一家羽毛球拍公司的生产经理希望检查所生产的羽毛球拍的平均质量是否如设计规格中所述为 800g。羽毛球拍的总体标准差为 20g。他对由 $n$ 个羽毛球拍组成的大样本进行了假设检验。样本平均质量为 807g，且原假设在 2% 的显著性水平下被拒绝。

1. State the null and alternative hypotheses.

Let $\mu$ be the population mean mass of the badminton rackets in grams.
$H_0: \mu = 800$
$H_1: \mu \neq 800$

2. State the test statistic and its distribution.

Since $n$ is large, by the Central Limit Theorem, the sample mean $\bar{X}$ is approximately normally distributed under $H_0$.
Test statistic: $Z = \frac{\bar{X} - \mu_0}{\sigma/\sqrt{n}} = \frac{\bar{X} - 800}{20/\sqrt{n}} \sim N(0, 1)$ approximately.

3. Determine the critical value for rejection at the 2% level of significance.

For a 2-tailed test at the $2\%$ level of significance, we find the critical value $z$ where the upper tail probability is $0.01$.
$P(Z > z) = 0.01 \implies z \approx 2.3263$

4. Set up the rejection inequality using the given sample mean.

The null hypothesis is rejected and $\bar{x} = 807 > 800$, so the calculated test statistic must fall in the upper rejection region:
$Z_{calc} \ge 2.3263$
$\frac{807 - 800}{20/\sqrt{n}} \ge 2.3263$

5. Solve the inequality algebraically for $n$.

$\frac{7\sqrt{n}}{20} \ge 2.3263$
$\sqrt{n} \ge \frac{20 \times 2.3263}{7}$
$\sqrt{n} \ge 6.6466$
$n \ge 44.177$

6. State the least integer value of $n$.

Since $n$ must be an integer, the least value of $n$ is $45$.

1. State the conclusion and justify it using the Central Limit Theorem.

No. Since the sample size $n$ is large, by the Central Limit Theorem, the sample mean $\bar{X}$ is approximately normally distributed regardless of the population distribution of the masses.