S2 June 2014 (R) Q1
1. Before Roger will use a tennis ball he checks it using a “bounce” test. The probability that a ball from Roger’s usual supplier fails the bounce test is 0.2. A new supplier claims that the probability of one of their balls failing the bounce test is less than 0.2. Roger checks a random sample of 40 balls from the new supplier and finds that 3 balls fail the bounce test.
Stating your hypotheses clearly, use a 5% level of significance to test the new supplier’s claim. (5)
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : p = 0.2 \qquad \mathrm{H}_1 : p \lt 0.2\) | B1 |
| \(\left[X \sim \mathrm{B}(40, 0.2)\right] \qquad \mathrm{P}(X \leqslant 3) = 0.0285\) or CR of \(X \leqslant 3\) | M1A1 |
| [0.0285 < 0.05] significant, reject \(\mathrm{H}_0\) | M1dep |
| There is evidence to support the supplier’s claim or The probability of a ball failing the bounce test is less than 0.2 | A1cso |
| (5) |
Notes
1st B1 for both \(\mathrm{H}_0\) and \(\mathrm{H}_1\) must use \(p\) or \(\pi\)
1st M1 for writing or using B(40, 0.2), may be implied by correct answer
1st A1 awrt 0.0285 or CR of \(X \leqslant 3\) as their final answer
2nd M1 dependent on the previous method mark being awarded. A correct statement (this may be contextual) comparing “their probability” and 0.05 (or comparing 3 with their critical region). Do not allow conflicting statements.
2nd A1cso This is cso so can only be awarded for a fully correct solution. A correct contextualised conclusion (to include the words underlined in bold)