题目

Problem

A research student is investigating the number of children who are girls in families with 4 children.

The table below shows her results for 200 such families.

Number of girls	0	1	2	3	4
Frequency	15	68	69	38	10

The research student suggests that a binomial distribution with $p=\frac12$ could be a suitable model for the number of children who are girls in a family of 4 children.

(a) Using her results and a 5% significance level, test the research student’s claim. You should state your hypotheses, expected frequencies, test statistic and the critical value used.

(8)

The research student decides to refine the model and retains the idea of using a binomial distribution but does not specify the probability that the child is a girl.

(b) Use the data in the table to show that the probability that a child is a girl is 0.45

(2)

The research student uses the probability from part (b) to calculate a new set of expected frequencies, none of which are less than 5

\sum \frac{(O-E)^2}{E}

is evaluated and found to be 2.47

(c) Test, at the 5% significance level, whether using a binomial distribution is suitable to model the number of children who are girls in a family of 4 children. You should state your hypotheses and the critical value used.

(4)

(Total for Question 4 is 14 marks)

题目中文翻译

一位研究生正在研究有 4 个孩子的家庭中女孩的人数。

下表给出了她对 200 个此类家庭的调查结果。

女孩人数	0	1	2	3	4
频数	15	68	69	38	10

研究生认为，二项分布且 $p=\frac12$ 可能适合作为有 4 个孩子的家庭中女孩人数的模型。

(a) 利用她的结果并在 5% 显著性水平下，检验她的说法。你应写出原假设、期望频数、检验统计量和所用临界值。

研究生决定改进模型，仍采用二项分布，但不指定孩子是女孩的概率。

(b) 利用表中的数据证明一个孩子是女孩的概率为 0.45。

研究生用 (b) 小题得到的概率计算出一组新的期望频数，其中没有一个小于 5。

\sum \frac{(O-E)^2}{E}

被算出为 2.47。

解答

(a)

解法一

思路

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在原假设下，女孩人数 $X\sim\mathrm B(4,0.5)$ 。先用二项分布计算 $X=0,1,2,3,4$ 的概率并乘以 200 得到期望频数，再进行卡方拟合优度检验。这里没有从样本估计参数，所以自由度为类别数减 1。

答题过程

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The hypotheses are:

\begin{aligned} H_0:&\ \mathrm B(4,0.5)\text{ is a suitable model},\\ H_1:&\ \mathrm B(4,0.5)\text{ is not a suitable model}. \end{aligned}

Under $H_0$ ,

\operatorname P(X=r) =\binom4r(0.5)^r(0.5)^{4-r} =\binom4r(0.5)^4.

Hence the expected frequencies are:

Number of girls, $r$	0	1	2	3	4
Expected frequency	12.5	50	75	50	12.5

The test statistic is

\begin{align*} \chi^2 =&\,\sum\frac{(O-E)^2}{E}\\ =&\,\frac{(15-12.5)^2}{12.5}\\ &\,\hspace{2pt}+\frac{(68-50)^2}{50}\\ &\,\hspace{4pt}+\frac{(69-75)^2}{75}\\ &\,\hspace{6pt}+\frac{(38-50)^2}{50}\\ &\,\hspace{8pt}+\frac{(10-12.5)^2}{12.5}\\ =&\,0.5+6.48+0.48+2.88+0.5\\ =&\,10.84. \end{align*}

Since no parameters have been estimated from the sample,

\nu=5-1=4.

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

\chi^2_4(0.05)=9.488.

Since $10.84>9.488$ , the test statistic lies in the critical region, so $H_0$ is rejected. There is sufficient evidence at the $5\%$ significance level that $\mathrm B(4,0.5)$ is not a suitable model, so the research student’s claim is not supported.

(b)

解法一

思路

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先由“每个家庭的女孩人数乘对应家庭数”求出样本中的女孩总数，再除以所有 200 个家庭的孩子总数 $200\times4$ ，得到女孩概率的样本估计。

答题过程

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The total number of girls in the 200 families is

\begin{align*} G =&\,0(15)+1(68)+2(69)\\ &\,\hspace{2pt}+3(38)+4(10)\\ =&\,360. \end{align*}

There are $200\times4=800$ children altogether. Therefore, the estimated probability that a child is a girl is

\widehat p=\frac{360}{800}=\boxed{0.45},

as required.

(c)

解法一

思路

展开

仍使用卡方拟合优度检验，但这次概率参数 $p$ 是由样本估计的，因此自由度要在“类别数减 1”的基础上再减去 1 个已估参数，即 $\nu=3$ 。将题目给出的统计量 2.47 与相应临界值比较即可。

答题过程

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The hypotheses are:

\begin{aligned} H_0:&\ \text{A binomial distribution is a suitable model},\\ H_1:&\ \text{A binomial distribution is not a suitable model}. \end{aligned}

There are five classes and one parameter, $p$ , has been estimated from the sample. Therefore,

\nu=5-1-1=3.

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

\chi^2_3(0.05)=7.815.

Since $2.47<7.815$ , 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 that a binomial distribution is not a suitable model for the number of girls in a family of four children.

IAL 2023 Jan S3 Q4

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程

(c)

解法一

思路

答题过程