从随机变量 $X$ 的取值中随机抽取 $200$ 个，得到下列结果。

(a) 求 $X$ 的总体均值的一个 $95\%$ 置信区间。

(b) 指出你在 (a) 的解答中哪一步用到了中心极限定理。

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。

平移不改变方差，均值则需要加回二；无偏方差用分母一百九十九。

Put $Y=X-2$. Its sample mean is $60/200=0.3$, so $\overline x=2.3$. Since translation does not change variance, the unbiased variance estimate for $X$ is

By the Central Limit Theorem, the sample mean is approximately normal for the large random sample. An approximate $95\%$ confidence interval is

Hence the interval is $\boxed{(2.29,\ 2.31)}$ to $3$ significant figures.

明确指出使用正态临界值的步骤，不只说总体未给正态。

The Central Limit Theorem justifies the normal approximation for the sample mean, and hence the use of the normal critical value $\boxed{z=1.96}$ in the confidence-interval formula.