S4 June 2013 Q1
1. George owns a garage and he records the mileage of cars, \(x\) thousands of miles, between services. The results from a random sample of 10 cars are summarised below.
\[\sum x = 113.4 \qquad \sum x^2 = 1414.08\]The mileage of cars between services is normally distributed and George believes that the standard deviation is 2.4 thousand miles.
Stating your hypotheses clearly, test, at the 5% level of significance, whether or not these data support George’s belief. (7)
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \sigma^2 = 2.4^2 \qquad \mathrm{H}_1 : \sigma^2 \neq 2.4^2\) | B1 |
| \(s^2 = \dfrac{1414.08 - 10 \times \left(\dfrac{113.4}{10}\right)^2}{9} = 14.236\) | M1 A1 |
| \(\chi^2 = \dfrac{9s^2}{\sigma^2} = \dfrac{9 \times 14.236}{2.4^2} = 22.24375\) | M1 A1 |
| Critical Value \(\chi_9^{\,2}(0.025) = 19.023\) | B1 |
| Significant result, there is evidence of a change in standard deviation or the data do not support George’s belief | A1cso |
| (7) | |
| (7 marks) |
Notes
1st B1 Both hypotheses, must use \(\sigma\). Allow \(\mathrm{H}_0 : \sigma = 2.4 \quad \mathrm{H}_1 : \sigma \neq 2.4\)
1st M1 correct method used
1st A1 awrt 14.2
2nd M1 \(\chi^2 = \dfrac{9 \times \text{"their }s^2\text{"}}{2.4^2}\)
2nd A1 awrt 22.2
2nd B1 for critical value, this should be compatible with their alternative hypothesis (16.919 for one tail test)
3rd A1ft fully correct solution only