S4 June 2006 Q1
1. Historical records from a large colony of squirrels show that the weight of squirrels is normally distributed with a mean of 1012 g. Following a change in the diet of squirrels, a biologist is interested in whether or not the mean weight has changed.
A random sample of 14 squirrels is weighed and their weights \(x\), in grams, recorded. The results are summarised as follows:
\[\Sigma x = 13\,700, \qquad \Sigma x^2 = 13\,448\,750.\]Stating your hypotheses clearly test, at the 5% level of significance, whether or not there has been a change in the mean weight of the squirrels. (7)
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \mu = 1012 \qquad \mathrm{H}_1 : \mu \neq 1012\) both | B1 |
| \(\bar{x} = \dfrac{13700}{14}\ (= 978.57\ldots)\) | M1 |
| \(s_x^2 = \dfrac{13\,448\,750 - 14\bar{x}^2}{13}\ (= 3255.49\ldots)\ (s_x = 57.056\ldots)\) \(s^2\) or \(s\) | M1 |
| \(t_{13} = \dfrac{\bar{x} - \mu}{s/\sqrt{n}} = \dfrac{978.6 - 1012}{57.06/\sqrt{14}} = -2.19\ldots\) | M1 |
| awrt \(-2.19\) | A1 |
| \(t_{13}(5\%)\) 2 tail c.v. \(= -2.160\) \((\pm)\) | B1 |
| Significant result – there is evidence of a change in mean weight of squirrels (condone decrease) must mention weight | A1ft |
| (7 marks) |
Notes
The scheme links the 3rd M1 and the final A1ft with an arrow.