S3 June 2015 Q2

EdexcelOld spec10 marksIncludes hypothesis testingt-Tests & Further Tests

2. A researcher believes that the mean weight loss of those people using a slimming plan as part of a group is more than 1.5 kg a year greater than the mean weight loss of those using the plan on their own. The mean weight loss of a random sample of 80 people using the plan as part of a group is 8.7 kg with a standard deviation of 2.1 kg. The mean weight loss of a random sample of 65 people using the plan on their own is 6.6 kg with a standard deviation of 1.4 kg.

(a) Stating your hypotheses clearly, test the researcher’s claim. Use a 1% level of significance. (8)

(b) For the test in part (a), state whether or not it is necessary to assume that the weight loss of a person using this plan has a normal distribution. Give a reason for your answer. (2)

Mark scheme (a)

Scheme	Marks
\(\mathrm{H_0}\): \(\mu_g - \mu_s = 1.5\) [\(g\) = in a group, \(s\) = on their own]	B1
\(\mathrm{H_1}\): \(\mu_g - \mu_s \gt 1.5\)	B1
s.e. \(= \sqrt{\dfrac{2.1^2}{80} + \dfrac{1.4^2}{65}} = \left[\sqrt{0.08527\ldots}\right] = [0.292]\)	M1
\(z = \dfrac{8.7 - 6.6 - 1.5}{\text{"}\sqrt{\frac{2.1^2}{80} + \frac{1.4^2}{65}}\text{"}}\)	dM1
\(= 2.0546\ldots\) awrt 2.05(5)	A1
cv 1% one tailed = 2.3263	B1
Not significant, accept \(\mathrm{H_0}\)	dM1
Insufficient evidence that using plan as part of a group leads to weight loss of more than 1.5 kg than using plan on one’s own or researcher’s belief not supported	A1ft
	(8)

Notes

1st & 2nd B1 for hypotheses. Accept \(\mu_1, \mu_2\) or \(\mu_A, \mu_B\) etc if there is some indication of which is which e.g. \(G \sim \mathrm{N}(\mu_g, 8.7)\)

1st M1 for an attempt at se with 3 out of 4 values correct. Condone switching 2.1 and 1.4 \(\sqrt{\dfrac{2.1^2 \text{ or } 1.4^2}{80} + \dfrac{1.4^2 \text{ or } 2.1^2}{65}}\)

2nd dM1 dependent on 1st M1 for a correct numerator (must have \(-1.5\)) and ft their se.

1st A1 for awrt 2.05

3rd B1 for \(\pm 2.3263\) or better seen or probability of awrt 0.02

3rd dM1 dep. on 1st M1 for a correct statement based on their normal cv and their test statistic

2nd A1ft for correct comment in context. Must mention “plan” and “group or individual” and “1.5” or “researcher” and “belief or claim”

NB Use of cv for difference in means \(D\) will have \(D = 1.5 + 2.3263 \times \text{s.e.}\) = awrt 2.18 and requires sight of \(d = 2.1\) with a comment for the 3rd M1

Mark scheme (b)

Scheme	Marks
Since sample is large Central Limit Theorem (CLT) applies	B1
No need to assume normal distribution	dB1
	(2)
	(10 marks)

Notes

1st B1 for mentioning “large samples” and “CLT”

2nd dB1 dependent on 1st B1 for stating no need to assume normality (since CLT assures it)