一家公司声称，其工人在施工现场的平均工作时间每天不超过 8 小时。从 150 名工人中随机抽样，记录每名工人的工作时间为 $x$ 小时，汇总数据如下： $\sum x = 1305,\ \sum x^2 = 12751.$ 在 1% 的显著性水平下，检验该公司的声称是否成立。

1. State the hypotheses.

Let $\mu$ be the population mean time clocked by workers, in hours.
$H_0: \mu = 8$
$H_1: \mu > 8$
Significance level: $\alpha = 0.01$

2. Calculate the sample mean and unbiased estimate of the population variance.

Sample mean, $\bar{x} = \frac{1305}{150} = 8.7$

Unbiased estimate of population variance, $s^2 = \frac{1}{n-1}\left(\sum x^2 - \frac{(\sum x)^2}{n}\right)$
$s^2 = \frac{1}{149}\left(12751 - \frac{1305^2}{150}\right) = \frac{1397.5}{149} = \frac{2795}{298} \approx 9.3792$

3. Determine the test statistic and compute the $p$-value.

Under $H_0$, since the sample size $n = 150$ is large, by the Central Limit Theorem, $\bar{X}$ is approximately normally distributed.
Test statistic: $Z = \frac{\bar{X} - 8}{s / \sqrt{150}} \approx N(0, 1)$

Using a graphing calculator (1-tail $Z$-test):
$z \approx 2.799$
$p\text{-value} \approx 0.00256$ (to 3 s.f.)

4. State the conclusion in the context of the problem.

Since $p\text{-value} = 0.00256 < 0.01$, we reject $H_0$.
There is sufficient evidence at the $1\%$ level of significance to conclude that the population mean time clocked is more than $8$ hours. Therefore, the firm's claim is not valid.