S3 June 2013 Q7
7. Lambs are born in a shed on Mill Farm. The birth weights, \(x\) kg, of a random sample of 8 newborn lambs are given below.
4.12 5.12 4.84 4.65 3.55 3.65 3.96 3.40
A further random sample of 32 lambs is chosen and the unbiased estimates of the mean and variance of the birth weight of lambs from this sample are 4.55 and 0.25 respectively.
The owner of Mill Farm researches the breed of lamb and discovers that the population of birth weights is normally distributed with standard deviation 0.67 kg.
| Scheme | Marks |
|---|---|
| \(\hat{\mu} = \bar{x} = \dfrac{33.29}{8} = 4.16125\) ( awrt 4.16) | B1 |
| \(\hat{\sigma}^2 = s^2 = \dfrac{4.12^2 + 5.12^2 + \ldots - 8 \times \bar{x}^2}{7}\) | M1 |
| \(\hat{\sigma}^2 = s^2 = \dfrac{141.4035 - 138.528013}{7} = 0.41078\ldots\) (awrt 0.411) | A1 |
| (3) |
Notes
M1 for an attempt at \(s^2\): correct denom, clear attempt at \(\sum x^2\) and ft their \(\bar{x}\) Ans only 2/2
| Scheme | Marks |
|---|---|
| \(\sum x = 33.29 + 32 \times 4.55 = 178.89\), (awrt 179) | B1 |
| \(\sum x^2 =\) “141.4035”\(+ 31 \times 0.25 + 32 \times 4.55^2 (= 811.6335)\) (awrt 812) | M1A1 |
| Combined sample: \(s^2 = \dfrac{811.6335 - \dfrac{178.89^2}{40}}{39} = 0.29724865\ldots\) (awrt 0.297) | M1A1 |
| \(\dfrac{s}{\sqrt{n}} = \dfrac{\sqrt{0.297\ldots.}}{\sqrt{40}} = 0.0862\) (awrt 0.0862) | M1A1 |
| (7) |
Notes
B1 for correct sum or mean or fully correct expression (accept mean = awrt 4.47) May be in (c)
1st M1 for their \(141.4035 + 31 \times 0.25 + 32 \times 4.55^2\) or “141.4035” + 7.75+ 662.48 (accept 3sf)
Beware: 32(0.25 + 4.552) + “141.4035” = awrt 812 but scores M0A0.
1st A1 for a fully correct expression (all to 3sf or better) or answer only = awrt 812
2nd M1 for a correct expression using their values
3rd M1 dependent on using a changed \(s^2\) (not their 0.411 or 0.25) for \(\dfrac{\sqrt{\text{"}0.297\text{"}}}{\sqrt{40}}\)
This \(s^2\) must be based on a combination of their 0.411 and 0.25 e.g. 0.661
| Scheme | Marks |
|---|---|
| \(\bar{x} \pm 1.96\dfrac{\sigma}{\sqrt{n}} = \dfrac{178.89}{40} \pm 1.96\dfrac{0.67}{\sqrt{40}}\) | M1B1 |
| \(= (4.2646\ldots, 4.67988\ldots)\) awrt (4.26[or 4.265], 4.68) | A1 |
| (3) | |
| (13 marks) |
Notes
M1 for \(\bar{x} \pm z \times \dfrac{\sigma}{\sqrt{n}}\) for any \(z\) ( > 1.5) and ft their \(\bar{x}\) based on combining their 4.16 and 4.55, do not award for simply using 4.55 or their 4.16. Condone \(\sigma = \sqrt{\text{their } 0.297}\) or their (b)
B1 for \(z = 1.96\) used in an attempt at a CI, may for example miss \(\sqrt{n}\)
A1 for both limits awrt 3sf. Allow lower limit of 4.265