题目

Problem

A doctor believes that the diet of her patients and their health are not independent.

She takes a random sample of 200 patients and records whether they are in good health or poor health and whether they have a good diet or a poor diet. The results are summarised in the table below.

	Good health	Poor health
Good diet	86	8
Poor diet	91	15

Stating your hypotheses clearly, test the doctor’s belief using a 5% level of significance. Show your working for your test statistic and state your critical value clearly.

(9)

(Total for Question 2 is 9 marks)

题目中文翻译

一位医生认为，她的病人的饮食与健康状况并不是相互独立的。

她随机抽取 200 名病人，记录他们是健康状况良好还是较差，以及他们饮食是良好还是较差。结果汇总如下表。

	健康状况良好	健康状况较差
饮食良好	86	8
饮食较差	91	15

清楚写出假设，在 5% 显著性水平下检验医生的看法。写出你的检验统计量计算过程，并清楚写出临界值。

解答

解法一

思路

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这是一个 $2\times2$ 列联表的卡方独立性检验。先由行总数、列总数与总人数计算四个期望频数，再求卡方统计量。自由度为 $(2-1)(2-1)=1$ ，最后用 $5\%$ 临界值作出与医生观点相关的结论。

答题过程

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The hypotheses are

\begin{aligned} H_0:&\ \text{diet and health are independent},\\ H_1:&\ \text{diet and health are not independent}. \end{aligned}

The row and column totals are

	Good health	Poor health	Total
Good diet	86	8	94
Poor diet	91	15	106
Total	177	23	200

Under $H_0$ , each expected frequency is calculated using

E=\frac{\text{row total}\times\text{column total}}{200}.

Therefore,

Diet	Health	Expected frequency, $E$
Good	Good	$94(177)/200=83.19$
Good	Poor	$94(23)/200=10.81$
Poor	Good	$106(177)/200=93.81$
Poor	Poor	$106(23)/200=12.19$

The test statistic is

\begin{align*} \chi^2 =&\,\sum\frac{(O-E)^2}{E}\\ =&\,\frac{(86-83.19)^2}{83.19}\\ &\,\hspace{2pt}+\frac{(8-10.81)^2}{10.81}\\ &\,\hspace{4pt}+\frac{(91-93.81)^2}{93.81}\\ &\,\hspace{6pt}+\frac{(15-12.19)^2}{12.19}\\ =&\,1.557\ldots\\ \approx&\,1.56. \end{align*}

The number of degrees of freedom is

\nu=(2-1)(2-1)=1.

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

\chi_1^2(0.05)=3.841.

Since

1.557\ldots<3.841,

the test statistic does not lie in the critical region, so $H_0$ is not rejected. There is insufficient evidence at the $5\%$ significance level of an association between diet and health. Therefore, these data do not support the doctor’s belief.

IAL 2021 June S3 Q2

题目

解答

解法一

思路

答题过程