The production manager of a badminton racket company wishes to check that the mean mass of the badminton rackets produced is 800g as stated in the design specifications. The population standard deviation of the badminton rackets is 20g. He carries out a hypothesis test on a large sample of $n$ badminton rackets. The sample mean mass is 807g and the null hypothesis is rejected at the 2% level of significance.

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.