题目

Problem

A scientist is investigating the lengths of the leaves of a certain type of plant. The scientist assumes that the lengths of the leaves of this type of plant are normally distributed. He measures the lengths, $x$ cm, of the leaves of a random sample of 8 plants of this type. His results are as follows.

\begin{gathered} 3.5 \qquad 4.2 \qquad 3.8 \qquad 5.2 \\ 2.9 \qquad 3.7 \qquad 4.1 \qquad 3.2 \end{gathered}

Find a 90% confidence interval for the population mean length of leaves of this type of plant.

[4]

题目中文翻译

一名科学家正在研究某种植物叶子的长度。该科学家假设这种植物叶子的长度服从正态分布。他测量了这种植物的 8 株随机样本的叶子长度，记为 $x$ cm。结果如下。

\begin{gathered} 3.5 \qquad 4.2 \qquad 3.8 \qquad 5.2 \\ 2.9 \qquad 3.7 \qquad 4.1 \qquad 3.2 \end{gathered}

求这种植物叶子总体平均长度的 90% 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

展开

小样本来自题设的正态总体，方差未知。用样本方差和自由度为 $7$ 的 $t$ 临界值构造区间。

答题过程

展开

There are $8$ measurements, with $\sum x=30.6$ and $\sum x^2=120.52$ . Thus $\bar x=30.6/8=3.825$ and

s^2=\frac{120.52-30.6^2/8}{7}=0.496428\ldots.

The two-sided $90\%$ interval uses $t_{0.95,7}=1.895$ . It is

\begin{align*} &\,3.825\pm1.895\sqrt{\frac{s^2}{8}}\\ =&\,(3.35294\ldots,4.29705\ldots). \end{align*}

Thus the confidence interval is $(3.35,4.30)$ cm.

CIE 9231 2024 Nov Paper 42 Q1

题目

解答

解法一

思路

答题过程