S4 June 2014 (R) Q3

EdexcelOld spec12 marksIncludes hypothesis testingt-Tests & Further Tests

3. A farmer is investigating the milk yields of two breeds of cow. He takes a random sample of 9 cows of breed \(A\) and an independent random sample of 12 cows of breed \(B\). For a 5 day period he measures the amount of milk, \(x\) gallons, produced by each cow. The results are summarised in the table below.

Breed	Sample size	Mean (\(\bar{x}\))	Standard deviation (\(s_x\))
\(A\)	9	6.23	2.98
\(B\)	12	7.13	2.33

The amount of milk produced by each cow can be assumed to follow a normal distribution.

(a) Use a two-tail test to show, at the 10% level of significance, that the variances of the yields of the two breeds can be assumed to be equal. State your hypotheses clearly. (4)

(b) Stating your hypotheses clearly, test, at the 5% level of significance, whether or not there is a difference in the mean yields of the two breeds of cow. (7)

Mark scheme (a)

Scheme	Marks
\(\mathrm{H}_0 : {\sigma_A}^2 = {\sigma_B}^2 \qquad \mathrm{H}_1 : {\sigma_A}^2 \ne {\sigma_B}^2\)	B1
\(\left(F_{8,11} =\right) \dfrac{2.98^2}{2.33^2} = (1.6357\ldots)\)	M1
\(F_{8,11}\) 10% (two-tail) cv = 2.95 (or prob. = awrt 0.22)	B1
Not significant so can accept the assumption that variances are equal.	A1
	(4)

Notes

1st B1 allow \(\sigma\) or \(\sigma^2\)
M1 for use of the correct test statistic

Mark scheme (b)

Scheme	Marks
\(\mathrm{H}_0 : \mu_A = \mu_B \qquad \mathrm{H}_1 : \mu_A \ne \mu_B\)	B1
\({s_p}^2 = \dfrac{8 \times 2.98^2 + 11 \times 2.33^2}{19},= 6.88216\ldots\) or \(s_p = 2.62338\ldots\)	M1, A1
\(\left(t_{19} =\right)(\pm)\dfrac{7.13 - 6.23}{s_p\sqrt{\frac{1}{9} + \frac{1}{12}}} = (\pm)0.7780047\ldots\) = awrt 0.778	M1 A1
\(t_{19}(0.05)\) two-tail cv = 2.093	B1
[Not significant] Insufficient evidence of a difference in mean milk yields between the two breeds	A1
	(7)

Notes

1st M1 for attempting \(s_p\) or \({s_p}^2\)
1st A1 for awrt 6.90 or 2.63
2nd M1 for use of a correct test statistic
2nd A1 for awrt 0.77 (accept \(\pm\))
2nd B1 for 2.093 (allow \(\pm\) 1.729 for one-tailed \(\mathrm{H}_1\))