| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2019 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Interpret test conclusion or error |
| Difficulty | Standard +0.3 This is a standard AS-level hypothesis testing question requiring students to identify common errors (wrong alternative hypothesis direction, using P(X=8) instead of P(X≥8)), then perform a routine critical region calculation. While it tests understanding of hypothesis test mechanics, it involves straightforward binomial probability calculations with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| The alternative hypothesis should be \(H_1: p > 0.15\) | B1 | Identifying that \(\geq\) should be \(>\) in the alternative hypothesis |
| The calculation of the test statistic should be \(P(X \geq 8)\ [= 0.0698]\) | B1 | Identifying that \(P(X=8)\) should be \(P(X \geq 8)\); stating \(P(X=8)\) is incorrect on its own is insufficient; check for errors identified and corrected next to the question |
| Answer | Marks | Guidance |
|---|---|---|
| These will affect the conclusion (as the null hypothesis should not be rejected) since \(P(X \geq 8)\ [= 0.0698]\) is greater than 0.05 | B1 | Will affect conclusion and correct supporting reason |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X \leq 8) = 0.9722\ldots > 0.95\) or \(P(X \geq 9) = 0.0277\ldots < 0.05\) | M1 | For use of tables to find probability associated with critical value \([P(X \leq 8)\) or \(P(X \geq 9)]\) with \(B(30, 0.15)\) (may be implied by either correct probability awrt 0.97 or awrt 0.03) or by the correct CR |
| \(\text{CR: } \{X \geq 9\}\) | A1 | \([30\geq] X \geq 9\) o.e. e.g. \(X > 8\); allow '9 or more' or 'CR \(\geq 9\)' |
| Answer | Marks | Guidance |
|---|---|---|
| awrt \(\mathbf{0.0278}\) | B1ft | awrt 0.0278 (allow awrt 2.78%); or correct ft their one-tailed upper CR from \(B(30, 0.15)\) to 3 s.f. |
# Question 5:
## Part (a):
The alternative hypothesis should be $H_1: p > 0.15$ | B1 | Identifying that $\geq$ should be $>$ in the alternative hypothesis
The calculation of the test statistic should be $P(X \geq 8)\ [= 0.0698]$ | B1 | Identifying that $P(X=8)$ should be $P(X \geq 8)$; stating $P(X=8)$ is incorrect on its own is insufficient; check for errors identified and corrected next to the question
## Part (b):
These will affect the conclusion (as the null hypothesis should not be rejected) since $P(X \geq 8)\ [= 0.0698]$ is greater than 0.05 | B1 | Will affect conclusion **and** correct supporting reason
## Part (c):
$P(X \leq 8) = 0.9722\ldots > 0.95$ or $P(X \geq 9) = 0.0277\ldots < 0.05$ | M1 | For use of tables to find probability associated with critical value $[P(X \leq 8)$ or $P(X \geq 9)]$ with $B(30, 0.15)$ (may be implied by either correct probability awrt 0.97 or awrt 0.03) or by the correct CR
$\text{CR: } \{X \geq 9\}$ | A1 | $[30\geq] X \geq 9$ o.e. e.g. $X > 8$; allow '9 or more' or 'CR $\geq 9$'
## Part (d):
awrt $\mathbf{0.0278}$ | B1ft | awrt 0.0278 (allow awrt 2.78%); or correct ft their one-tailed upper CR from $B(30, 0.15)$ to 3 s.f.\begin{enumerate}
\item Past records show that $15 \%$ of customers at a shop buy chocolate. The shopkeeper believes that moving the chocolate closer to the till will increase the proportion of customers buying chocolate.
\end{enumerate}
After moving the chocolate closer to the till, a random sample of 30 customers is taken and 8 of them are found to have bought chocolate.
Julie carries out a hypothesis test, at the 5\% level of significance, to test the shopkeeper's belief.\\
Julie's hypothesis test is shown below.\\
$\mathrm { H } _ { 0 } : p = 0.15$\\
$\mathrm { H } _ { 1 } : p \geqslant 0.15$\\
Let $X =$ the number of customers who buy chocolate.\\
$X \sim \mathrm {~B} ( 30,0.15 )$\\
$\mathrm { P } ( X = 8 ) = 0.0420$\\
$0.0420 < 0.05$ so reject $\mathrm { H } _ { 0 }$\\
There is sufficient evidence to suggest that the proportion of customers buying chocolate has increased.\\
(a) Identify the first two errors that Julie has made in her hypothesis test.\\
(b) Explain whether or not these errors will affect the conclusion of her hypothesis test. Give a reason for your answer.\\
(c) Find, using a 5\% level of significance, the critical region for a one-tailed test of the shopkeeper's belief. The probability in the tail should be less than 0.05\\
(d) Find the actual level of significance of this test.
\hfill \mbox{\textit{Edexcel AS Paper 2 2019 Q5 [6]}}