| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2014 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Two-sample t-test equal variance |
| Difficulty | Standard +0.8 This S4 question requires conducting both an F-test for variance equality and a pooled two-sample t-test, interpreting critical values from tables, and explaining the connection between tests. While the calculations are systematic, it demands understanding of multiple hypothesis testing procedures, correct use of degrees of freedom, and conceptual insight into why variance homogeneity matters for the t-test—going beyond routine single-test questions. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Breed | Sample size | Mean \(( \overline { \boldsymbol { x } } )\) | Standard deviation \(\left( \boldsymbol { s } _ { \boldsymbol { x } } \right)\) |
| \(A\) | 9 | 6.23 | 2.98 |
| \(B\) | 12 | 7.13 | 2.33 |
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Please provide the extracted mark scheme content and I'll clean it up according to your specifications (converting unicode to LaTeX math notation, preserving marking annotations, formatting one point per line).3. A farmer is investigating the milk yields of two breeds of cow. He takes a random sample of 9 cows of breed $A$ and an independent random sample of 12 cows of breed $B$. For a 5 day period he measures the amount of milk, $x$ gallons, produced by each cow. The results are summarised in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
Breed & Sample size & Mean $( \overline { \boldsymbol { x } } )$ & Standard deviation $\left( \boldsymbol { s } _ { \boldsymbol { x } } \right)$ \\
\hline
$A$ & 9 & 6.23 & 2.98 \\
\hline
$B$ & 12 & 7.13 & 2.33 \\
\hline
\end{tabular}
\end{center}
The amount of milk produced by each cow can be assumed to follow a normal distribution.
\begin{enumerate}[label=(\alph*)]
\item Use a two-tail test to show, at the $10 \%$ level of significance, that the variances of the yields of the two breeds can be assumed to be equal. State your hypotheses clearly.
\item Stating your hypotheses clearly, test, at the $5 \%$ level of significance, whether or not there is a difference in the mean yields of the two breeds of cow.
\item Explain briefly the importance of the test in part (a) for the test in part (b).
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2014 Q3 [12]}}