Applied Statistics and Probability for Engineers

7. Computations: Since , s 0.02456,  0  0.82, and n 15, we have


8. Conclusions: Since , we reject H 0 and conclude at the 0.05 level of
   significance that the mean coefficient of restitution exceeds 0.82.
   Minitab will conduct the one-sample t-test. The output from this software package is in the
   following display:

t 0 2.721.761

t 0 

0.83725 0.82

0.02456 

115

2.72

x0.83725

9-3 TESTS ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE UNKNOWN 303

Notice that Minitab computes both the test statistic T 0 and a 95% lower confidence bound for

the coefficient of restitution. Because the 95% lower confidence bound exceeds 0.82, we

would reject the hypothesis that H 0 : 0.82 and conclude that the alternative hypothesis

is true. Minitab also calculates a P-value for the test statistic T 0. In the next

section we explain how this is done.

9-3.2 P-Value for a t-Test

The P-value for a t-test is just the smallest level of significance at which the null hypothesis

would be rejected. That is, it is the tail area beyond the value of the test statistic t 0 for a one-

sided test or twice this area for a two-sided test. Because the t-table in Appendix Table IV

contains only 10 critical values for each tdistribution, computation of the exact P-value

directly from the table is usually impossible. However, it is easy to find upper and lower

bounds on the P-value from this table.

To illustrate, consider the t-test based on 14 degrees of freedom in Example 9-6. The

relevant critical values from Appendix Table IV are as follows:

Critical Value: 0.258 0.692 1.345 1.761 2.145 2.624 2.977 3.326 3.787 4.140

Tail Area: 0.40 0.25 0.10 0.05 0.025 0.01 0.005 0.0025 0.001 0.0005

Notice that t 0  2.72 in Example 9-6, and that this is between two tabulated values, 2.624 and

2.977. Therefore, the P-value must be between 0.01 and 0.005. These are effectively the up-

per and lower bounds on the P-value.

Example 9-6 is an upper-tailed test. If the test is lower-tailed, just change the sign of t 0 and

proceed as above. Remember that for a two-tailed test the level of significance associated with a

particular critical value is twice the corresponding tail area in the column heading. This consider-

ation must be taken into account when we compute the bound on the P-value. For example, sup-

pose that t 0  2.72 for a two-tailed alternate based on 14 degrees of freedom. The value

(corresponding to  0.02) and (corresponding to  0.01), so the

lower and upper bounds on the P-value would be 0.01P0.02for this case.

t 0 2.624 t 0 2.977

H 1 : 0.82

One-Sample T: COR

Test of mu  0.82 vs mu0.82

Variable N Mean StDev SE Mean

COR 15 0.83725 0.02456 0.00634

Variable 95.0% Lower Bound T P

COR 0.82608 2.72 0.008

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