| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2016 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Two-sample z-test large samples |
| Difficulty | Challenging +1.2 This is a standard two-sample t-test question requiring calculation of a test statistic and critical value comparison (part i), followed by working backwards to find a sample size (part ii). While part (ii) requires algebraic manipulation and solving an inequality, the techniques are routine for S3 level with no novel conceptual insight needed. The non-zero null hypothesis (0.04 kg difference) adds minor complexity but is a standard variation. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Year | No. of babies |
|
| ||||
| 1970 | 285 | 3.303 | 0.2043 | ||||
| 2010 | 260 | 3.352 | 0.2323 |
6 The masses at birth, in kg, of random samples of babies were recorded for each of the years 1970 and 2010. The table shows the sample mean and an unbiased estimate of the population variance for each year.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
Year & No. of babies & \begin{tabular}{ c }
Sample \\
mean \\
\end{tabular} & \begin{tabular}{ c }
Unbiased estimate of \\
population variance \\
\end{tabular} \\
\hline
1970 & 285 & 3.303 & 0.2043 \\
\hline
2010 & 260 & 3.352 & 0.2323 \\
\hline
\end{tabular}
\end{center}
(i) A researcher tests the null hypothesis that babies born in 2010 are 0.04 kg heavier, on average, than babies born in 1970, against the alternative hypothesis that they are more than 0.04 kg heavier on average. Show that, at the $5 \%$ level of significance, the null hypothesis is not rejected.\\
(ii) Another researcher chooses samples of equal size, $n$, for the two years. Using the same hypothesis as in part (i), she finds that the null hypothesis is rejected at the $5 \%$ level of significance. Assuming that the sample means and unbiased estimates of population variance for the two years are as given in the table, find the smallest possible value of $n$.
\hfill \mbox{\textit{OCR S3 2016 Q6 [11]}}