| Exam Board | AQA |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Two-sample t-test (unknown variances) |
| Difficulty | Standard +0.3 This is a standard two-sample t-test confidence interval question with all necessary information provided in a clear table format. Students need to apply the formula for difference of means with unequal variances (pooled or unpooled), look up a critical value, and interpret the result. While it requires careful calculation and understanding of confidence intervals, it's a routine textbook exercise with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| \cline { 3 - 5 } \multicolumn{2}{c|}{} | Weight of raspberries in a punnet (grams) | |||
| \cline { 3 - 5 } \multicolumn{2}{c|}{} | Sample size | Sample mean | Sample standard deviation, \(\boldsymbol { s }\) | |
| \multirow{2}{*}{Stallholder} | Rodney | 50 | 225 | 5 |
| \cline { 2 - 5 } | Derrick | 75 | 219 | 8 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\bar{x}_R - \bar{x}_D = 225 - 219 = 6\) | B1 | Correct difference of means |
| \(SE = \sqrt{\frac{s_R^2}{n_R} + \frac{s_D^2}{n_D}} = \sqrt{\frac{25}{50} + \frac{64}{75}}\) | M1 | Correct structure of standard error |
| \(= \sqrt{0.5 + 0.8533...} = \sqrt{1.3533} = 1.1633...\) | A1 | Correct value |
| \(z = 2.576\) for 99% CI | B1 | Correct \(z\) value |
| \(6 \pm 2.576 \times 1.1633 = 6 \pm 2.997...\) | M1 | Correct structure of CI |
| \((3.00, 9.00)\) or \((3.003, 8.997)\) | A1 | Correct interval (allow awrt) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Interval does not contain 0 | M1 | |
| Evidence that Rodney's punnets are heavier on average than Derrick's (mean weight significantly greater) | A1 | Conclusion in context |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Any valid factor e.g. price / quality / appearance / location of stall | B1 | Accept any reasonable factor |
## Question 2:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\bar{x}_R - \bar{x}_D = 225 - 219 = 6$ | B1 | Correct difference of means |
| $SE = \sqrt{\frac{s_R^2}{n_R} + \frac{s_D^2}{n_D}} = \sqrt{\frac{25}{50} + \frac{64}{75}}$ | M1 | Correct structure of standard error |
| $= \sqrt{0.5 + 0.8533...} = \sqrt{1.3533} = 1.1633...$ | A1 | Correct value |
| $z = 2.576$ for 99% CI | B1 | Correct $z$ value |
| $6 \pm 2.576 \times 1.1633 = 6 \pm 2.997...$ | M1 | Correct structure of CI |
| $(3.00, 9.00)$ or $(3.003, 8.997)$ | A1 | Correct interval (allow awrt) |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Interval does not contain 0 | M1 | |
| Evidence that Rodney's punnets are heavier on average than Derrick's (mean weight significantly greater) | A1 | Conclusion in context |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Any valid factor e.g. price / quality / appearance / location of stall | B1 | Accept any reasonable factor |
---2 Rodney and Derrick, two independent fruit and vegetable market stallholders, sell punnets of locally-grown raspberries from their stalls during June and July.
The following information, based on independent random samples, was collected as part of an investigation by Trading Standards Officers.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\cline { 3 - 5 }
\multicolumn{2}{c|}{} & \multicolumn{2}{c|}{Weight of raspberries in a punnet (grams)} & \\
\cline { 3 - 5 }
\multicolumn{2}{c|}{} & Sample size & Sample mean & Sample standard deviation, $\boldsymbol { s }$ \\
\hline
\multirow{2}{*}{Stallholder} & Rodney & 50 & 225 & 5 \\
\cline { 2 - 5 }
& Derrick & 75 & 219 & 8 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Construct a $99 \%$ confidence interval for the difference between the mean weight of raspberries in a punnet sold by Rodney and the mean weight of raspberries in a punnet sold by Derrick.
\item What can be concluded from your confidence interval?
\item In addition to weight, state one other factor that may influence whether customers buy raspberries from Rodney or from Derrick.
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{b855b5b3-097e-4894-aaec-d77f515949b0-05_2484_1709_223_153}
\end{center}
\begin{center}
\begin{tabular}{|l|l|}
\hline
& \begin{tabular}{l}
\end{enumerate}
\end{tabular}
\end{tabular}
\end{center}
\hfill \mbox{\textit{AQA S3 2010 Q2 [8]}}