从服从正态分布的随机变量 $X$ 中随机抽取的 $9$ 个观测值如下。

$X$ 的总体均值为 $\mu$。

在 $2.5\%$ 的显著性水平下，检验原假设 $\mu=20$，备择假设为 $\mu>20$。

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。

总体正态而方差未知，用单样本 $t$ 检验。备择假设为均值大于 $20$，所以取右尾。

The sample gives $\sum x=192.2$ and $\sum x^2=4127.64$. Thus

The test statistic for $H_0\colon\mu=20$ against $H_1\colon\mu>20$ is

The upper $2.5\%$ critical value with $8$ degrees of freedom is $2.306$. Since the statistic is larger, we reject $H_0$. There is sufficient evidence at the $2.5\%$ level that $\mu>20$.