22.1. The hypotheses. The null hypothesis says that the coin is balanced (p = 0.5). We do not suspect a bias in a specific direction before we see the data, so the alternative hypothesis is just “the coin is not balanced.” The two hypotheses are

H₀ : p = 0.5

H_a : p ≠ 0.5

The sampling distribution. If the null hypothesis is true, the sample proportion of heads has approximately the Normal distribution with

mean = p = 0.5

= 0.0707

22.2. The data. The sample proportion is . The standard score for this outcome is

= −1.13

The P-value. To use Table B, round the standard score to −1.1. This is the 13.57 percentile of a Normal distribution. So the area to the left of −1.1 is 0.1357. The area to the left of −1.1 and to the right of 1.1 is double this, or 0.2714. This is our approximate P-value.

Conclusion. The large P-value gives no reason to think that the true proportion of heads differs from 0.5.

22.3. The hypotheses. The null hypothesis is “no difference” from the population mean of 100. The alternative is two-sided because we did not have a particular direction in mind before examining the data. So, the hypotheses about the unknown mean μ of the middle-school girls in the district are

633

The sampling distribution. If the null hypothesis is true, the sample mean has approximately the Normal distribution with mean μ = 100 and standard deviation

The data. The sample mean is . The standard score for this outcome is

= 2.26

The P-value. To use Table B, round the standard score to 2.3. This is the 98.93 percentile of a normal distribution. So the area to the right of 2.3 is 1 − 0.9893 = 0.0107. The area to the left of −2.3 and to the right of 2.3 is double this, or 0.0214. This is our approximate P-value.

Conclusion. The small P-value gives some reason to think that the mean IQ score for middle-school girls in this district differs from 100.