| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2018 |
| Session | Specimen |
| 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 t-test with clearly structured parts. Part (a) requires standard formulas for unbiased estimates (mean and variance), part (b) is a routine hypothesis test with given summary statistics, and parts (c)-(d) test understanding of assumptions. All steps are procedural with no novel insight required, making it slightly easier than average for S3 level. |
| Spec | 2.01a Population and sample: terminology5.05b Unbiased estimates: of population mean and variance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Calculation of at least 3 widths and probabilities \(P(a \leq X < b)\): \(\frac{1}{12}, \frac{1}{12}, \frac{1}{6}, \frac{1}{6}, \frac{1}{4}, \frac{1}{4}\) | M1 | For 1:2:3 ratio seen |
| Correct probabilities | A1 | |
| Expected frequencies: 19, 19, 38, 38, 57, 57 all correct | A1 | |
| \(H_0\): continuous uniform distribution is a good fit \(H_1\): continuous uniform distribution is not a good fit | B1 | |
| At least 3 correct \(\frac{(O-E)^2}{E}\) or \(\frac{O^2}{E}\) expressions | M1 | Follow through their \(E_i\) provided not all \(= 38\) |
| \(\sum \frac{(O_i - E_i)^2}{E_i} = \frac{313}{114} = 2.75\) awrt 2.75 | dM1A1 | Dependent on 2nd M1; correct sum (must see at least 3 terms and \(+\)) |
| \(\nu = 6 - 1 = 5\) | B1 | |
| \(\chi^2_5(0.05) = 11.070\) | B1ft | ft their \(\nu\) |
| \(2.75 < 11.070\), insufficient evidence to reject \(H_0\) | M1 | Correct statement (test stat \(> 1\), cv \(> 3.8\)); contradictory statements score M0 |
| Continuous uniform distribution is a suitable model | A1 | Correct contextualised comment; no ft |
# Question 6:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Calculation of at least 3 widths and probabilities $P(a \leq X < b)$: $\frac{1}{12}, \frac{1}{12}, \frac{1}{6}, \frac{1}{6}, \frac{1}{4}, \frac{1}{4}$ | M1 | For 1:2:3 ratio seen |
| Correct probabilities | A1 | |
| Expected frequencies: 19, 19, 38, 38, 57, 57 all correct | A1 | |
| $H_0$: continuous uniform distribution is a good fit $H_1$: continuous uniform distribution is not a good fit | B1 | |
| At least 3 correct $\frac{(O-E)^2}{E}$ or $\frac{O^2}{E}$ expressions | M1 | Follow through their $E_i$ provided not all $= 38$ |
| $\sum \frac{(O_i - E_i)^2}{E_i} = \frac{313}{114} = 2.75$ awrt **2.75** | dM1A1 | Dependent on 2nd M1; correct sum (must see at least 3 terms and $+$) |
| $\nu = 6 - 1 = 5$ | B1 | |
| $\chi^2_5(0.05) = 11.070$ | B1ft | ft their $\nu$ |
| $2.75 < 11.070$, insufficient evidence to reject $H_0$ | M1 | Correct statement (test stat $> 1$, cv $> 3.8$); contradictory statements score M0 |
| Continuous uniform distribution is a suitable model | A1 | Correct contextualised comment; no ft |
---6. As part of an investigation, a random sample was taken of 50 footballers who had completed an obstacle course in the early morning. The time taken by each of these footballers to complete the obstacle course, $x$ minutes, was recorded and the results are summarised by
$$\sum x = 1570 \quad \text { and } \quad \sum x ^ { 2 } = 49467.58$$
\begin{enumerate}[label=(\alph*)]
\item Find unbiased estimates for the mean and variance of the time taken by footballers to complete the obstacle course in the early morning.
An independent random sample was taken of 50 footballers who had completed the same obstacle course in the late afternoon. The time taken by each of these footballers to complete the obstacle course, $y$ minutes, was recorded and the results are summarised as
$$\bar { y } = 30.9 \quad \text { and } \quad s _ { y } ^ { 2 } = 3.03$$
\item Test, at the $5 \%$ level of significance, whether or not the mean time taken by footballers to complete the obstacle course in the early morning, is greater than the mean time taken by footballers to complete the obstacle course in the late afternoon. State your hypotheses clearly.
\item Explain the relevance of the Central Limit Theorem to the test in part (b).
\item State an assumption you have made in carrying out the test in part (b).
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\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2018 Q6 [13]}}