一个机构开设课程，培训学生成为工程师。这些学生以每组 8 人进行教学。该机构负责人声称，平均而言，每组中有 60% 的学生及格。随机抽取 150 个由 8 名学生组成的小组。下表给出观测频数，以及使用适当二项分布得到的一些期望频数。

(a) 求 $p$、$q$ 和 $r$ 的值，答案精确到 3 位小数。

(b) 在 10% 显著性水平下进行拟合优度检验，检验是否有证据拒绝该负责人的说法。

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。

每组的通过人数服从题设二项模型。把相应概率乘以 $150$ 组。

Under the proposed model the expected count at $j$ passes is

For $j=0,5,6$, this gives respectively $p=0.098$, $q=41.804$ and $r=31.353$, to three decimal places.

低端两格合并后仍不足 $5$，因此合并前三格。高端将 $7$ 和 $8$ 合并，再用剩下的格数确定自由度。

Test $H_0$ that $\operatorname{Binomial}(8,0.6)$ fits the pass counts against $H_1$ that it does not fit. Combine $0,1,2$ and combine $7,8$. The pooled observed frequencies are $8,24,45,36,26,11$, and the expected frequencies are approximately $7.471,18.579,34.836,41.804,31.353,15.956$.

Using unrounded expectations, the statistic is

There are $6-1=5$ degrees of freedom, since the pass probability is specified. The upper $10\%$ critical value is $9.236$. Since $7.84326\ldots<9.236$, we fail to reject $H_0$. There is insufficient evidence at the $10\%$ level to reject the director’s proposed pass model.