AS June 2019 Q2

EdexcelAS paperCurrent spec7 marksIncludes hypothesis testingChi-Squared Tests

2. A spinner used for a game is designed to give scores with the following probabilities

Score	1	2	3	4	6
Probability	\(\dfrac{3}{10}\)	\(\dfrac{1}{10}\)	\(\dfrac{1}{10}\)	\(\dfrac{2}{5}\)	\(\dfrac{1}{10}\)

The spinner is spun 80 times and the results are as follows

Score	1	2	3	4	6
Frequency	15	4	12	41	8

Test, at the 10% level of significance, whether or not the spinner is giving scores as it is designed to do. Show your working and state your hypotheses clearly. (7)

Mark scheme

Scheme

Marks

\(\mathrm{H}_0\): Spinner is working as designed (o.e.)
\(\mathrm{H}_1\): Spinner is not working as designed (o.e.)

1.2

\(E_i\)	24	8	8	32	8
\(O_i\)	15	4	12	41	8
\(\dfrac{(O_i - E_i)^2}{E_i}\)	\(\dfrac{81}{24}\)	\(\dfrac{16}{8}\)	\(\dfrac{16}{8}\)	\(\dfrac{81}{32}\)	0
\(\dfrac{O_i^2}{E_i}\)	\(\dfrac{225}{24}\)	\(\dfrac{16}{8}\)	\(\dfrac{144}{8}\)	\(\dfrac{1681}{32}\)	\(\dfrac{64}{8}\)

M1
A1

M1

3.4
1.1b

1.1b

\(\displaystyle\sum \frac{(O_i - E_i)^2}{E_i} = 3.375 + 2 + 2 + 2.53125 + 0 = 9.90625\)
or \(\displaystyle\sum \frac{O_i^2}{E_i} - N = 9.375 + 2 + 18 + 52.53125 + 8 - 80 = 9.90625\)

1.1b

\(\nu = 5 - 1 = 4\) so \(\chi^2_4(10\%)\) cv \(= 7.779\) or better

3.4

Result is significant so there is evidence that the spinner is not operating as designed

A1cso

3.5a

(7 marks)

Notes

1^st B1 for both hypotheses given in suitable context

1^st M1 for using the model to find at least 2 correct expected frequencies

1^st A1 for all correct \(E_i\)

2^nd M1 for attempt to find test statistic (at least two correct expressions, fractions or decimals)

2^nd A1 for a correct test statistic (awrt 9.91) [ accept \(\tfrac{317}{32}\) ]

2^nd B1 for correct critical value (allow 7.78)
NB \(p\)- value \(= 0.042036\ldots\) so allow awrt 0.042

3^rd A1cso dep on all previous marks for a correct conclusion in context
(can be in terms of model or spinner’s design) Must mention spinner and scores or design
Accept “spinner is not accurate”