S4 June 2015 Q1

EdexcelOld spec9 marksIncludes hypothesis testingt-Tests & Further Tests

1. The Sales Manager of a large chain of convenience stores is studying the sale of lottery tickets in her stores. She randomly selects 8 of her stores. From these stores she collects data for the total sales of lottery tickets in the previous January and July. The data are shown below

Store	A	B	C	D	E	F	G	H
January ticket sales (£)	1080	1639	710	1108	915	1066	1322	819
July ticket sales (£)	1113	1702	831	1048	861	1090	1303	852

(a) Use a paired \(t\)-test to determine whether or not there is evidence, at the 5% level of significance, that the mean sales of lottery tickets in this chain’s stores are higher in July than in January. You should state your hypotheses and show your working clearly. (8)

(b) State what assumption the Sales Manager needs to make about the sales of lottery tickets in her stores for the test in part (a) to be valid. (1)

Mark scheme (a)

Scheme

Marks

Store	A	B	C	D	E	F	G	H
Difference July-Jan	33	63	121	−60	−54	24	−19	33

\(\bar{d} = \dfrac{141}{8} = (\pm)17.625\)

\({s_d}^2 = \dfrac{8}{7}\left(\dfrac{28241}{8} - 17.625^2\right) = 3679.4\ldots\)
or
\({s_d}^2 = \dfrac{1}{7}\left(28241 - \dfrac{141^2}{8}\right) = 3679.4\ldots\)

To test \(\mathrm{H}_0 : \mu_d = 0\) against \(\mathrm{H}_1 : \mu_d \gt 0\) (o.e.)

Test stat \(t = \dfrac{17.625 - 0}{\sqrt{\frac{3679.4\ldots}{8}}} = 0.8218\ldots\)

M1A1cso

Critical value, \(t_7 = 1.895\)

Not in critical region therefore insufficient reason to reject \(\mathrm{H}_0\)
No significant evidence that on average stores sell more lottery tickets in July than in January

A1ft

(8)

Notes

1st B1 for differences all correct (o.e.)

1st M1 attempt to find \(\bar{d} = \dfrac{\sum \text{"their } d\text{"}}{8}\)

2nd M1 attempting \(s_d\) or \({s_d}^2\ \dfrac{1}{7}\left(\sum \text{"their } d^2\text{"} - \dfrac{\left(\sum \text{"their } d\text{"}\right)^2}{8}\right)\)

2nd B1 both correct in terms of \(\mu\) or \(\mu_d\) (allow a defined symbol) condone \(\mu_{July-Jan}\)

3rd M1 for attempting the correct test statistic \(\dfrac{\bar{d}}{s_d/\sqrt{8}}\)

1st A1cso awrt 0.822 with no errors.

3rd B1 alternate method, \(p\) value of 0.219. Allow 2.365 for 2-tail test

Final A1 need conclusion in context, need tickets July and January, ft their test stat and critical value

NB difference of 2 means test gains no marks

Mark scheme (b)

Scheme	Marks
Need assumption that the underlying distribution of the difference in sales in July and in January is normally distributed.	B1
	(1)
	(9 marks)

Notes

B1 need differences to be normally distributed, not just normal distribution