一位科学家正在调查某地区某种蝴蝶的数量。他声称，每平方米发现的这种蝴蝶的数量可以用均值为 $2.5$ 的泊松分布建模。他随机抽取 $120$ 块面积均为一平方米的区域，数出每块区域内这种蝴蝶的数量。下表列出了观测频数，以及利用这位科学家的泊松分布计算出的部分期望频数。

(a) 求 $p$ 和 $q$ 的值，精确到小数点后 $2$ 位。

(b) 在 $10\%$ 的显著性水平下进行拟合优度检验，检验这位科学家的声称。

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。

恰好四只的期望频数用泊松概率公式。至少七只的期望频数用补事件。

For $N\sim\operatorname{Poisson}(2.5)$, the expected counts are

and

Thus $p=16.03$ and $q=1.70$ to two decimal places.

合并 $6$ 与至少 $7$ 两格，使合并格的期望频数大于 $5$。均值已给定，所以自由度为格数减一。

Test $H_0$ that $\operatorname{Poisson}(2.5)$ fits the butterfly counts per square metre against $H_1$ that it does not. Combining the last two categories gives observed count $1$ and expected count $5.04252\ldots$. There are seven categories, all with expected counts above $5$. Using unrounded expected counts,

With $6$ degrees of freedom the upper $10\%$ critical value is $10.645$. The statistic is smaller, so we fail to reject $H_0$. There is insufficient evidence at the $10\%$ level that the scientist’s Poisson model fails to fit the counts.