Standard +0.3 This is a straightforward hypothesis testing question requiring students to find the critical region for a two-tailed binomial test and calculate the significance level. While it involves multiple steps (finding critical values in both tails, calculating probabilities using binomial distribution), these are standard procedures taught explicitly in Further Maths Statistics. The question guides students by stating what to show, making it a verification exercise rather than requiring problem-solving insight. It's slightly above average difficulty due to the two-tailed nature and need for careful probability calculations, but remains a routine textbook-style question.
4 The proportion, \(p\), of people in a particular town who use the local supermarket is unknown.
A random sample of 30 people in the town is taken and each person is asked if they use the local supermarket.
The manager of the supermarket claims that 35\% of the people in the town use the local supermarket.
The random sample is used to conduct a hypothesis test at the \(5 \%\) level of significance with the hypotheses
$$\begin{aligned}
& \mathrm { H } _ { 0 } : p = 0.35 \\
& \mathrm { H } _ { 1 } : p \neq 0.35
\end{aligned}$$
Show that the probability that a Type I error is made is 0.0356 , correct to four decimal places.
AWRT 0.023; and correct value or calculation of \(P(X \geq 17)\) AWRT 0.012
\(P(X \leq 15) = 0.96992\)
—
—
\(P(X \geq 16) = 1 - P(X \leq 15) = 1 - 0.96992\)
—
—
\(P(X \geq 16) = 0.03008 > 0.025\)
M1
Adds probability of lower tail to probability of upper tail
\(P(X \leq 16) = 0.98764\)
—
—
\(P(X \geq 17) = 1 - P(X \leq 16) = 1 - 0.98764\)
—
—
\(P(X \geq 17) = 0.01236 < 0.025\)
—
—
Probability of Type I error \(= 0.02326 + 0.01236 = 0.03562 = 0.0356\) (4 dp)
R1
Correct values rounded/truncated of \(P(X \leq 5)\), \(P(X \leq 6)\), \(P(X \leq 15)\), \(P(X \leq 16)\), \(P(X \geq 16)\), \(P(X \geq 17)\) all needed; probabilities in sum correct to at least 5 decimal places
## Question 4:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $X \sim B(30, 0.35)$ | M1 | Uses Binomial model to calculate any probability |
| $P(X \leq 6) = 0.05857 > 0.025$ | — | — |
| $P(X \leq 5) = 0.02326 < 0.025$ | A1 | AWRT 0.023; and correct value or calculation of $P(X \geq 17)$ AWRT 0.012 |
| $P(X \leq 15) = 0.96992$ | — | — |
| $P(X \geq 16) = 1 - P(X \leq 15) = 1 - 0.96992$ | — | — |
| $P(X \geq 16) = 0.03008 > 0.025$ | M1 | Adds probability of lower tail to probability of upper tail |
| $P(X \leq 16) = 0.98764$ | — | — |
| $P(X \geq 17) = 1 - P(X \leq 16) = 1 - 0.98764$ | — | — |
| $P(X \geq 17) = 0.01236 < 0.025$ | — | — |
| Probability of Type I error $= 0.02326 + 0.01236 = 0.03562 = 0.0356$ (4 dp) | R1 | Correct values rounded/truncated of $P(X \leq 5)$, $P(X \leq 6)$, $P(X \leq 15)$, $P(X \leq 16)$, $P(X \geq 16)$, $P(X \geq 17)$ all needed; probabilities in sum correct to at least 5 decimal places |
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4 The proportion, $p$, of people in a particular town who use the local supermarket is unknown.
A random sample of 30 people in the town is taken and each person is asked if they use the local supermarket.
The manager of the supermarket claims that 35\% of the people in the town use the local supermarket.
The random sample is used to conduct a hypothesis test at the $5 \%$ level of significance with the hypotheses
$$\begin{aligned}
& \mathrm { H } _ { 0 } : p = 0.35 \\
& \mathrm { H } _ { 1 } : p \neq 0.35
\end{aligned}$$
Show that the probability that a Type I error is made is 0.0356 , correct to four decimal places.\\
\hfill \mbox{\textit{AQA Further AS Paper 2 Statistics 2023 Q4 [4]}}