S4 June 2017 Q1
1. The times taken by children to run 150 m are normally distributed. The times taken, \(x\) seconds, by a random sample of 9 boys and an independent random sample of 6 girls are recorded. The following statistics are obtained.
| Number of children | Sample mean \(\bar{x}\) | \(\sum x^2\) | |
|---|---|---|---|
| Boys | 9 | 22.8 | 4693.60 |
| Girls | 6 | 29.5 | 5236.12 |
The Headteacher claims that the mean time taken for the girls is more than 5 seconds greater than the mean time taken for the boys.
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \sigma_B^2 = \sigma_G^2, \quad \mathrm{H}_1 : \sigma_B^2 \ne \sigma_G^2\) | B1 |
| \(\left[s_B^2 =\right] \tfrac{1}{8}\left(4693.6 - 9 \times 22.8^2\right) = 1.88\) | M1 |
| \(\left[s_G^2 =\right] \tfrac{1}{5}\left(5236.12 - 6 \times 29.5^2\right) = 2.924\) awrt 2.92 | A1 |
| \(\dfrac{s_G^2}{s_B^2} = 1.555\ldots\ [0.643]\) | M1 A1 |
| critical value \(F_{5,\,8} = 3.69\ [0.271]\) | B1 |
| not significant, variances are the same | A1 cso |
| (7) |
Notes
B1 both hypotheses. Must use \(\sigma\) or \(\sigma^2\) and make clear which is \(\mathrm{H}_0\) and which is \(\mathrm{H}_1\). Do not allow in words
M1 correct method for either \(s_B^2\) or \(s_G^2\)
A1 Both \(s_B^2\) and \(s_G^2\) correct to 3sf allow sd’s
M1 allow use of \(s_B\) and \(s_G\) instead of \(s_B^2\) or \(s_G^2\)
A1 awrt 1.56 or 0.643
B1 correct CV for their \(F\) or a correct comparison if use \(p\)
A1 cso – All previous marks must be awarded. Variances are the same or var are not different
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \mu_G = \mu_B + 5, \quad \mathrm{H}_1 : \mu_G \gt \mu_B + 5\) | B1 |
| pooled estimate of variance \(s_p^{\,2} = \dfrac{8 \times 1.88 + 5 \times 2.924}{13} = 2.2815\ldots\) or \(s_p = 1.51046\ldots\) | M1 |
| test statistic \(t = \pm\left(\dfrac{29.5 - 22.8 - 5}{s\sqrt{\frac{1}{9} + \frac{1}{6}}}\right) = \pm\) awrt 2.14 or \(p = 0.0262\) | M1 M1A1 |
| critical value \(t_{13}(1\%) = \pm 2.650\) or \(0.0262 \gt 0.01\) | B1 |
| Insufficient evidence to support Headteacher’s claim or The time taken for girls is not more than 5 seconds greater than for boys | A1 cso |
| (7) | |
| (14 marks) |
Notes
B1 both hypotheses using \(\mu\). Do not allow \(\geqslant\) sign instead of >. May use different letters eg \(A\) and \(B\) but they must be defined.
M1 only allow use of \(s_B\) and \(s_G\) instead of \(s_B^2\) or \(s_G^2\) – May be seen in part (a)
M1 use of correct formula with their \(s_p\) – condone missing 5
M1 use of correct formula with their \(s_p\). (which must have been attempted)
B1 correct CV but must match \(t\)-value or a correct comparison if use \(p\)
A1 A correct statement with either the word Headteacher/Teacher/Head or time and not more than 5 oe do not allow contradicting statements.