S4 June 2017 Q4

EdexcelOld spec12 marksIncludes hypothesis testingt-Tests & Further Tests

4. A coach believes that the average score in the final round of a golf tournament is more than one point below the average score in the first round. To test this belief, the scores of 8 randomly selected players are recorded. The results are given in the table below.

Player	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
First round	76	80	72	78	83	88	81	72
Final round	70	78	75	75	79	84	83	69

(a)

(i) State why a paired \(t\)-test is suitable for use with these data.

(ii) State an assumption that needs to be made in order to carry out a paired \(t\)-test in this case. (2)

(b) Test, at the 5% level of significance, whether or not there is evidence to support the coach’s belief. Show your working clearly. (8)

Mark scheme (a)

Scheme	Marks
(i) The data is collected in pairs or samples not independent	B1
(ii) The differences are normally distributed	B1
	(2)

Notes

(i) B1 Allow because the same person has been used. Do not allow 2 data sets.

(ii) B1 for a comment that mentions “differences” and “normal” distribution

Mark scheme (b)

Scheme	Marks
\(d\): 6 2 \(-3\) 3 4 4 \(-2\) 3	M1
\(\left(\Sigma d = 17,\ \Sigma d^2 = 103\right) \quad \bar{d} = \pm 2.125,\ s_d = 3.09\ldots\ \ (\text{Var} = 9.55\ldots)\)	M1 M1
\(\mathrm{H}_0 : \mu_d = 1,\ \mathrm{H}_1 : \mu_d \gt 1\) (\(\mathrm{H}_0 : \mu_d = -1 \quad \mathrm{H}_1 : \mu_d \lt -1\) if differences are \(-6, -2, 3\) etc)	B1
\(t = \pm\left(\dfrac{2.125 - 1}{3.09/\sqrt{8}}\right) = \pm 1.02947\ldots\)	M1A1
Critical value \(t_7(5\%) = \pm 1.895\) (1 tail)	B1
Not significant. Insufficient evidence to support that the score in the final round is more than 1 below the score in the first round or insufficient evidence to support the coach’s belief.	A1ft
	(8)

Notes

SC for two sample test they may get M0M0 M0B1M0A0B1A0
\(\mathrm{H}_0 : \mu_{\mathit{first}} = \mu_{\mathit{final}} + 1, \quad \mathrm{H}_1 : \mu_{\mathit{first}} \gt \mu_{\mathit{final}} + 1, \quad \pm 1.761\)

M1 for attempting the \(d\)s, at least 2 correct implied by the figures (\(\Sigma d = 17\), \(\Sigma d^2 = 103\), \(\bar{d} = \pm 2.125\), \(s_d = 3.09\ldots\))

M1 for attempting \(\bar{d}\)

M1 for \(s_d\) or \(s_d^{\,2}\)

B1 for both hypotheses correct in terms of \(\mu\) or \(\mu_d\) (allow a defined symbol) Must match their differences

M1 for attempting the correct test statistic \(\dfrac{\bar{d} - 1}{s_d/\sqrt{8}}\)

A1 awrt 1.03

B1 awrt 1.895 sign must match their \(t\)-value

A1ft ft \(t\)-value if awarded both B marks. A correct comment in context – bold words needed.

Mark scheme (c)

Scheme	Marks
The idea that “the coach’s belief is rejected when it is in fact true”	B1 B1
	(2)
	(12 marks)

Notes

B1 for \(\mathrm{H}_1\) is rejected when it is in fact true

B1 Correct contextual statement. (corrected from the printed mark scheme: the scheme prints “B2” here, but the part is marked B1 B1)