科学家正在研究运动对 LDL 血液胆固醇水平的影响。在三个月期间，一大组人每天运动 20 分钟。从这些人中随机选取 10 人，分别在三个月开始和结束时测量 LDL 血液胆固醇水平。结果用合适单位表示如下。

(a) 在 2.5% 显著性水平下，检验是否有证据表明三个月后总体平均 LDL 血液胆固醇水平降低了超过 2 个单位。

(b) 写出你在(a)中所作的任何假设。

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。

对同一个人的前后测量取差。题目要求平均减少超过 $2$，所以零假设的差值均值用 $2$。

Let $D$ be the beginning level minus the end level for one person. Test $H_0\colon\mu_D=2$ against $H_1\colon\mu_D>2$. The differences are $8,8,15,-2,-3,20,-3,0,5,4$. Thus $\sum d=52$, $\sum d^2=816$, and

The paired $t$ statistic is $(5.2-2)/\sqrt{s_D^2/10}=1.29967\ldots$. With $9$ degrees of freedom the upper $2.5\%$ critical value is $2.262$. Since the statistic is smaller, we fail to reject $H_0$. There is insufficient evidence at the $2.5\%$ level that the population mean LDL level decreased by more than $2$ units.

配对检验的正态假设针对同一个人前后测量的差值。

The population distribution of the paired differences in LDL blood cholesterol level is assumed to be normal.