题目

Problem

The lengths of the leaves of a particular type of tree are normally distributed with mean $\mu$ cm. The lengths, $x$ cm, of a random sample of 12 leaves of this type are recorded. The results are summarised as follows.

\sum x = 91.2 \qquad \sum x^2 = 695.8

Find a 95% confidence interval for $\mu$ .

[4]

题目中文翻译

某种树的叶片长度服从正态分布，均值为 $\mu$ cm。随机抽取这种树的 12 片叶子，记录其长度 $x$ cm。结果汇总如下。

\sum x = 91.2 \qquad \sum x^2 = 695.8

求 $\mu$ 的 95% 置信区间。

(4)

原卷通用作答要求

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。

解答

解法一

思路

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由汇总值求样本均值和无偏方差。总体正态、方差未知，使用小样本 $t$ 区间。

答题过程

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The sample mean is $91.2/12=7.6$ , and the unbiased variance estimate is

s^2=\frac{695.8-91.2^2/12}{11}=\frac{67}{275}.

The two-sided $95\%$ critical value for $11$ degrees of freedom is $2.201$ . Hence the interval is

\begin{align*} &\,7.6\pm2.201\sqrt{\frac{67/275}{12}}\\ =&\,(7.28638\ldots,7.91361\ldots). \end{align*}

The $95\%$ confidence interval is $(7.29,7.91)$ cm.

CIE 9231 2023 June Paper 41 Q1

题目

解答

解法一

思路

答题过程