题目

Problem

Jasmine is researching the heights of pine trees in forests in two regions $A$ and $B$ . She chooses a random sample of 50 pine trees in region $A$ and records their heights, $x$ m. She also chooses a random sample of 60 pine trees in region $B$ and records their heights, $y$ m. Her results are summarised as follows.

\begin{gathered} \sum x = 1625,\qquad \sum x^2 = 53200,\\ \sum y = 1854,\qquad \sum y^2 = 57900. \end{gathered}

Find a 95% confidence interval for the difference between the population mean heights of pine trees in regions $A$ and $B$ .

[7]

题目中文翻译

Jasmine 正在研究两个地区 $A$ 和 $B$ 的森林中松树的高度。她在地区 $A$ 随机抽取 50 棵松树，并记录它们的高度，记为 $x$ m。她还在地区 $B$ 随机抽取 60 棵松树，并记录它们的高度，记为 $y$ m。结果汇总如下。

\begin{gathered} \sum x = 1625,\qquad \sum x^2 = 53200,\\ \sum y = 1854,\qquad \sum y^2 = 57900. \end{gathered}

求地区 $A$ 和 $B$ 松树总体平均高度之差的 95% 置信区间。

(7)

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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分别用两组的汇总数据求均值和无偏方差。用大样本正态近似构造均值差的区间。

答题过程

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The sample means are $\bar x=1625/50=32.5$ and $\bar y=1854/60=30.9$ . The unbiased variance estimates are

\begin{align*} &\,s_A^2=\frac{53200-1625^2/50}{49}=7.90816\ldots,\\ &\,s_B^2=\frac{57900-1854^2/60}{59}=10.36271\ldots. \end{align*}

The estimated variance of the difference of independent sample means is $s_A^2/50+s_B^2/60=0.33087\ldots$ . The $95\%$ confidence interval for $\mu_A-\mu_B$ is

\begin{align*} &\,1.6\pm1.96\sqrt{s_A^2/50+s_B^2/60}\\ =&\,(0.47257\ldots,2.72742\ldots). \end{align*}

Thus the interval is $(0.473,2.73)$ m to three significant figures.

CIE 9231 2022 Nov Paper 43 Q1

题目

解答

解法一

思路

答题过程