题目
A garage sold 250 new cars which work using one of three types of engine: petrol, diesel or electric.
For each car, the garage recorded the month it first experienced an engine problem, if it occurred during the first year after being sold.
The results for the 250 cars are shown in the table.
| Month 1-6 | Month 7-12 | None | Total | |
|---|---|---|---|---|
| Petrol engine | 12 | 25 | 95 | 132 |
| Diesel engine | 10 | 8 | 17 | 35 |
| Electric engine | 14 | 10 | 59 | 83 |
| Total | 36 | 43 | 171 | 250 |
The results are to be used to investigate whether or not there is an association between when the car first experienced an engine problem and the type of engine.
(a) Calculate the expected frequency for a diesel engine that first experienced an engine problem during month 1-6. Given that for the other 8 classes ∑((O - E)^2 / E) = 7.444 to 3 decimal places,
(b) test, at a 5% level of significance, whether or not there is evidence of an association between when the car first experienced an engine problem and the type of engine. You should state the hypotheses, the test statistic, the degrees of freedom and the critical value used for this test.
题目中文翻译
一家车行售出 250 辆新车，这些车使用三种发动机中的一种：汽油、柴油或电动。
对于每辆车，车行记录了其首次出现发动机故障的月份，如果故障发生在售出后的第一年内。
结果如下表所示。
| 第 1–6 月 | 第 7–12 月 | 没有故障 | 总计 | |
|---|---|---|---|---|
| 汽油发动机 | 12 | 25 | 95 | 132 |
| 柴油发动机 | 10 | 8 | 17 | 35 |
| 电动发动机 | 14 | 10 | 59 | 83 |
| 总计 | 36 | 43 | 171 | 250 |
这些结果将用于检验：汽车首次出现发动机问题的时间与发动机类型之间是否存在关联。
(a) 计算一辆柴油车在第 1–6 月首次出现发动机问题的期望频数。 已知另外 8 个格子的 ∑((O - E)^2 / E) = 7.444，精确到 3 位小数。
(b) 在 5% 的显著性水平下检验汽车首次出现发动机问题的时间与发动机类型之间是否有关联。 你应写出原假设、检验统计量、自由度和所用临界值。