| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2008 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Unknown variance confidence intervals |
| Difficulty | Standard +0.3 This is a standard two-sample z-test with large samples (n=100, 200), requiring routine application of hypothesis testing procedures. The central limit theorem part (b) is straightforward recall. Slightly above average difficulty due to being a two-sample test rather than one-sample, but still a textbook S3 question with no novel insight required. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| \(n\) | mean | s.d. | |
| Female teenagers | 100 | \(\pounds 5.48\) | \(\pounds 3.62\) |
| Male teenagers | 200 | \(\pounds 6.86\) | \(\pounds 4.51\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \mu_F = \mu_M\) | B1 | |
| \(H_1: \mu_F \neq \mu_M\) (Allow \(\mu_1\) and \(\mu_2\)) | ||
| \(z = \frac{6.86 - 5.48}{\sqrt{\frac{4.51^2}{200} + \frac{3.62^2}{100}}}\) | M1 A1 | |
| \(= 2.860...\) | A1 | awrt \((\pm)2.86\) |
| 2 tail 5% critical value \((\pm) 1.96\) | B1 | (or probability awrt 0.0021–0.0022) |
| Significant result or reject the null hypothesis (o.e.) | M1 | |
| There is evidence of a difference in the (mean) amount spent on junk food by male and female teenagers | A1ft | (7 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| CLT enables us to assume \(\bar{F}\) and \(\bar{M}\) are normally distributed | B1 | (1 mark) |
**Part (a)**
$H_0: \mu_F = \mu_M$ | B1 |
$H_1: \mu_F \neq \mu_M$ (Allow $\mu_1$ and $\mu_2$) | |
$z = \frac{6.86 - 5.48}{\sqrt{\frac{4.51^2}{200} + \frac{3.62^2}{100}}}$ | M1 A1 |
$= 2.860...$ | A1 | awrt $(\pm)2.86$
2 tail 5% critical value $(\pm) 1.96$ | B1 | (or probability awrt 0.0021–0.0022)
Significant result or reject the null hypothesis (o.e.) | M1 |
There is evidence of a difference in the (mean) amount spent on junk food by male and female teenagers | A1ft | (7 marks)
**Part (b)**
CLT enables us to assume $\bar{F}$ and $\bar{M}$ are normally distributed | B1 | (1 mark)
**Total: 8 marks**
**Guidance Notes:**
(a) 1st M1 for an attempt at $\frac{a-b}{\sqrt{\frac{c}{100 \text{ or } 200} + \frac{d}{100 \text{ or } 200}}}$ with 3 of $a, b, c$ or $d$ correct
1st A1 for a fully correct expression
2nd B1 for $\pm 1.96$ but only if their $H_1$ is two-tail (it may be in words so B0B1 is OK)
If $H_1$ is one-tail this is automatically B0 too.
2nd M1 for a correct statement based on comparison of their $z$ with their cv. May be implied
3rd A1 for a correct conclusion in context based on their $z$ and 1.96.
Must mention junk food or money and male vs female.
(b) B1 for $\bar{F}$ or $\bar{M}$ mentioned. Allow "mean (amount spent on junk food) is normally distributed"
Read the whole statement e.g. "original distribution is normal so mean is..." scores B0
---\begin{enumerate}
\item A sociologist is studying how much junk food teenagers eat. A random sample of 100 female teenagers and an independent random sample of 200 male teenagers were asked to estimate what their weekly expenditure on junk food was. The results are summarised below.
\end{enumerate}
\begin{center}
\begin{tabular}{ | l | c | c | c | }
\hline
& $n$ & mean & s.d. \\
\hline
Female teenagers & 100 & $\pounds 5.48$ & $\pounds 3.62$ \\
\hline
Male teenagers & 200 & $\pounds 6.86$ & $\pounds 4.51$ \\
\hline
\end{tabular}
\end{center}
(a) Using a 5\% significance level, test whether or not there is a difference in the mean amounts spent on junk food by male teenagers and female teenagers. State your hypotheses clearly.\\
(b) Explain briefly the importance of the central limit theorem in this problem.\\
\hfill \mbox{\textit{Edexcel S3 2008 Q7 [8]}}