题目

Problem

Andy has some apple trees. Over many years she has graded each apple from her trees as A, B, C, D or E according to the quality of the apple, with A being the highest quality and E being the lowest quality.

She knows that the proportion of apples in each grade produced by her trees is as follows.

Grade	A	B	C	D	E
Proportion	4%	28%	52%	10%	6%

Raj advises Andy to add potassium to the soil around her apple trees.

Andy believes that adding potassium will not affect the distribution of grades for the quality of the apples.

To test her belief Andy adds potassium to the soil around her trees. The following year she counts the number of apples in each grade. The number of apples in each grade is shown in the table below.

Grade	A	B	C	D	E
Frequency	9	71	136	21	3

Test Andy’s belief using a 5% level of significance. Show your working clearly, stating your hypotheses, expected frequencies and degrees of freedom.

(8)

(Total 8 marks)

题目中文翻译

Andy 有一些苹果树。多年来，她会根据苹果质量把树上每个苹果分成 A、B、C、D 或 E 级，其中 A 级最高，E 级最低。

她知道她的苹果树所生产苹果在各等级中的比例如下。

等级	A	B	C	D	E
比例	4%	28%	52%	10%	6%

Raj 建议 Andy 在苹果树周围的土壤中加入钾。

Andy 认为加入钾不会影响苹果质量等级的分布。

为检验她的想法，Andy 在苹果树周围加入钾。第二年她统计了每个等级的苹果数量。各等级数量如下表。

等级	A	B	C	D	E
频数	9	71	136	21	3

在 5% 显著性水平下检验 Andy 的想法。清楚写出你的工作过程，说明假设、期望频数和自由度。

解答

解法一

思路

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这是检验实际等级频数是否仍符合原有比例的卡方拟合优度检验。先用总数 $240$ 乘各等级的原有比例求期望频数，再计算检验统计量。五个比例均已给定，没有从样本估计参数，因此自由度为 $5-1=4$ 。

答题过程

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The hypotheses are

\begin{aligned} H_0:&\ \text{adding potassium has no effect on the distribution}\\ &\ \text{of the quality grades of the apples},\\ H_1:&\ \text{adding potassium has an effect on the distribution}\\ &\ \text{of the quality grades of the apples}. \end{aligned}

The total number of apples is

9+71+136+21+3=240.

The expected frequencies under $H_0$ are therefore

Grade	Expected frequency, $E$
A	$0.04(240)=9.6$
B	$0.28(240)=67.2$
C	$0.52(240)=124.8$
D	$0.10(240)=24.0$
E	$0.06(240)=14.4$

The test statistic is

\begin{align*} \chi^2 =&\,\sum\frac{(O-E)^2}{E}\\ =&\,\frac{(9-9.6)^2}{9.6}\\ &\,\hspace{2pt}+\frac{(71-67.2)^2}{67.2}\\ &\,\hspace{4pt}+\frac{(136-124.8)^2}{124.8}\\ &\,\hspace{6pt}+\frac{(21-24.0)^2}{24.0}\\ &\,\hspace{8pt}+\frac{(3-14.4)^2}{14.4}\\ =&\,10.6575\ldots\\ \approx&\,10.7. \end{align*}

There are $5$ classes and no parameters have been estimated from the sample, so

\nu=5-1=4.

At the $5\%$ significance level, the critical value is

\chi_4^2(0.05)=9.488.

Since

10.6575\ldots>9.488,

the test statistic lies in the critical region, so $H_0$ is rejected. There is sufficient evidence at the $5\%$ significance level to suggest that adding potassium affects the distribution of the quality grades of the apples. Therefore, the data do not support Andy’s belief.

IAL 2021 Oct S3 Q2

题目

解答

解法一

思路

答题过程