| Exam Board | AQA |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2015 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | Two-sample t-test with summary statistics |
| Difficulty | Challenging +1.2 This is a multi-part hypothesis testing question requiring understanding of variance of linear combinations, one-tailed z-tests, and Type II error calculations. While it involves several steps and concepts (variance derivation, hypothesis test setup, critical value verification, power calculation), each component follows standard S3 procedures without requiring novel insight. The algebraic manipulation is straightforward, and the question structure guides students through the process systematically. It's moderately harder than average due to the multi-stage nature and Type II error component, but remains a typical exam question for this module. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{Var}(\bar{L} - 2\bar{S}) = \text{Var}(\bar{L}) + 4\text{Var}(\bar{S})\) | M1 | Attempt to find variance of linear combination |
| \(= \frac{\sigma^2}{n} + \frac{4\sigma^2}{n}\) | A1 | Both terms correct |
| \(= \frac{5\sigma^2}{n}\) | A1 | Simplified final answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0: \mu_L = 2\mu_S\), \(H_1: \mu_L > 2\mu_S\) | B1 | Both hypotheses correct |
| Under \(H_0\): \(\bar{L} - 2\bar{S} \sim N\left(0, \frac{5 \times 64}{25}\right)\) i.e. \(N(0, 12.8)\) | M1 | Using \(\sigma^2 = 64\), \(n = 25\) with their variance from (a) |
| \(z = \frac{(522 - 2 \times 258)}{\sqrt{12.8}} = \frac{6}{\sqrt{12.8}}\) | M1 | Standardising with \(\bar{l} - 2\bar{s} = 522 - 516 = 6\) |
| \(z = 1.677...\) | A1 | Correct \(z\) value |
| Critical value \(z = 1.282\) (10% one-tail) | B1 | Correct critical value stated |
| \(1.677 > 1.282\), reject \(H_0\) | M1 | Correct comparison and conclusion |
| Significant evidence that mean volume of large bottle is more than twice that of small bottle | A1 | Conclusion in context |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Critical value of \(\bar{L} - 2\bar{S}\): \(1.282 \times \sqrt{12.8} = 4.585\) | M1 | \(1.282 \times \sqrt{12.8}\) seen |
| \(= 4.585\) (to 3 d.p.) | A1 | Correctly shown/verified |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| If \(\mu_L - 2\mu_S = 10\), then \(\bar{L} - 2\bar{S} \sim N(10, 12.8)\) | M1 | New distribution with mean 10 |
| \(P(\text{Type II error}) = P(\bar{L} - 2\bar{S} < 4.585)\) | M1 | Using critical value from (ii) |
| \(= P\left(Z < \frac{4.585 - 10}{\sqrt{12.8}}\right) = P(Z < -1.513...)\) | A1 | Correct standardisation |
| \(= 1 - \Phi(1.513) = 1 - 0.9349 = 0.0651\) | A1 | Correct probability (accept awrt 0.065) |
# Question 6:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{Var}(\bar{L} - 2\bar{S}) = \text{Var}(\bar{L}) + 4\text{Var}(\bar{S})$ | M1 | Attempt to find variance of linear combination |
| $= \frac{\sigma^2}{n} + \frac{4\sigma^2}{n}$ | A1 | Both terms correct |
| $= \frac{5\sigma^2}{n}$ | A1 | Simplified final answer |
## Part (b)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: \mu_L = 2\mu_S$, $H_1: \mu_L > 2\mu_S$ | B1 | Both hypotheses correct |
| Under $H_0$: $\bar{L} - 2\bar{S} \sim N\left(0, \frac{5 \times 64}{25}\right)$ i.e. $N(0, 12.8)$ | M1 | Using $\sigma^2 = 64$, $n = 25$ with their variance from (a) |
| $z = \frac{(522 - 2 \times 258)}{\sqrt{12.8}} = \frac{6}{\sqrt{12.8}}$ | M1 | Standardising with $\bar{l} - 2\bar{s} = 522 - 516 = 6$ |
| $z = 1.677...$ | A1 | Correct $z$ value |
| Critical value $z = 1.282$ (10% one-tail) | B1 | Correct critical value stated |
| $1.677 > 1.282$, reject $H_0$ | M1 | Correct comparison and conclusion |
| Significant evidence that mean volume of large bottle is more than twice that of small bottle | A1 | Conclusion in context |
## Part (b)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Critical value of $\bar{L} - 2\bar{S}$: $1.282 \times \sqrt{12.8} = 4.585$ | M1 | $1.282 \times \sqrt{12.8}$ seen |
| $= 4.585$ (to 3 d.p.) | A1 | Correctly shown/verified |
## Part (b)(iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| If $\mu_L - 2\mu_S = 10$, then $\bar{L} - 2\bar{S} \sim N(10, 12.8)$ | M1 | New distribution with mean 10 |
| $P(\text{Type II error}) = P(\bar{L} - 2\bar{S} < 4.585)$ | M1 | Using critical value from (ii) |
| $= P\left(Z < \frac{4.585 - 10}{\sqrt{12.8}}\right) = P(Z < -1.513...)$ | A1 | Correct standardisation |
| $= 1 - \Phi(1.513) = 1 - 0.9349 = 0.0651$ | A1 | Correct probability (accept awrt 0.065) |6
\begin{enumerate}[label=(\alph*)]
\item The independent random variables $S$ and $L$ have means $\mu _ { S }$ and $\mu _ { L }$ respectively, and a common variance of $\sigma ^ { 2 }$.
The variable $\bar { S }$ denotes the mean of a random sample of $n$ observations on $S$ and the variable $\bar { L }$ denotes the mean of a random sample of $n$ observations on $L$.
Find a simplified expression, in terms of $\sigma ^ { 2 }$, for the variance of $\bar { L } - 2 \bar { S }$.
\item A machine fills both small bottles and large bottles with shower gel. It is known that the volume of shower gel delivered by the machine is normally distributed with a standard deviation of 8 ml .
\begin{enumerate}[label=(\roman*)]
\item A random sample of 25 small bottles filled by the machine contained a mean volume of $\bar { s } = 258 \mathrm { ml }$ of shower gel.
An independent random sample of 25 large bottles filled by the machine contained a mean volume of $\bar { l } = 522 \mathrm { ml }$ of shower gel.
Investigate, at the $10 \%$ level of significance, the hypothesis that the mean volume of shower gel in a large bottle is more than twice that in a small bottle.\\[0pt]
[7 marks]
\item Deduce that, for the test of the hypothesis in part (b)(i), the critical value of $\bar { L } - 2 \bar { S }$ is 4.585 , correct to three decimal places.\\[0pt]
[2 marks]
\item In fact, the mean volume of shower gel in a large bottle exceeds twice that in a small bottle by 10 ml .
Determine the probability of a Type II error for a test of the hypothesis in part (b)(i) at the 10\% level of significance, based upon random samples of 25 small bottles and 25 large bottles.\\[0pt]
[4 marks]
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S3 2015 Q6 [16]}}