S3 January 2006 Q5
5. Upon entering a school, a random sample of eight girls and an independent random sample of eighty boys were given the same examination in mathematics. The girls and boys were then taught in separate classes. After one year, they were all given another common examination in mathematics.
The means and standard deviations of the boys’ and the girls’ marks are shown in the table.
| Examination marks | ||||
|---|---|---|---|---|
| Upon entry | After 1 year | |||
| Mean | Standard deviation | Mean | Standard deviation | |
| Boys | 50 | 12 | 59 | 6 |
| Girls | 53 | 12 | 62 | 6 |
You may assume that the test results are normally distributed.
(a) Test, at the 5% level of significance, whether or not the difference between the means of the boys’ and girls’ results was significant when they entered school. (7)
(b) Test, at the 5% level of significance, whether or not the mean mark of the boys is significantly less than the mean mark of the girls in the ‘After 1 year’ examination. (5)
(c) Interpret the results found in part (a) and part (b). (1)
| Scheme | Marks |
|---|---|
| \(\mu_b\) = mean mark of boys, \(\mu_g\) = mean mark of girls. \(\mathrm{H}_0 : \mu_b = \mu_g\) \(\mathrm{H}_1 : \mu_b \neq \mu_g\) both | B1 |
| \(z = \dfrac{53 - 50}{\sqrt{\dfrac{144}{80} + \dfrac{144}{80}}}\) | M1 A1 |
| \(= 1.58\) | A1 |
| Critical region \(z \geqslant 1.96\) | B1 |
| \(1.58 \lt 1.96\) insufficient evidence to reject \(\mathrm{H}_0\). | M1 |
| No diff. between mean scores of boys and girls. | A1 |
| (7) |
Notes
The printed scheme uses 80 for the sample size of the girls as well as the boys; the question paper says “a random sample of eight girls”.
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \mu_b = \mu_g\) \(\mathrm{H}_1 : \mu_b \lt \mu_g\) | B1 |
| \(z = \dfrac{62 - 59}{\sqrt{\dfrac{36}{80} + \dfrac{36}{80}}}\) | M1 |
| \(= 3.16\) | A1 |
| Critical region \(z \geqslant 1.6449\) (accept 1.645) | B1 |
| \(3.16 \gt 1.6449\) sufficient evidence to reject \(\mathrm{H}_0\). the mean mark for boys is less than the mean mark of the girls. | A1 |
| (5) |
| Scheme | Marks |
|---|---|
| Girls have improved more than boys or girls performed better than boys after 1 year | B1 |
| (1) | |
| (13 marks) |