| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2014 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample t-test |
| Difficulty | Standard +0.3 This is a straightforward one-sample t-test with clearly stated hypotheses (H₀: μ = 100, H₁: μ < 100), given summary statistics, and standard significance level. Students need to calculate the test statistic, find the critical value from t-tables with 7 df, and make a conclusion. It's slightly above average difficulty due to being S4 content and requiring proper hypothesis test structure, but it's a textbook application with no complications or novel elements. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0: \mu = 100\), \(H_1: \mu < 100\) | B1 | Both hypotheses correct |
| \(t = \frac{92.875 - 100}{\frac{8.3055}{\sqrt{8}}}\) | M1 | Correct structure of test statistic |
| \(t = \frac{-7.125}{2.9369...} = -2.426\) | A1 | Correct value (allow -2.43) |
| Critical value: \(t_7 = -1.895\) (one-tailed, 5%) | B1 | Correct critical value |
| Since \(-2.426 < -1.895\), reject \(H_0\). There is sufficient evidence to support Malcolm's belief that the mean is less than 100 ml | A1 | Correct conclusion in context |
# Question 1:
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: \mu = 100$, $H_1: \mu < 100$ | B1 | Both hypotheses correct |
| $t = \frac{92.875 - 100}{\frac{8.3055}{\sqrt{8}}}$ | M1 | Correct structure of test statistic |
| $t = \frac{-7.125}{2.9369...} = -2.426$ | A1 | Correct value (allow -2.43) |
| Critical value: $t_7 = -1.895$ (one-tailed, 5%) | B1 | Correct critical value |
| Since $-2.426 < -1.895$, reject $H_0$. There is sufficient evidence to support Malcolm's belief that the mean is less than 100 ml | A1 | Correct conclusion in context |
---\begin{enumerate}
\item A production line is designed to fill bottles with oil. The amount of oil placed in a bottle is normally distributed and the mean is set to 100 ml .
\end{enumerate}
The amount of oil, $x \mathrm { ml }$, in each of 8 randomly selected bottles is recorded, and the following statistics are obtained.
$$\bar { x } = 92.875 \quad s = 8.3055$$
Malcolm believes that the mean amount of oil placed in a bottle is less than 100 ml .\\
Stating your hypotheses clearly, test, at the $5 \%$ significance level, whether or not Malcolm's belief is supported.\\
\hfill \mbox{\textit{Edexcel S4 2014 Q1 [5]}}