| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2011 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Two-tailed test critical region |
| Difficulty | Standard +0.3 This is a standard S2 hypothesis testing question with two routine parts: (a) exact binomial test with n=30, and (b) normal approximation with n=120. Both require standard procedure (hypotheses, test statistic, critical region/p-value, conclusion) with no novel insight. Slightly above average difficulty due to the two-part structure and need to apply normal approximation correctly, but well within typical S2 scope. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.02d Binomial: mean np and variance np(1-p)5.04b Linear combinations: of normal distributions5.05a Sample mean distribution: central limit theorem5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: p = 0.15\), \(H_1: p \neq 0.15\) | B1 B1 | Both hypotheses must use \(p\) |
| \(X \sim B(30, 0.15)\); \(P(X \leq 1) = 0.0480\) or CR: \(X = 0\) | M1 A1 | M1 for writing or using \(B(30, 0.15)\); A1 0.0480 or \(X=0\); ignore upper CR |
| \((0.0480 > 0.025)\) not significant / do not reject \(H_0\) / not in CR | M1 | Correct statement (see table); do not allow non-contextual conflicting statements |
| There is no evidence of a change in the proportion of customers buying an item from the display | A1ft | Need idea of change/decrease in number of customers buying from display |
| Answer | Marks | Guidance |
|---|---|---|
| Two tail \(0.025 < p < 0.975\) or One tail \(0.05 < p < 0.95\) | Two tail \(p < 0.025\) or \(p > 0.975\) or One tail \(p < 0.05\) or \(p > 0.95\) | |
| \(2^\text{nd}\) M1 | not significant/accept \(H_0\)/Not in CR or contextual | significant/reject \(H_0\)/In CR or contextual |
| \(2^\text{nd}\) 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 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: p = 0.2\), \(H_1: p > 0.2\) | B1 | Both hypotheses correct — must use \(p\) |
| Let \(S\) = number who buy sandwiches, \(S \sim B(120, 0.2)\) | ||
| \(S \approx W \sim N\!\left(24, \sqrt{19.2}^2\right)\) | M1 A1 | M1 for normal approx; A1 for correct mean and sd |
| \(P(S \geq 31) = P(W \geq 30.5)\) | M1 | \(2^\text{nd}\) M1 for continuity correction, either 30.5 or 31.5 or \(x \pm 0.5\) seen |
| \(= P\!\left(Z > \frac{30.5 - 24}{\sqrt{19.2}}\right)\) or \(\frac{x - 0.5 - 24}{\sqrt{19.2}} = 1.2816\) | M1 | \(3^\text{rd}\) M1 standardising with mean, sd and 30.5, 31 or 31.5 or \(x\) or \((x \pm 0.5)\) |
| \([= P(Z > 1.48\ldots)]\) | ||
| \(= 1 - 0.9306 = 0.0694\) | M1 A1 | \(4^\text{th}\) M1 for 1 — tables value or 1.2816; \(2^\text{nd}\) A1 for awrt 0.069 or \(x = 30.1\) |
| \(x = 30.1\) | ||
| \(< 0.10\) so significant result, there is evidence that more customers are purchasing sandwiches or the shopkeeper's claim is correct | B1ft | Correct conclusion in context using probability and 0.1; need idea of *more* customers buying sandwiches |
## Question 6:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: p = 0.15$, $H_1: p \neq 0.15$ | B1 B1 | Both hypotheses must use $p$ |
| $X \sim B(30, 0.15)$; $P(X \leq 1) = 0.0480$ or CR: $X = 0$ | M1 A1 | M1 for writing or using $B(30, 0.15)$; A1 0.0480 or $X=0$; ignore upper CR |
| $(0.0480 > 0.025)$ not significant / do not reject $H_0$ / not in CR | M1 | Correct statement (see table); do not allow non-contextual conflicting statements |
| There is no evidence of a change in the proportion of customers buying an item from the display | A1ft | Need idea of change/decrease in number of customers buying from display |
**Significance table for part (a):**
| | Two tail $0.025 < p < 0.975$ or One tail $0.05 < p < 0.95$ | Two tail $p < 0.025$ or $p > 0.975$ or One tail $p < 0.05$ or $p > 0.95$ |
|---|---|---|
| $2^\text{nd}$ M1 | not significant/accept $H_0$/Not in CR or contextual | significant/reject $H_0$/In CR or contextual |
| $2^\text{nd}$ 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 |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: p = 0.2$, $H_1: p > 0.2$ | B1 | Both hypotheses correct — must use $p$ |
| Let $S$ = number who buy sandwiches, $S \sim B(120, 0.2)$ | | |
| $S \approx W \sim N\!\left(24, \sqrt{19.2}^2\right)$ | M1 A1 | M1 for normal approx; A1 for correct mean and sd |
| $P(S \geq 31) = P(W \geq 30.5)$ | M1 | $2^\text{nd}$ M1 for continuity correction, either 30.5 or 31.5 or $x \pm 0.5$ seen |
| $= P\!\left(Z > \frac{30.5 - 24}{\sqrt{19.2}}\right)$ or $\frac{x - 0.5 - 24}{\sqrt{19.2}} = 1.2816$ | M1 | $3^\text{rd}$ M1 standardising with mean, sd and 30.5, 31 or 31.5 or $x$ or $(x \pm 0.5)$ |
| $[= P(Z > 1.48\ldots)]$ | | |
| $= 1 - 0.9306 = 0.0694$ | M1 A1 | $4^\text{th}$ M1 for 1 — tables value or 1.2816; $2^\text{nd}$ A1 for awrt 0.069 or $x = 30.1$ |
| $x = 30.1$ | | |
| $< 0.10$ so significant result, there is evidence that more customers are purchasing sandwiches or the shopkeeper's claim is correct | B1ft | Correct conclusion in context using probability and 0.1; need idea of *more* customers buying sandwiches |
**SC** using $P(X<31.5) - P(X<30.5)$ can get B1M1 A1 M1 M1M0A0B0
---\begin{enumerate}
\item 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.\\
(a) Stating your hypotheses clearly, and using a $5 \%$ level of significance, test whether or not there has been a change in the proportion of customers buying an item from the display next to the till.
\end{enumerate}
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.\\
(b) Using a suitable approximation and stating your hypotheses clearly, test the shopkeeper's claim. Use a $10 \%$ level of significance.\\
\hfill \mbox{\textit{Edexcel S2 2011 Q6 [14]}}