| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Two-sample z-test large samples |
| Difficulty | Standard +0.3 This is a straightforward two-sample hypothesis test with standard calculations. Part (a) requires routine application of unbiased estimator formulas (mean and variance from summary statistics), while part (b) is a standard z-test for difference in means with clearly defined steps. The question is slightly easier than average because all necessary values are provided or easily calculated, requiring no conceptual insight beyond textbook procedures for S3 level. |
| Spec | 5.02b Expectation and variance: discrete random variables5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Part (a): \(\hat{\mu} = \bar{V} = \frac{10367}{80} = 129.6\) cm | M1 A1 | |
| \(\hat{\sigma}^2 = s^2 = \frac{80}{79}(\frac{1350314}{80} - 129.5875^2) = 87.09\) | M2 A1 | |
| Part (b): \(H_0: \mu_v = \mu_M\); \(H_1: \mu_v \neq \mu_M\) | B1 | |
| 1% level \(\therefore\) C.R. is \(z < -2.5758\) or \(z > 2.5758\) | B1 | |
| test statistic \(= \frac{129.6-130.5}{\sqrt{\frac{87.09}{80} + \frac{90.24}{80}}} = -0.7520\) | M2 A2 | |
| not in C.R. do not reject \(H_0\); no evidence of difference in mean heights | M1 A1 | (13 marks total) |
**Part (a):** $\hat{\mu} = \bar{V} = \frac{10367}{80} = 129.6$ cm | M1 A1 |
$\hat{\sigma}^2 = s^2 = \frac{80}{79}(\frac{1350314}{80} - 129.5875^2) = 87.09$ | M2 A1 |
**Part (b):** $H_0: \mu_v = \mu_M$; $H_1: \mu_v \neq \mu_M$ | B1 |
1% level $\therefore$ C.R. is $z < -2.5758$ or $z > 2.5758$ | B1 |
test statistic $= \frac{129.6-130.5}{\sqrt{\frac{87.09}{80} + \frac{90.24}{80}}} = -0.7520$ | M2 A2 |
not in C.R. do not reject $H_0$; no evidence of difference in mean heights | M1 A1 | (13 marks total)
---A researcher collects data on the height of boys aged between nine and nine and-a-half years and their diet.
The data on the height, $V$ cm, of the 80 boys who had always eaten a vegetarian diet is summarised by
$$\Sigma V = 10\,367, \quad \Sigma V^2 = 1\,350\,314.$$
\begin{enumerate}[label=(\alph*)]
\item Calculate unbiased estimates of the mean and variance of $V$. [5]
\end{enumerate}
The researcher calculates unbiased estimates of the mean and variance of the height of boys whose diet has included meat from a sample of size 280, giving values of 130.5 cm and 96.24 cm$^2$ respectively.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Stating your hypotheses clearly, test at the 1% level whether or not there is a significant difference in the heights of boys of this age according to whether or not they have a vegetarian diet. [8]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 Q6 [13]}}