Rosie 在 150 个花盆中每盆播种 5 粒种子。记录每盆中发芽的种子数量。结果汇总如下表。

Rosie 认为发芽的种子数量服从二项分布 $\mathrm{B}(5, p)$。

(a) 利用 Rosie 的结果说明 $p = 0.42$。

(b) 在 10% 显著性水平下进行拟合优度检验，检验分布 $\mathrm{B}(5, 0.42)$ 是否很好地拟合这些数据。

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。

把每盆发芽数乘以对应盆数，得到发芽总数。再除以全部播下的种子数。

The total number germinating is

There are $150(5)=750$ seeds, so the estimated germination probability is $315/750=0.42$.

根据二项模型算期望频数，并把 $4$ 与 $5$ 合并。估计过一个参数，所以自由度再减 $1$。

Let $H_0$ be that the fitted $\operatorname{Binomial}(5,0.42)$ model describes the germination counts, and $H_1$ that it does not. The expected frequency for $j$ germinations is

The expected count for $5$ is $1.960\ldots<5$, so combine the categories $4$ and $5$. The pooled observed counts are $12,40,43,35,20$, and the expected counts are approximately $9.845,35.647,51.627,37.385,15.496$. Using unrounded values,

There are $5-1-1=3$ degrees of freedom, because $p$ was estimated from the data in (a). The upper $10\%$ critical value is $6.251$. As $3.90567\ldots<6.251$, we fail to reject $H_0$. There is insufficient evidence at the $10\%$ level that the fitted binomial distribution fails to describe the germination counts.