| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2020 |
| Session | October |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Two-sample t-test (unknown variances) |
| Difficulty | Standard +0.3 This is a standard two-sample t-test with routine calculations: computing unbiased variance estimates from summary statistics, performing a hypothesis test with clearly stated hypotheses, and explaining the large-sample approximation. All steps are textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 5.05b Unbiased estimates: of population mean and variance5.05c Hypothesis test: normal distribution for population mean |
| \(n\) | \(\bar { x }\) | \(\sum x ^ { 2 }\) | |
| Type \(A\) oranges | 40 | 140.4 | 790258 |
| Type \(B\) oranges | 32 | 134.7 | 581430 |
5. A greengrocer is investigating the weights of two types of orange, type $A$ and type $B$. She believes that on average type $A$ oranges weigh greater than 5 grams more than type $B$ oranges. She collects a random sample of 40 type $A$ oranges and 32 type $B$ oranges and records the weight, $x$ grams, of each orange.
The table shows a summary of her data.
\begin{center}
\begin{tabular}{ | l | c | c | c | }
\hline
& $n$ & $\bar { x }$ & $\sum x ^ { 2 }$ \\
\hline
Type $A$ oranges & 40 & 140.4 & 790258 \\
\hline
Type $B$ oranges & 32 & 134.7 & 581430 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate unbiased estimates for the variance of the weights of the population of type $A$ oranges and the variance of the weights of the population of type $B$ oranges.
\item Test, at the $5 \%$ level of significance, the greengrocer's belief. You should state the hypotheses and the critical value used for this test.
\item Explain how you have used the fact that the sample sizes are large in your answer to part (b).
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2020 Q5 [12]}}