| Date | May 2016 | Marks available | 4 | Reference code | 16M.1.hl.TZ0.4 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Find and Write down | Question number | 4 | Adapted from | N/A |
Question
All members of a large athletics club take part in an annual shotput competition.
The following data give the distances achieved, in metres, by a random selection of 10 members of the club in the 2016 competition
11.8, 14.3, 13.8, 10.3, 14.9, 14.7, 12.4, 13.9, 14.0, 11.7
The president of the club wishes to test whether these data provide evidence that distances achieved have increased since the 2015 competition, when the mean result for the club was 12.4 m. You may assume that the distances achieved follow a normal distribution with mean \(\mu \), variance \({\sigma ^2}\), and that the membership of the club has not changed from 2015 to 2016.
State suitable hypotheses.
(i) Give a reason why a \(t\) test is appropriate and write down its degrees of freedom.
(ii) Find the critical region for testing at each of the 5% and 10% significance levels.
(i) Find unbiased estimates of \(\mu \) and \({\sigma ^2}\).
(ii) Find the value of the test statistic.
State the conclusions that the president of the club should reach from this test, giving reasons for your answer.
Markscheme
\({H_0}:{\text{ }}\mu = 12.4;{\text{ }}{H_1}:{\text{ }}\mu > 12.4\) A1
[1 mark]
(i) \(t\) test is appropriate because the variance (standard deviation) is unknown R1
\(v = 9\) A1
(ii) \(t \geqslant 1.83{\text{ }}(5\% );{\text{ }}t \geqslant 1.38{\text{ }}(10\% )\) A1A1
Note: Accept strict inequalities.
[4 marks]
(i) unbiased estimate of \(\mu \) is 13.18 A1
Note: Accept 13.2.
unbiased estimate of \({\sigma ^2}\) is 2.34 \(({1.531^2})\) A1
(ii) \({t_{{\text{calc}}}} = \left( {\frac{{13.18 - 12.4}}{{\frac{{1.531}}{{\sqrt {10} }}}}} \right) = 1.61{\text{ or }}1.65\) A1
[3 marks]
as \(1.38 < 1.61 < 1.83\) R1
evidence to accept \({H_0}\) at the 5% level, but not at the 10% level A1
Note: Accept the use of the \(p\)-value \( = 0.0708\).
[2 marks]
Examiners report
Most candidates had an understanding of how to start the question, but only a small number were able to gain full marks. It appeared that many candidates were used to finding \(p\)-values, but showed a lack of understanding when asked to find the critical regions and test a \(t\)-value. The conclusions required in part (d) were often too brief and/or poorly expressed.
Most candidates had an understanding of how to start the question, but only a small number were able to gain full marks. It appeared that many candidates were used to finding \(p\)-values, but showed a lack of understanding when asked to find the critical regions and test a \(t\)-value. The conclusions required in part (d) were often too brief and/or poorly expressed.
Most candidates had an understanding of how to start the question, but only a small number were able to gain full marks. It appeared that many candidates were used to finding \(p\)-values, but showed a lack of understanding when asked to find the critical regions and test a \(t\)-value. The conclusions required in part (d) were often too brief and/or poorly expressed.
Most candidates had an understanding of how to start the question, but only a small number were able to gain full marks. It appeared that many candidates were used to finding \(p\)-values, but showed a lack of understanding when asked to find the critical regions and test a \(t\)-value. The conclusions required in part (d) were often too brief and/or poorly expressed.
