一名工程师正在比较由两台机器 A 和 B 制造的钢棒的抗拉强度。该工程师从机器 A 随机选取 8 根钢棒，从机器 B 随机选取 6 根钢棒。以适当单位表示的抗拉强度如下表所示。

你应假设这两个分布均为正态分布，并且具有相同的总体方差。

在 5% 显著性水平下使用 t 检验，检验来自两台机器的钢棒平均抗拉强度是否存在差异。

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。

两个样本独立，题目假设总体方差相等。先合并组内平方离差，再用双侧两样本 $t$ 检验。

Test $H_0\colon\mu_A=\mu_B$ against $H_1\colon\mu_A\ne\mu_B$, where $\mu_A$ and $\mu_B$ are the population mean tensile strengths. The sample means are $3264/8=408$ and $2406/6=401$. The two within-sample sums of squares are

The pooled variance is $s_p^2=(136+158)/(8+6-2)=24.5$. The statistic is

For $12$ degrees of freedom the two-sided $5\%$ critical values are $\pm2.179$. Since $|t|>2.179$, we reject $H_0$. There is sufficient evidence at the $5\%$ level that the two machines produce rods with different mean tensile strengths.