在 90 天内，每天记录某一路段发生故障的次数。有人提出，故障次数服从均值为 3.5 的泊松分布。数据汇总在下表中，同时给出由所提出的泊松分布得到的一些期望频数。

(a) 完成表格。

(b) 在 10% 显著性水平下进行拟合优度检验，判断 $\operatorname{Po}(3.5)$ 是否很好地拟合该数据。

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。

把泊松分布在 $3$ 和 $6$ 的概率分别乘以 $90$，得到缺失的期望频数。

For $N\sim\operatorname{Poisson}(3.5)$, the expected count at $j$ is $90\mathrm e^{-3.5}3.5^j/j!$. At $j=3$ and $j=6$, this gives $19.421$ and $6.939$ respectively to three decimal places.

两端分别合并小期望频数的格。均值是给定值，合并后只因总频数约束扣除一个自由度。

Test $H_0$ that $\operatorname{Poisson}(3.5)$ fits the daily breakdown counts against $H_1$ that it does not fit. Combine $0$ with $1$, giving observed and expected frequencies $5$ and $12.22994\ldots$. Combine $7$ with $8$ or more, giving $9$ and $5.87592\ldots$. The resulting seven expected frequencies all exceed $5$.

The statistic calculated with unrounded expectations is

There are $7-1=6$ degrees of freedom, giving an upper $10\%$ critical value of $10.645$. Since the statistic is smaller, we fail to reject $H_0$. There is insufficient evidence at the $10\%$ level that the proposed Poisson distribution fails to fit the daily counts.