Jade 是一所体育学院的游泳教练。她声称，经过一次强化训练课程后，学生游 50 米所用的平均时间减少了超过 1 秒。她随机抽取 10 名学生。训练课程前后所用的时间如下表，单位为秒。

(a) 在 10% 显著性水平下，检验 Jade 的声称是否合理。

(b) 说明使该检验有效所需要的一个假设。

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。

取每位学生训练前减训练后的时间。检验平均减少是否超过 $1$ 秒，所以分子是样本平均减少量减 $1$。

Let $\mu_D$ be the population mean reduction in swim time. Test $H_0\colon\mu_D=1$ against $H_1\colon\mu_D>1$. The paired reductions are $4.1,1.1,-0.4,0.2,3.9,2.6,-0.6,3.5,0.1,0$, giving

The test statistic is

With $9$ degrees of freedom, the upper $10\%$ critical value is $1.383$. Since the statistic is smaller, we fail to reject $H_0$. There is insufficient evidence at the $10\%$ level that the population mean swimming time was reduced by more than one second.

配对 $t$ 检验的正态假设针对训练前后时间之差。

The population distribution of the paired differences in swimming times is assumed to be normal.