A2 June 2025 Q3
3. A scientist investigated the effects of nitrogen-fixing bacteria on cereal crop yield, \(x\,\mathrm{kg\,ha^{-1}}\)
The data come from independent random samples from normal distributions.
The results are summarised in the table below.
| \(n\) | \(\bar{x}\) | \(s_x\) | |
|---|---|---|---|
| Without nitrogen-fixing bacteria | 6 | 1648 | 143.8 |
| With nitrogen-fixing bacteria | 5 | 1910 | 64.0 |
You should use a 10% level of significance and state your hypotheses and critical value clearly. (4)
A suitable test to determine whether there is an increase in mean cereal crop yield has a \(p\)-value of 0.002
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{H}_0 : \sigma_1^2 = \sigma_2^2 \qquad \mathrm{H}_1 : \sigma_1^2 \ne \sigma_2^2\) | B1 | 2.5 |
| \(F = \dfrac{143.8^2}{64.0^2} = (5.0484\ldots)\) | M1 | 3.4 |
| Critical value (5% one-tail) \(F_{5,4} = 6.26\) | B1 | 1.1b |
| (Not significant) insufficient evidence of a difference in variances | A1 | 2.2b |
| (4) |
Notes
B1: for both hypotheses correct in terms of \(\sigma\) or \(\sigma^2\)
May use labelling of e.g. 1 and 2 or e.g. “with” and “without”. Condone \(x\) and \(y\).
M1: for a correct numerical expression for \(F\) (must have squares but does not need to be evaluated) May be implied by awrt 5.05
Could be reciprocal.
B1: for 6.26 (or better)
If reciprocal (0.198…) is used as test statistic then need \(\dfrac{1}{5.19}\) or awrt 0.193
A1: dep on a critical value such that awrt \(5.05 \lt \text{cv} \lt 7\) for a correct test statistic awrt 5.05, and correct conclusion. Condone e.g. variances are the same. Do not accept conclusions which do not refer to the variances. If an incorrect comparison or statement is made A0
| Scheme | Marks | AO |
|---|---|---|
| (i) Need to assume variances are the same to carry out the test for means (the test showed that the variances could be assumed to be equal) | B1 | 2.4 |
| (ii) There is evidence it increases mean yield (but does not effect variance) | B1 | 2.2b |
| (2) | ||
| (6 marks) |
Notes
(i) B1: for recognising that the test (for difference of means) requires the populations to have the same variance oe. Must indicate that it is a requirement (not just that you can assume the variances are equal). Must be in words.
Allow explanations suggesting that it enables a pooled estimate for \(\sigma^2\) or a better estimate of \(\sigma^2\) to be found (so that a \(t\)-test can be carried out on the difference between means of two independent distributions with unknown variances)
Do not allow references to the normal distribution being used.
(ii) B1: for a correct conclusion in context mentioning increase in mean yield.
Does not require the comment about variances.