S3 June 2014 Q3
3. A number of males and females were asked to rate their happiness under the headings “not happy”, “fairly happy” and “very happy”.
The results are shown in the table below
| Happiness | Total | ||||
|---|---|---|---|---|---|
| Not happy | Fairly happy | Very happy | |||
| Gender | Female | 9 | 43 | 34 | 86 |
| Male | 13 | 25 | 16 | 54 | |
| Total | 22 | 68 | 50 | 140 | |
Stating your hypotheses, test at the 5% level of significance, whether or not there is evidence of an association between happiness and gender. Show your working clearly. (10)
| Scheme | Marks | ||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| M1 A1 | ||||||||||||||||||||||||||||
| \(\mathrm{H_0}\): Happiness and gender are independent/ not associated \(\mathrm{H_1}\): Happiness and gender are not independent/ associated | B1 | ||||||||||||||||||||||||||||
| dM1 A1 | ||||||||||||||||||||||||||||
| \(\displaystyle\sum \frac{(O - E)^2}{E} = 4.91\) or \(\displaystyle\sum \frac{O^2}{E} - N = 144.91 - 140 = 4.91\) | A1 | ||||||||||||||||||||||||||||
| \(\nu = (3 - 1)(2 - 1) = 2\) | B1 | ||||||||||||||||||||||||||||
| \(\displaystyle\sum \frac{(O - E)^2}{E} \lt 5.991\) | B1ft | ||||||||||||||||||||||||||||
| 4.91 < 5.991 so ‘insufficient evidence to reject \(\mathrm{H_0}\)’ or ‘Accept \(\mathrm{H_0}\)’ | M1 | ||||||||||||||||||||||||||||
| No association between gender and happiness. | A1 | ||||||||||||||||||||||||||||
| (10 marks) | |||||||||||||||||||||||||||||
Notes
(Corrected from the printed mark scheme: the degrees of freedom line is printed as \(\nu = (3 - 2)(2 - 1) = 2\).)
1st M1 for some use of \(\dfrac{\text{Row Total} \times \text{Column Total}}{\text{Grand Total}}\). May be implied by at least 1 correct \(E_i\)
1st A1 awrt 13.5, 41.8, 30.7, 8.5, 26.2 and 19.3 Allow M1A0 for \(E_i\) rounded to integers
1st B1 for both hypotheses. Must mention “happiness” and “gender” at least once.
Use of “relationship” or “correlation” or “connection” is B0
2nd dM1 for at least 2 correct terms (in 3rd or 4th columns) or correct expressions with their \(E_i\)
Dependent on 1st M1. Accept 2sf accuracy for the M mark.
2nd A1 for all correct terms (2sf or better). May be implied by a correct ans
Allow truncation e.g. 44.2…
3rd A1 awrt 4.91 . Condone 4.915
2nd B1 for correct degrees of freedom (may be implied by a cv of 5.991)
3rd B1ft for cv that follows from their degrees of freedom
3rd M1 for a correct statement linking their test statistic and their cv
Contradictory statements score M0 e.g. “significant, do not reject \(\mathrm{H_0}\)”
Condone “reject \(\mathrm{H_1}\)”
4th A1 for a correct comment in context - must mention “gender” and “happiness”
Condone “relationship” or “connection” here but not “correlation”.
e.g. “There is no evidence of a relationship between gender and happiness”
No follow through. If e.g hypotheses are the wrong way around A0 here.
SC Use of calculator with no working may get M0A0B1M1A0A1B1B1M1A1