| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| Session | October |
| Marks | 10 |
| 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 requiring routine application of binomial critical regions and normal approximation. Part (a) involves finding critical values from tables (standard procedure), part (b) is direct calculation, and part (c) is a textbook normal approximation test. All techniques are algorithmic with no novel insight required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: p = 0.35\), \(H_1: p \neq 0.35\) | B1 | Both hypotheses in terms of \(p\) or \(\pi\) |
| \(P(X \leq 8) = \text{awrt } 0.0303\) or \(P(X \geq 21) = \text{awrt } 0.0173\) or \(P(X \leq 20) = \text{awrt } 0.9827\) | M1 | One of the correct probability statements. Implied by a correct critical region |
| \(P(X \leq 8) = \text{awrt } 0.0303\) and \(P(X \geq 21) = \text{awrt } 0.0173\) | A1 | awrt 0.0303 and awrt 0.0173 |
| CR: \(X \leq 8\) and \(X \geq 21\) | A1 | Both parts of the critical region given. Allow alternative notation e.g. \(X < 9\) and \(X > 20\). Do not allow as probability statements |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(0.0476\) | B1ft | For 0.0476. Allow awrt 0.0475. ft their two critical regions provided probabilities are seen in part (a) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: p = 0.028\), \(H_1: p > 0.028\) | B1 | Both hypotheses in terms of \(p\) or \(\pi\) |
| \(Y \sim B(250, 0.028) \Rightarrow Y \sim Po(7)\) | M1 | \(Po(7)\) written or used |
| \(P(Y \geq 11) = 1 - P(Y \leq 10)\) or \(P(Y \geq 13) = 1 - 0.973\) | M1 | Writing or using \(1 - P(Y \leq 10)\) or \(1 - 0.9015\) or \(1 - 0.973\) |
| \(= 0.0985\) or Critical region \(Y \geq 13\) | A1 | awrt 0.0985 or CR: \(Y \geq 13\) provided \(Po(7)\) seen or used |
| There is insufficient evidence to suggest that the proportion of sunflower seeds that grow to a height of more than 3 metres is now greater than 0.028 | A1 | Independent of hypotheses but dependent on previous M1A1. A correct conclusion in context |
# Question 3:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: p = 0.35$, $H_1: p \neq 0.35$ | B1 | Both hypotheses in terms of $p$ or $\pi$ |
| $P(X \leq 8) = \text{awrt } 0.0303$ or $P(X \geq 21) = \text{awrt } 0.0173$ or $P(X \leq 20) = \text{awrt } 0.9827$ | M1 | One of the correct probability statements. Implied by a correct critical region |
| $P(X \leq 8) = \text{awrt } 0.0303$ and $P(X \geq 21) = \text{awrt } 0.0173$ | A1 | awrt 0.0303 and awrt 0.0173 |
| CR: $X \leq 8$ and $X \geq 21$ | A1 | Both parts of the critical region given. Allow alternative notation e.g. $X < 9$ and $X > 20$. Do not allow as probability statements |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $0.0476$ | B1ft | For 0.0476. Allow awrt 0.0475. ft their two critical regions provided probabilities are seen in part (a) |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: p = 0.028$, $H_1: p > 0.028$ | B1 | Both hypotheses in terms of $p$ or $\pi$ |
| $Y \sim B(250, 0.028) \Rightarrow Y \sim Po(7)$ | M1 | $Po(7)$ written or used |
| $P(Y \geq 11) = 1 - P(Y \leq 10)$ or $P(Y \geq 13) = 1 - 0.973$ | M1 | Writing or using $1 - P(Y \leq 10)$ or $1 - 0.9015$ or $1 - 0.973$ |
| $= 0.0985$ or Critical region $Y \geq 13$ | A1 | awrt 0.0985 or CR: $Y \geq 13$ provided $Po(7)$ seen or used |
| There is insufficient evidence to suggest that the proportion of sunflower seeds that grow to a height of more than 3 metres is now greater than 0.028 | A1 | Independent of hypotheses but dependent on previous M1A1. A correct conclusion in context |
---\begin{enumerate}
\item A company produces packets of sunflower seeds. Each packet contains 40 seeds. The company claims that, on average, only 35\% of its sunflower seeds do not germinate.
\end{enumerate}
A packet is selected at random.\\
(a) Using a $5 \%$ level of significance, find an appropriate critical region for a two-tailed test that the proportion of sunflower seeds that do not germinate is 0.35 You should state your hypotheses clearly and state the probability, which should be as close as possible to $2.5 \%$, for each tail of your critical region.\\
(b) Write down the actual significance level of this test.
Past records suggest that $2.8 \%$ of the company's sunflower seeds grow to a height of more than 3 metres.\\
A random sample of 250 of the company's sunflower seeds is taken and 11 of them grow to a height of more than 3 metres.\\
(c) Using a suitable approximation test, at the $5 \%$ level of significance, whether or not there is evidence that the proportion of sunflower seeds that grow to a height of more than 3 metres is now greater than $2.8 \%$\\
State your hypotheses clearly.
\hfill \mbox{\textit{Edexcel S2 2022 Q3 [10]}}