Margaret 开设了一门新的复习课程，帮助学生提高数学能力。每位学生在参加课程前做一次数学测试，参加课程后再做一次难度相近的测试。Margaret 声称，该课程将使学生的分数平均提高超过 $5$ 分。随机抽取 $8$ 名学生，他们在复习课程前后的测试分数如下表所示。

(a) 在 $10\%$ 的显著性水平下，使用 $t$ 检验考察 Margaret 的声称。

(b) 写出使 5(a) 中的检验有效所必需的一项假设。

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。

每位学生的前后测验形成一对。取后测减前测，并检验总体均值是否超过 $5$。

Let $\mu_D$ be the population mean after-minus-before mark difference. Test $H_0\colon\mu_D=5$ against $H_1\colon\mu_D>5$. The differences are $6,24,12,8,5,26,13,2$. Their sum is $96$ and their sum of squares is $1694$, so

The statistic is

With $7$ degrees of freedom, the upper $10\%$ critical value is $1.415$. Since the statistic exceeds it, we reject $H_0$. There is sufficient evidence at the $10\%$ level that the revision course increases population mean scores by more than $5$ marks.

配对检验要求同一学生的前后成绩差来自正态总体。

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