A2 June 2022 Q4

EdexcelCurrent spec8 marksIncludes hypothesis testingt-Tests & Further Tests

4. A doctor believes that a four-week exercise programme can reduce the resting heart rate of her patients. She takes a random sample of 7 patients and records their resting heart rate before the exercise programme and again after the exercise programme.

Patient	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)
Resting heart rate before	65	68	77	79	80	88	92
Resting heart rate after	63	65	73	76	80	84	80

(a) Using a 5% level of significance, carry out an appropriate test of the doctor’s belief.
You should state your hypotheses, test statistic and critical value. (7)

(b) State the assumption made about the resting heart rates that was required to carry out the test. (1)

Mark scheme (a)

Scheme	Marks	AO
\(d\): 2 3 4 3 0 4 12	M1	3.1b
\(\bar{d} = \pm 4 \qquad s_\mathrm{d} = \sqrt{\tfrac{1}{6}\left(\text{‘}198\text{’} - 7(\text{‘}4\text{’})^2\right)} = \sqrt{14.333\ldots} = 3.7859\ldots\)	M1	1.1b
\(\mathrm{H}_0 : \mu_\mathrm{d} = 0 \qquad \mathrm{H}_1 : \mu_\mathrm{d} \gt 0\)	B1	2.5
\(t = \pm\dfrac{\text{“}\pm 4\text{”}}{\frac{\text{“}3.78\ldots\text{”}}{\sqrt{7}}}\)	M1	1.1b
\(= \pm 2.795\ldots\) awrt \(\pm 2.80\)	A1	1.1b
Critical value \(t_6 = \pm 1.943\)	B1	1.1b
[\(2.80 \gt 1.943\),] therefore there is sufficient evidence to support the doctor’s belief/evidence to suggest resting heart rate is reduced.	A1	2.2b
	(7)

Notes

M1: For understanding paired \(t\)-test is required and attempting to find differences at least 5 correct, allow \(\pm\) (may be implied by correct \(\bar{d}\) and \(s_\mathrm{d}\))

M1: Complete method for \(\bar{d}\) and \(s_\mathrm{d}\) or \((s_\mathrm{d})^2\)

B1: Correct model for differences with both hypotheses correct in terms of \(\mu\) / \(\mu_\mathrm{d}\) (sign of \(\mathrm{H}_1\) must be compatible with their \(\bar{d}\))

M1: Method for finding test statistic with their values

A1: awrt \(\pm 2.80\) (allow awrt \(\pm 2.8\) from correct working)

B1: Correct critical \(\pm 1.943\) (or better) with compatible sign

A1: Correct comparison to deduce that the doctor’s belief is supported. Must be consistent with their CV and their test statistic (dependent upon all M marks).

SC: Difference of means test apply scheme but also allow 2^nd B1 for \(t_{12} = 1.782\)