题目
A supermarket sells apples in trays. Each tray contains 10 apples.
Trays are inspected for rotten apples daily. The following table shows the numbers of rotten apples in a random sample of 150 trays on one day.
| Number of rotten apples | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| Frequency | 25 | 47 | 37 | 24 | 8 | 6 | 3 |
The manager believes that these data can be modelled by a binomial distribution B(10, p).
(a) Use these data to estimate the value of the parameter p for this model. Give your answer to 2 decimal places.
The manager uses the estimated value of p to 2 decimal places and calculates expected frequencies to 2 decimal places as follows.
| Number of rotten apples | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| Expected frequency | 20.62 | 45.26 | 44.71 | 26.17 | 10.05 | r | s |
(b) Show that the value of r is 2.65 to 2 decimal places.
(c) Find the value of s to 2 decimal places.
The manager says that the cells for 4, 5 and > 6 need to be combined into a single cell.
(d) Explain why it is necessary for the manager to do this.
The value of χ^2 for the first 4 frequencies given in each table is 2.51.
(e) Using a 10% significance level, test whether or not a binomial distribution is a suitable model. You must state your hypotheses, test statistic and the critical value used.
题目中文翻译
一家超市出售装有 10 个苹果的苹果筐。
每天都会检查这些苹果筐中的腐烂苹果数。 下表给出了某一天随机抽取的 150 个苹果筐中腐烂苹果的数量。
| 腐烂苹果数 | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| 频数 | 25 | 47 | 37 | 24 | 8 | 6 | 3 |
经理认为这些数据可以用二项分布 B(10, p) 来建模。
(a) 用这些数据估计参数 p 的值，答案保留 2 位小数。
经理将 p 估计到 2 位小数后，计算出如下期望频数。
| 腐烂苹果数 | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| 期望频数 | 20.62 | 45.26 | 44.71 | 26.17 | 10.05 | r | s |
(b) 证明 r 的值为 2.65，保留到 2 位小数。
(c) 求 s 的值，保留到 2 位小数。
经理说必须把 4、5 和大于 6 的格子合并成一个格子。
(d) 解释为什么有必要这样做。
对两张表中前 4 个频数计算得到的 χ^2 值为 2.51。
(e) 在 10% 的显著性水平下，检验二项分布是否适合作为模型。你必须写出原假设、检验统计量和所用临界值。