一组 10 名学生被要求估计某个锐角三角形中角 $\theta^\circ$ 的大小。他们的估计值（单位为度）如下。

$84 \quad 85 \quad 77 \quad 85 \quad 84 \quad 87 \quad 86 \quad 88 \quad 83 \quad 85$

(a) 说明你作出的任何假设，并计算 $\theta$ 的 95% 置信区间。

(b) 给出一个理由，说明为什么(a)中作出的假设在本情境下可能不合适。

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。

把真实角度视为估计值总体的均值。样本量为 $10$，总体方差未知，故用 $t$ 区间。

Assume that the estimates are an independent random sample from a normal population with mean $\theta$ and unknown variance. There are $10$ estimates, with $\sum x=844$ and $\sum x^2=71314$. The sample mean is $84.4$ and the unbiased variance estimate is

For a two-sided $95\%$ interval, the critical value with $9$ degrees of freedom is $2.262$. The interval is

Thus the interval for $\theta$ is $(82.3,86.5)$ degrees to three significant figures.

结合数据收集方式检查随机抽样假设。题目只说明选了一组学生。

The children are described as one group, with no indication that they were randomly selected. Their estimates may therefore fail to represent the population assumed in (a).