S3 June 2008 Q7
7. A sociologist is studying how much junk food teenagers eat. A random sample of 100 female teenagers and an independent random sample of 200 male teenagers were asked to estimate what their weekly expenditure on junk food was. The results are summarised below.
| \(n\) | mean | s.d. | |
|---|---|---|---|
| Female teenagers | 100 | £5.48 | £3.62 |
| Male teenagers | 200 | £6.86 | £4.51 |
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \mu_F = \mu_M \qquad \mathrm{H}_1 : \mu_F \neq \mu_M\) (Allow \(\mu_1\) and \(\mu_2\)) | B1 |
| \(z = \dfrac{6.86 - 5.48}{\sqrt{\dfrac{4.51^2}{200} + \dfrac{3.62^2}{100}}}\) | M1 A1 |
| \(= 2.860\ldots\) awrt (\(\pm\))2.86 | A1 |
| 2 tail 5% critical value (\(\pm\)) 1.96 (or probability awrt 0.0021~0.0022) | B1 |
| Significant result or reject the null hypothesis (o.e.) | M1 |
| There is evidence of a difference in the (mean) amount spent on junk food by male and female teenagers | A1ft |
| (7) |
Notes
1st M1 for an attempt at \(\dfrac{a - b}{\sqrt{\dfrac{c}{100 \text{ or } 200} + \dfrac{d}{100 \text{ or } 200}}}\) with 3 of \(a\), \(b\), \(c\) or \(d\) correct
1st A1 for a fully correct expression
2nd B1 for \(\pm\) 1.96 but only if their \(\mathrm{H}_1\) is two-tail (it may be in words so B0B1 is OK)
If \(\mathrm{H}_1\) is one-tail this is automatically B0 too.
2nd M1 for a correct statement based on comparison of their \(z\) with their cv. May be implied
3rd A1 for a correct conclusion in context based on their \(z\) and 1.96.
Must mention junk food or money and male vs female.
| Scheme | Marks |
|---|---|
| CLT enables us to assume \(\bar{F}\) and \(\bar{M}\) are normally distributed | B1 |
| (1) | |
| (8 marks) |
Notes
B1 for \(\bar{F}\) or \(\bar{M}\) mentioned. Allow “mean (amount spent on junk food) is normally distributed”
Read the whole statement e.g. “original distribution is normal so mean is…” scores B0