| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2008 |
| Session | January |
| Marks | 6 |
| 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 straightforward application of standard two-sample inference procedures with large samples. Part (i) requires simple variance formula substitution, part (ii) is routine confidence interval calculation, and part (iii) tests understanding of when normal approximation applies. The large sample sizes (80, 100) and given summary statistics make this easier than typical hypothesis testing questions, requiring no novel insight or complex multi-step reasoning. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.05d Confidence intervals: using normal distribution |
| Sample size | \(\bar { x }\) | \(s ^ { 2 }\) | |
| Previous season's crop \(( P )\) | 80 | 1.24 | 0.00356 |
| New crop \(( N )\) | 100 | 1.36 | 0.00340 |
1 A blueberry farmer increased the amount of water sprayed over his berries to see what effect this had on their weight. The farmer weighed each of a random sample of 80 berries of the previous season's crop and each of a random sample of 100 berries of the new crop. The results are summarised in the following table, in which $\bar { x }$ denotes the sample mean weight in grams, and $s ^ { 2 }$ denotes an unbiased estimate of the relevant population variance.
\begin{center}
\begin{tabular}{ | l | c | c | c | }
\hline
& Sample size & $\bar { x }$ & $s ^ { 2 }$ \\
\hline
Previous season's crop $( P )$ & 80 & 1.24 & 0.00356 \\
\hline
New crop $( N )$ & 100 & 1.36 & 0.00340 \\
\hline
\end{tabular}
\end{center}
(i) Calculate an estimate of $\operatorname { Var } \left( \bar { X } _ { N } - \bar { X } _ { P } \right)$.\\
(ii) Calculate a $95 \%$ confidence interval for the difference in population mean weights.\\
(iii) Give a reason why it is unnecessary to use a $t$-distribution in calculating the confidence interval.
\hfill \mbox{\textit{OCR S3 2008 Q1 [6]}}