Trueleaf 茶每包的平均质量应为 $500$ 克。一名检验员想检验这一数值是否正确。他随机抽取 $60$ 包，称量并记录每包的质量 $x$（以克为单位）。结果汇总如下。

在 $5\%$ 的显著性水平下检验总体平均质量是否为 $500$ 克。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

先从汇总数据估计总体均值和无偏方差；样本量较大，可对样本均值用正态近似。检验的是均值是否正确，采用双侧备择。

The unbiased estimates are

and

Let $\mu$ be the population mean mass of the tea packets. Test

For the large random sample, the Central Limit Theorem gives an approximately normal sample mean. Using $s^2$ to estimate the population variance,

At the $5\%$ significance level, the two-tailed critical values are $\pm1.96$. Since $|z|<1.96$, do not reject $H_0$.

There is insufficient evidence at the $5\%$ level that the population mean mass differs from $500\ \text{g}$.