超市中的鸡蛋按每盒六个出售。一位超市经理希望建立模型描述店内售出盒装鸡蛋中的破损鸡蛋数量。随机抽取 2000 盒，并记录破损鸡蛋的数量。观测频数如下表所示。

(a) 使用数据估计一个鸡蛋破损的概率。答案精确到 4 位有效数字。

决定在 0.5% 显著性水平下进行拟合优度检验，以判断二项分布是否适合这些数据。

观测频数和期望频数如下表所示。

(b) 证明 $a = 162.6$，精确到 1 位小数。

(c) 在 0.5% 显著性水平下进行拟合优度检验，检验二项分布是否是这些数据的合适模型。

(d) 给出一个理由说明为什么在这种情境下二项分布可能不是合适的模型。

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。

由每箱破损数与对应箱数求破损总数，再除以全部鸡蛋数。

There are $1(143)+2(11)+4(1)+6(1)=175$ broken eggs out of $2000(6)=12000$. Thus the estimated probability is

to four significant figures.

用 (a) 未舍入的估计概率计算一箱恰好破损一枚的概率，再乘样本箱数。

Under the fitted binomial model,

Thus $a=162.6$ to one decimal place.

期望频数的总和等于样本总数。用总数减去表中其余各格，即可求缺少的一格。

Using the expected frequencies given in the table,

Thus $a=162.6$ to one decimal place. The small difference from the direct binomial calculation is due to rounding in the table.

将低频的尾格并为 $2$ 枚或以上。参数 $p$ 来自同一数据，所以计算自由度时扣除一个参数。

Test $H_0$ that the fitted binomial distribution is a satisfactory model for the number of broken eggs per box against $H_1$ that it is not. Combine categories $2$ through $6$. The observed counts are then $1844,143,13$, with expected counts approximately $1831.2575,162.6064,6.1361$. They all exceed $5$. The statistic is

There are $3-1-1=1$ degrees of freedom. The upper $0.5\%$ critical value is $7.879$. Since the statistic exceeds this value, we reject $H_0$. There is sufficient evidence at the $0.5\%$ level that the fitted binomial distribution is unsuitable for these box counts.

二项模型要求同一箱内每枚蛋的破损事件独立。考虑整箱受到冲击的情况。

If a box is dropped or crushed, several eggs in it may break together. The breakages within a box may therefore be dependent, contrary to the binomial model.