S2 June 2011 Q6
6. A shopkeeper knows, from past records, that 15% of customers buy an item from the display next to the till. After a refurbishment of the shop, he takes a random sample of 30 customers and finds that only 1 customer has bought an item from the display next to the till.
During the refurbishment a new sandwich display was installed. Before the refurbishment 20% of customers bought sandwiches. The shopkeeper claims that the proportion of customers buying sandwiches has now increased. He selects a random sample of 120 customers and finds that 31 of them have bought sandwiches.
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : p = 0.15 \quad \mathrm{H}_1 : p \neq 0.15\) | B1 B1 |
| \(X \sim \mathrm{B}(30, 0.15)\) | M1 |
| \(\mathrm{P}(X \leqslant 1) = 0.0480\) or CR: \(X = 0\) | A1 |
| (0.0480 > 0.025) not a significant result or do not reject H0 or not in CR | M1 |
| there is no evidence of a change in the proportion of customers buying an item from the display. | A1ft |
| (6) |
Notes
1st B1 for H0 must use \(p\) 2nd B1 for H1 must use \(p\)
1st M1 for writing or using B(30,0.15) – may be implied by correct CR
1st A1 0.0480 or \(X = 0\). Allow \(X \leqslant 0\). Ignore upper CR. NB Allow CR \(X \leqslant 1\) if using one tail test.
2nd M1 A correct statement (see table below) Do not allow non-contextual conflicting statements eg“significant” and “accept H0”. Ignore comparisons
2nd A1 for a correct statement in context. For context we need idea of change/decrease in number of customers buying from display – may use different words. NB A correct contextual statement on its own scores M1A1
| Two tail \(0.025 \lt p \lt 0.975\) or One tail \(0.05 \lt p \lt 0.95\) | Two tail \(p \lt 0.025\) or \(p \gt 0.975\) or One tail \(p \lt 0.05\) or \(p \gt 0.95\) | |
|---|---|---|
| 2nd M1 | not significant/ accept H0/ Not in CR or contextual | significant/ reject H0/ In CR or contextual |
| 2nd A1 | There is no evidence of a change/decrease in the proportion of customers buying an item from the display | There is evidence of a change/decrease in the proportion of customers buying an item from the display. |
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : p = 0.2 \quad \mathrm{H}_1 : p \gt 0.2\) | B1 |
| Let \(S\) = the number who buy sandwiches, \(S \sim \mathrm{B}(120, 0.2)\), \(S \approx W \sim \mathrm{N}\left(24, \sqrt{19.2}^{\,2}\right)\) | M1 A1 |
| \(\mathrm{P}(S \geqslant 31) = \mathrm{P}(W \geqslant 30.5)\) | M1 |
| \(= \mathrm{P}\left(Z \gt \dfrac{30.5 - 24}{\sqrt{19.2}}\right)\) or \(\dfrac{x - 0.5 - 24}{\sqrt{19.2}} = 1.2816\) | M1 |
| [\(= \mathrm{P}(Z \gt 1.48..)\) ] \(= 1 - 0.9306\) | M1 |
| \(= 0.0694\) \(x = 30.1\) | A1 |
| < 0.10 so a significant result, there is evidence that more customers are purchasing sandwiches or the shopkeepers claim is correct. | B1ft |
| (8) | |
| (14 marks) |
Notes
1st B1 both hypotheses correct – must use \(p\).
1st M1 for a normal approx
1st A1 for correct mean and sd
2nd M1 for use of continuity correction, either 30.5 or 31.5 or (\(x \pm 0.5\)) seen
3rd M1 standardising with their mean and their sd and 30.5, 31 or 31.5 or \(x\) or (\(x \pm 0.5\)))
4th M1 for 1 - tables value or 1.2816
2nd A1 for awrt 0.069 or \(x = 30.1\)
2nd B1ft For a correct conclusion in context using their probability and 0.1 For context we need idea of more customers buying sandwiches – may use different words
| One tail \(0.1 \lt p \lt 0.9\) or Two tail \(0.05 \lt p \lt 0.95\) | One tail \(p \lt 0.1\) or \(p \gt 0.9\) or Two tail \(p \lt 0.05\) or \(p \gt 0.95\) | |
|---|---|---|
| 2nd M1 | not significant/ accept H0/ Not in CR or contextual | significant/ reject H0/ In CR or contextual |
| 2nd A1 | There is no evidence of an increase in the proportion of customers buying sandwiches | There is evidence of a change/increase in the proportion of customers buying sandwiches. |
SC using P(\(X\)<31.5) – P(\(X\)<30.5) can get B1M1 A1 M1 M1M0A0B0