题目

Problem

A farmer is investigating whether using a new fertiliser will increase the yield of tomato plants. The farmer selects 40 tomato plants at random and gives them the new fertiliser. The crop mass, $x$ kg, of each of these 40 plants is recorded. The farmer selects a further 60 tomato plants at random and gives them a standard fertiliser. The crop mass, $y$ kg, of each of these 60 plants is recorded. The results are summarised as follows.

\begin{align*} &\,\sum x = 168 \qquad \sum x^2 = 720\\ &\,\sum y = 228 \qquad \sum y^2 = 900 \end{align*}

Find a 90% confidence interval for the difference in mean crop mass associated with each type of fertiliser.

[7]

题目中文翻译

一位农民正在研究使用一种新肥料是否会提高番茄植株的产量。农民随机选取 40 株番茄植株，并给它们使用新肥料。记录这 40 株植株中每一株的作物质量 $x$ kg。农民又随机选取另外 60 株番茄植株，并给它们使用标准肥料。记录这 60 株植株中每一株的作物质量 $y$ kg。结果概括如下。

\begin{align*} &\,\sum x = 168 \qquad \sum x^2 = 720\\ &\,\sum y = 228 \qquad \sum y^2 = 900 \end{align*}

求两种肥料对应的总体平均作物质量之差的 90% 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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两个独立样本量都较大。分别估计总体方差，然后用正态近似构造均值差的区间。

答题过程

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The sample means are $168/40=4.2$ and $228/60=3.8$ . The unbiased variance estimates are

\begin{align*} &\,s_X^2=\frac{720-168^2/40}{39}=0.369230\ldots,\\ &\,s_Y^2=\frac{900-228^2/60}{59}=0.569491\ldots. \end{align*}

For the new-minus-standard mean difference, the estimated standard error is

\sqrt{\frac{s_X^2}{40}+\frac{s_Y^2}{60}}=0.136829\ldots.

The $90\%$ confidence interval is $0.4\pm1.645(0.136829\ldots)$ , or $(0.175,0.625)$ kg to three significant figures.

CIE 9231 2025 June Paper 43 Q2

题目

解答

解法一

思路

答题过程