AS June 2022 Q1
1. Stuart is investigating a treatment for a disease that affects fruit trees. He has 400 fruit trees and applies the treatment to a random sample of these trees. The remainder of the trees have no treatment. He records the number of years, \(y\), that each fruit tree remains free from this disease.
The results are summarised in the table below.
| Treatment | |||
|---|---|---|---|
| Applied | Not applied | ||
| Number of years free from this disease | \(y \lt 1\) | 15 | 25 |
| \(1 \leqslant y \lt 2\) | 35 | 61 | |
| \(2 \leqslant y\) | 124 | 140 | |
The data are to be used to determine whether or not there is an association between the application of the treatment and the number of years that a fruit tree remains free from this disease.
The value of \(\displaystyle\sum \frac{(O-E)^2}{E}\) for the other four classes is 2.642 to 3 decimal places.
You should state your hypotheses, test statistic, critical value and conclusion clearly. (5)
| Scheme | Marks | AO |
|---|---|---|
| (i) \(\dfrac{40 \times 174}{400}\) (ii) \(\dfrac{96 \times 226}{400}\) | M1 | 1.1b |
| \(= 17.4\) \(= 54.24\) | A1 | 1.1b |
| (2) |
Notes
M1 A correct method to work out either expected frequencies – or 1 correct
A1 17.4 and 54.24 (accept 54.2)
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{H}_0\): There is no association between the application of the treatment and the number of years that a fruit tree remains free from this disease. \(\mathrm{H}_1\): There is an association between the application of the treatment and the number of years that a fruit tree remains free from this disease. | B1 | 3.4 |
| \(\displaystyle\sum \frac{(O-E)^2}{E} = \frac{(15 - \text{“}17.4\text{”})^2}{\text{“}17.4\text{”}} + \frac{(61 - \text{“}54.24\text{”})^2}{\text{“}54.24\text{”}} + 2.642\) | M1 | 1.1b |
| \(= 3.815\ldots\) awrt 3.82 | A1 | 1.1b |
| [\(3.82 \lt\)] \(\chi^2_{2,(0.05)} = 5.991\) | B1 | 3.1b |
| There is no evidence of association between the application of the treatment and the number of years that a fruit tree remains free from this disease. | A1ft | 2.2b |
| (5) | ||
| (7 marks) |
Notes
B1: For both hypotheses in terms of "association" or independence" Must mention application/treatment and years in at least one and be connected correctly to \(\mathrm{H}_0\) and \(\mathrm{H}_1\)
[Use of link, relationship or connection. is B0 but allow for last A1ft]
M1: A correct method to find the total \(\chi^2\) value. ft their values from (a)
If no method shown at least 1 of the two missing \(\chi^2\) contributions must be correct
(\(0.331\ldots \left(\dfrac{48}{145}\right)\) and 0.8425… allow 2sf). Implied by awrt 3.82
A1: awrt 3.82 or awrt 3.83
B1: Using the degrees of freedom to find the \(\chi^2\) CV for the appropriate model. awrt 5.991 allow 5.9915
A1ft: Ft "their 3.82" and their CV or p-value. Correct conclusion in context. (application or treatment and years) This is independent of hypotheses ie if they should accept \(\mathrm{H}_0\) then they need eg there is no association between …. If they should reject \(\mathrm{H}_0\) then they need there is an association"… Allow relationship, link, connection for association BUT do not accept correlation or contradictory statements
NB If p-value [0.148388] given instead of CV could get B1M1A1B0A1 unless they give the CV as well