EXAMPLE 6.16
Significance test of the mean SATM score. In a discussion of SAT Mathematics (SATM) scores, someone comments: “Because only a select minority of California high school students take the test, the scores overestimate the ability of typical high school seniors. I think that if all seniors took the test, the mean score would be no more than 485.” You do not agree with this claim and decide to use the SRS of 500 seniors from Example 6.3 (page 344) to assess the degree of evidence against it. Those 500 seniors had a mean SATM score of . Is this strong enough evidence to conclude that this person’s claim is wrong?
Because the claim states that the mean is “no more than 485,” the alternative hypothesis is one-sided. The hypotheses are
H0: μ = 485
Ha: μ > 485
As we did in the discussion following Example 6.3, we assume that σ = 100. The z statistic is
= 2.24
Because Ha is one-sided on the high side, large values of z count against H0. From Table A, we find that the P-value is
P = P(Z ≥ 2.24) = 1 − 0.9875 = 0.0125
Figure 6.12 illustrates this P-value. A mean score as large as that observed would occur roughly 12 times in 1000 samples if the population mean were 485. This is convincing evidence that the mean SATM score for all California high school seniors is higher than 485. You can confidently tell this person that his or her claim is incorrect.
