June 2022 Paper 3 Q19
19 A bank runs a campaign to promote Internet banking accounts to their customers.
Before the campaign, 42% of their customers had an Internet banking account.
One week after the campaign started, 35 customers were surveyed at random and 18 of them were found to have registered for an Internet banking account.
Using a binomial distribution, carry out a hypothesis test at the 10% significance level to investigate the claim that, since the campaign, there has been an increase in the proportion of customers registered for an Internet banking account. [6 marks]
| Scheme | Marks | AO |
|---|---|---|
| States both hypotheses correctly for one-tailed test 0.42 OE | B1 | 2.5 |
| States or uses correct model PI by calculation of one of the probabilities below \(P(X \leqslant 16)\) = [0.73, 0.733] \(P(X \leqslant 17)\) = [0.83, 0.832] \(P(X \leqslant 18)\) = [0.90, 0.903] \(P(X \geqslant 18)\) = [0.168, 0.17] \(P(X \geqslant 19)\) = [0.097, 0.10] or critical value of 19 or region ≥ 19 | M1 | 3.3 |
| Obtains \(P(X \geqslant 18)\) = [0.168, 0.17] or Obtains a critical value 19 or critical region ≥ 19 | A1 | 1.1b |
| Evaluates binomial model by comparing their \(P(X \geqslant 18)\) with 0.10 or Evaluates binomial model by comparing their critical region with 18 | M1 | 3.5a |
| Infers \(H_0\) or null hypothesis not rejected Condone \(H_0\) accepted or \(H_1\) or alternative hypothesis rejected All figures must be correct | A1 | 2.2b |
| Concludes correctly in context that there is insufficient evidence to suggest an increase in the proportion of bank customers registered for an internet banking account To be awarded R1, marks M1A1M1A1 must be scored as the minimum | R1 | 3.2a |
| (6 marks) |
Typical solution
\(X\) = number of customers with an internet banking account
\[H_0: p = 0.42\]\[H_1: p \gt 0.42\]Under null hypothesis
\[X \sim \mathrm{B}(35, 0.42)\]\[P(X \geqslant 18) = 0.169\]\[0.169 \gt 0.10\]Do not reject \(H_0\)
There is insufficient evidence to suggest an increase in the proportion of bank customers registered for an internet banking account.