\mbox{SE}_1=S_p/\sqrt{n_1}$, $\mbox{SE}_2=S_p/\sqrt{n_2}$ t (df=n1+n2-2 )
$\hat{p}$ $\frac{\sqrt{\hat{p}(1-\hat{p})}}{\sqrt{n}}$ z
$\hat{p_1}-\hat{p_2}$ $\sqrt{ \mbox{SE}_1^2 + \mbox{SE}_2^2}$ where  
  $\;\;\; \mbox{SE}_1=\frac{\sqrt{\hat{p}_1(1-\hat{p}_1)}}{\sqrt{n_1}}$, $\mbox{SE}_2=\frac{\sqrt{\hat{p}_2(1-\hat{p}_2)}}{\sqrt{n_2}}$ 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 $\overline{X}$.

1. Hypotheses: H0: Parameter $\geq$ Hypothesized Value versus H1: Parameter < Hypothesized Value
2. Test Statistic = $\frac{\mbox{ESTIMATE - Hypothesized Value}}
{\mbox{SE of ESTIMATE}}$.
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 $\leq .05$ 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 up previous contents index
Next: Examples Up: Test of Significance involving Previous: The t-test is not

2003-09-08