S4 June 2018 Q3
3. A random sample of 8 students is selected from a school database.
Each student’s reaction time is measured at the start of the school day and again at the end of the school day. The reaction times, in milliseconds, are recorded below.
| Student | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) |
|---|---|---|---|---|---|---|---|---|
| Reaction time at the start of the school day | 10.8 | 7.2 | 8.7 | 6.8 | 9.4 | 10.9 | 11.1 | 7.6 |
| Reaction time at the end of the school day | 10 | 6.1 | 8.8 | 5.7 | 8.7 | 8.1 | 9.8 | 6.8 |
The random variable \(R\) is the reaction time at the start of the school day minus the reaction time at the end of the school day. The mean of \(R\) is \(\mu\).
John uses a paired \(t\)-test to test the hypotheses
\[\mathrm{H}_0 : \mu = m \qquad\qquad \mathrm{H}_1 : \mu \ne m\]Given that \(\mathrm{H}_0\) is rejected at the 5% level of significance but accepted at the 1% level of significance,
| Scheme | Marks |
|---|---|
| Need assumption that the underlying distribution of the difference in reaction times is normally distributed. | B1 |
| (1) |
Notes
B1 for a comment that mentions “differences” and “normal” distribution
| Scheme | Marks | ||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| M1 | ||||||||||||||||||
| \(\bar{d} = \dfrac{8.5}{8} = (\pm)1.0625\) | M1 | ||||||||||||||||||
| \(s^2 = \dfrac{8}{7}\left(\dfrac{13.73}{8} - 1.0625^2\right) = 0.67125\) or \(s^2 = \dfrac{1}{7}\left(13.73 - \dfrac{8.5^2}{8}\right) = 0.67125\) | M1 | ||||||||||||||||||
| Test stat \(t = \dfrac{\text{"1.0625"} - m}{\sqrt{\frac{\text{"0.67125"}}{8}}}\) | M1 | ||||||||||||||||||
| Critical value, \(t_7(2.5\%) = \pm 2.365 \qquad t_7(0.5\%) = \pm 3.499\) | B1 | ||||||||||||||||||
| \(\dfrac{1.0625 - m}{\sqrt{\frac{0.67125}{8}}} = \pm 2.365 \qquad \dfrac{1.0625 - m}{\sqrt{\frac{0.67125}{8}}} = \pm 3.499\) | M1d A1ft | ||||||||||||||||||
| \(0.049 \lt m \lt 0.377\) and \(1.748 \lt m \lt 2.076\) | A1 A1 | ||||||||||||||||||
| (9) | |||||||||||||||||||
| (10 marks) |
Notes
M1 attempting differences
M1 attempt to find \(\bar{d} = \dfrac{\sum \text{"their } d\text{"}}{8}\)
M1 attempting \(s\) or \(s^2\ \dfrac{1}{7}\left(\sum \text{"their } d^2\text{"} - \dfrac{\left(\sum \text{"their } d\text{"}\right)^2}{8}\right)\) \(s = 0.8192\ldots\)
M1 for attempting the correct test statistic \(\dfrac{\bar{d} - m}{s/\sqrt{8}}\), allow any letter
B1 Both critical values correct (ignore sign)
M1d dependent on previous M being awarded. Having a pair of equations with the same sign and one of each CV. Ft their test statistic and CV
A1ft ft their CV four equations, may be implied by both ranges correct
A1 awrt \(0.049 \lt m \lt\) awrt 0.377 allow \(\leqslant\) instead of <
A1 awrt \(1.75 \lt m \lt\) awrt 2.08 allow \(\leqslant\) instead of <
NB if test stat the wrong way round remove one of the A marks awarded at the end
(corrected from the printed mark scheme: the scheme prints \(t_7(0.005\%)\); 3.499 is the 0.5% point)