题目
The manager of a company making ice cream believes that the proportions of people in the population who prefer vanilla, chocolate, strawberry and other are in the ratio 10 : 5 : 2 : 3
The manager takes a random sample of 400 customers and records their age and favourite ice cream flavour. The results are shown in the table below.
| Age | Vanilla | Chocolate | Strawberry | Other | Total |
|---|---|---|---|---|---|
| Child | 95 | 25 | 13 | 25 | 158 |
| Teenager | 57 | 20 | 17 | 36 | 130 |
| Adult | 36 | 50 | 10 | 16 | 112 |
| Total | 188 | 95 | 40 | 77 | 400 |
(a) Use the data in the table to test, at the 5% level of significance, the manager’s belief. You should state your hypotheses, test statistic, critical value and conclusion clearly.
A researcher wants to investigate whether or not there is a relationship between the age of a customer and their favourite ice cream flavour. In order to test whether favourite ice cream flavour and age are related, the researcher plans to carry out a χ 2 test.
(b) Use the table to calculate expected frequencies for the group (i) teenagers whose favourite ice cream flavour is vanilla, (ii) adults whose favourite ice cream flavour is chocolate.
(c) Write down the number of degrees of freedom for this χ 2 test.
题目中文翻译
某冰淇淋公司的经理认为，人群中偏爱香草、巧克力、草莓和其他口味的比例之比为 10 : 5 : 2 : 3。
经理对 400 名顾客做了随机抽样，并记录他们的年龄与最喜欢的冰淇淋口味。结果如下表所示。
| 年龄 | 香草 | 巧克力 | 草莓 | 其他 | 总计 |
|---|---|---|---|---|---|
| 儿童 | 95 | 25 | 13 | 25 | 158 |
| 青少年 | 57 | 20 | 17 | 36 | 130 |
| 成年人 | 36 | 50 | 10 | 16 | 112 |
| 总计 | 188 | 95 | 40 | 77 | 400 |
(a) 利用表中的数据，在 5% 显著性水平下检验经理的看法。 你应清楚写出原假设、检验统计量、临界值和结论。
一位研究人员想调查顾客年龄与其最喜欢的冰淇淋口味之间是否存在关系。 为了检验最喜欢的冰淇淋口味与年龄是否相关，研究人员计划进行 χ 2 检验。
(b) 利用表格计算以下组别的期望频数： (i) 最喜欢香草口味的青少年， (ii) 最喜欢巧克力口味的成年人。
(c) 写出这项 χ 2 检验的自由度。