June 2023 Paper 3 Q17
17 A council found that 70% of its new local businesses made a profit in their first year.
The council introduced an incentive scheme for its residents to encourage the use of new local businesses.
At the end of the scheme, a random sample of 25 new local businesses was selected and it was found that 21 of them had made a profit in their first year.
Using a binomial distribution, investigate, at the 2.5% level of significance, whether there is evidence of an increase in the proportion of new local businesses making a profit in their first year. [6 marks]
| Scheme | Marks | AO |
|---|---|---|
| States both hypotheses correctly for one-tailed test 0.7 OE | B1 | 2.5 |
| States or uses correct model PI by calculation of one of the probabilities below \(\mathrm{P}(X \leqslant 19)\) = [0.806, 0.807] \(\mathrm{P}(X \leqslant 20)\) = [0.909, 0.91] \(\mathrm{P}(X \leqslant 21)\) = [0.966, 0.967] \(\mathrm{P}(X \geqslant 20)\) = [0.193, 0.1935] \(\mathrm{P}(X \geqslant 21)\) = [0.09, 0.091] \(\mathrm{P}(X \geqslant 22)\) = [0.033, 0.0333] \(\mathrm{P}(X \geqslant 23)\) = [0.0089, 0.00896] or critical value of 23 or critical region \(\geqslant 23\) condone missing or incorrect labels | M1 | 3.3 |
| Obtains [0.09, 0.091] or [0.909, 0.91] or obtains critical value 23 or critical region \(\geqslant 23\) | A1 | 1.1b |
| Evaluates binomial model by correctly comparing their \(\mathrm{P}(X \geqslant 21)\) or [0.09, 0.091] with 0.025 or evaluates binomial model by correctly comparing their \(\mathrm{P}(X \lt 21)\) with 0.975 or evaluates binomial model by correctly determining if 21 is in their critical region | M1 | 3.5a |
| Infers \(\mathrm{H}_0\) or null hypothesis not rejected Condone \(\mathrm{H}_0\) accepted All figures must be correct Ignore reference to \(\mathrm{H}_1\) | A1 | 2.2b |
| Concludes correctly in context that there is insufficient evidence of an increase in the proportion of local businesses that made a profit in their first year. To be awarded R1, marks M1A1M1A1 must be scored as the minimum Labels of probability calculations must be correct Conclusion must not be definite | R1 | 3.2a |
| (6 marks) |
Typical solution
\[\mathrm{H}_0 : p = 0.7\]\[\mathrm{H}_1 : p \gt 0.7\]Under null hypothesis
\[X \sim \mathrm{B}(25, 0.7)\]\[\begin{aligned}\mathrm{P}(X \geqslant 21) &= 1 - \mathrm{P}(X \leqslant 20) \\ &= 1 - 0.9095 \\ &= 0.0905\end{aligned}\]\[0.0905 \gt 0.025\]Do not reject \(\mathrm{H}_0\)
There is insufficient evidence of an increase in the proportion of local businesses that made a profit in their first year.