Solution to Exercise 8.31.

  Let $\mu$ denote the population mean time in sec. an inspector's eyes need to become used to the reduced amount of light. The prior belief is that this mean time was at least 7. We want to see if the data contradict this, so we set up the testing problem

$$H_0 : \mu \ge 7 , \quad vs. \quad H_1 : \mu < 7 .$$

The exercise says to use a t-test with level of significance .1. We will reject if the t-statistic satisfies

$$t \; \le \; -t_{.1,n-1} \; = \; -t_{.1,8} \; = \; -1.397 .$$

The observed value of the t-statistic is

$$t \; = \; \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \; = \; \frac{6.32 - 7}{1.65/\sqrt{9}} \; = \; -1.236364 .$$

As the observed value does not fall inthe rejection region, we do not reject the null hypothesis.