两家公司 P 和 Q 生产某种油漆刷。一名独立检查员将所生产油漆刷的质量评为差、合格或好。他从每家公司随机抽取油漆刷样本。检查员的评分概括如下表。

(a) 在 5% 显著性水平下，检验油漆刷质量是否与公司相互独立。

(b) 比较两家公司生产的油漆刷质量。

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。

先算两个样本的行总数，再根据独立假设求每格期望频数。自由度由表格行列数决定。

Test $H_0$ that brush quality is independent of company against $H_1$ that it is associated with company. The row totals are $125,75$, the column totals are $40,65,95$, and the overall total is $200$. Using row total times column total divided by $200$, the expected counts are

They all exceed $5$. The statistic is

There are $(2-1)(3-1)=2$ degrees of freedom. The upper $5\%$ critical value is $5.991$. Since $6.55762\ldots>5.991$, we reject $H_0$. There is sufficient evidence at the $5\%$ level that brush quality is associated with company.

两家公司样本量不同，比较各等级的比例。也可结合 (a) 看低质量刷子的频数偏向哪一家公司。

Company P’s brushes tend to be of better quality. In the samples, $18/125=14.4\%$ of P’s brushes were poor compared with $22/75\approx29.3\%$ of Q’s, and $64/125=51.2\%$ of P’s brushes were good compared with $31/75\approx41.3\%$ of Q’s.