S4 June 2016 Q2
2. The weights of piglets at birth, \(M\) kg, are normally distributed \(\mathrm{N}(\mu, \sigma^2)\)
A random sample of 9 piglets is taken and their weights at birth, \(m\) kg, are recorded. The results are summarised as
\[\sum m = 11.6 \qquad \sum m^2 = 15.2\]Stating your hypotheses clearly, test at the 5% level of significance
(a) whether or not the mean weight of piglets at birth is greater than 1.2 kg, (7)
(b) whether or not the standard deviation of the weights of piglets at birth is different from 0.3 kg. (6)
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \mu = 1.2 \quad \mathrm{H}_1 : \mu \gt 1.2\) | B1 |
| \(t_8(5\%) = 1.860\) | B1 |
| \(\bar{m} = 1.28888\ldots\) | B1 |
| \(t = \dfrac{1.28\ldots - 1.2}{\sqrt{\dfrac{0.031111}{9}}} = 1.511\) awrt 1.51 | M1 A1ft A1 |
| Not significant. There is not sufficient evidence that the mean weight of piglets is greater than 1.2 kg | A1 |
| (7) |
Notes
B1 both hypotheses
M1 for attempting the correct statistic
A1ft follow through their s2
A1 awrt 1.51
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \sigma^2 = 0.09 \quad \mathrm{H}_1 : \sigma^2 \ne 0.09 \qquad [\mathrm{H}_0 : \sigma = 0.3 \ \ \mathrm{H}_1 : \sigma \ne 0.3]\) | B1 |
| \(s^2 = \dfrac{15.2 - 9 \times \left(\dfrac{11.6}{9}\right)^2}{8} = 0.031111\) | B1 |
| \([\chi^2_8(0.025) = 17.535] \quad \chi^2_8(0.975) = 2.18\) | B1 |
| Critical region \(\dfrac{(n-1)s^2}{\sigma^2} \sim \chi^2_8\) test statistic = 2.7654… awrt 2.77 | M1A1 |
| 2.77 is not in the critical region. There is no evidence that the standard deviation of the weights of piglets is different to 0.3 | A1 |
| (6) | |
| (13 marks) |
Notes
B1 both hypotheses, must be two tail
B1 awrt 0.0311
B1 NB allow 2.733 for one tail hypotheses. (no hypotheses gains B0)
M1 for a correct test statistic
NB one tail test can get B0 B1 B1 (2.733)B0 M1 A1 A1
(Corrected from the printed mark scheme: the upper critical value is printed as \(\chi^2_8(0.25) = 17.535\); it is the upper 2.5% point, \(\chi^2_8(0.025)\).)