S4 June 2010 Q5
5. A car manufacturer claims that, on a motorway, the mean number of miles per gallon for the Panther car is more than 70. To test this claim a car magazine measures the number of miles per gallon, \(x\), of each of a random sample of 20 Panther cars and obtained the following statistics.
\[\bar{x} = 71.2 \qquad s = 3.4\]The number of miles per gallon may be assumed to be normally distributed.
The standard deviation of the number of miles per gallon for the Tiger car is 4.
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \mu = 70\) [accept \(\leqslant 70\)], \(\mathrm{H}_1 : \mu \gt 70\) | B1 |
| \(t = \dfrac{71.2 - 70}{3.4/\sqrt{20}} = 1.58\) | M1A1 |
| critical value \(t_{19}(5\%) = 1.729\) | B1 |
| not significant, insufficient evidence to confirm manufacturer’s claim | A1 ft |
| (5) |
Notes
B1 both hypotheses using \(\mu\)
M1 \(\dfrac{71.2 - 70}{3.4/\sqrt{20}}\)
A1 awrt 1.58 (corrected from the printed mark scheme: printed as “A1 awrt 1.73”; 1.58 is the test statistic in the scheme above) CHECK
A1 correct conclusion ft their \(t\) value and CV
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \sigma^2 = 16,\ \mathrm{H}_1 : \sigma^2 \neq 16\) | B1 |
| test statistic \(\dfrac{(n-1)s^2}{\sigma^2} =,\ \dfrac{219.64}{16} = 13.7..\) | M1 A1 |
| critical values \(\begin{aligned}&\chi^2_{19}(5\%)\text{ upper tail} = 32.852\\ &\chi^2_{19}(5\%)\text{ lower tail} = 8.907\end{aligned}\) not significant | B1 B1 |
| Insufficient evidence to suggest that the variance of the miles per gallon of the panther is different from that of the Tiger. | A1ft |
| (6) | |
| (11 marks) |
Notes
B1 both hypotheses and 16. accept \(\sigma = 4\) and \(\sigma \neq 4\)
M1 \(\dfrac{(19) \times 3.4^2}{16}\) allow \(\dfrac{(19) \times 3.4^2}{4}\)
A1 awrt 13.7
B1 32.852
B1 8.907
A1 correct contextual comment
NB those who use \(\sigma^2 = 4\) throughout can get B0 M1 A0B1 B1 A1