| Date | November 2012 | Marks available | 2 | Reference code | 12N.2.sl.TZ0.4 |
| Level | SL only | Paper | 2 | Time zone | TZ0 |
| Command term | Show that | Question number | 4 | Adapted from | N/A |
Question
A store recorded their sales of televisions during the 2010 football World Cup. They looked at the numbers of televisions bought by gender and the size of the television screens.
This information is shown in the table below; S represents the size of the television screen in inches.
The store wants to use this information to predict the probability of selling these sizes of televisions for the 2014 football World Cup.
Use the table to find the probability that
(i) a television will be bought by a female;
(ii) a television with a screen size of 32 < S ≤ 46 will be bought;
(iii) a television with a screen size of 32 < S ≤ 46 will be bought by a female;
(iv) a television with a screen size greater than 46 inches will be bought, given that it is bought by a male.
The manager of the store wants to determine whether the screen size is independent of gender. A Chi-squared test is performed at the 1 % significance level.
Write down the null hypothesis.
The manager of the store wants to determine whether the screen size is independent of gender. A Chi-squared test is performed at the 1 % significance level.
Show that the expected frequency for females who bought a screen size of 32 < S ≤ 46, is 79, correct to the nearest integer.
The manager of the store wants to determine whether the screen size is independent of gender. A Chi-squared test is performed at the 1 % significance level.
Write down the number of degrees of freedom.
The manager of the store wants to determine whether the screen size is independent of gender. A Chi-squared test is performed at the 1 % significance level.
Write down the \({\chi ^2}\) calculated value.
The manager of the store wants to determine whether the screen size is independent of gender. A Chi-squared test is performed at the 1 % significance level.
Write down the critical value for this test.
The manager of the store wants to determine whether the screen size is independent of gender. A Chi-squared test is performed at the 1 % significance level.
Determine if the null hypothesis should be accepted. Give a reason for your answer.
Markscheme
(i) \(\frac{{220}}{{500}}\left( {\frac{{11}}{{25}},{\text{ 0}}{\text{.44, 44}}\% } \right)\) (A1)(G1)
(ii) \(\frac{{180}}{{500}}\left( {\frac{{9}}{{25}},{\text{ 0}}{\text{.36, 36}}\% } \right)\) (A1)(G1)
(iii) \(\frac{{40}}{{500}}\left( {\frac{{2}}{{25}},{\text{ 0}}{\text{.08, 8}}\% } \right)\) (A1)(A1)(G2)
(iv) \(\frac{{55}}{{500}}\left( {\frac{{11}}{{56}},{\text{ 0}}{\text{.196, 19.6}}\% } \right)\) (A1)(A1)(G2)
Note: Award (A1) for numerator, (A1) for denominator. Award (A0)(A0) if answers are given as incorrect reduced fractions without working.
[6 marks]
“The size of the television screen is independent of gender.” (A1)
Note: Accept “not associated”, do not accept “not correlated”.
[1 mark]
\(\frac{{180}}{{500}} \times \frac{{220}}{{500}} \times 500\) OR \(\frac{{180 \times 220}}{{500}}\) (M1)
= 79.2 (A1)
= 79 (AG)
Note: Both the unrounded and the given answer must be seen for the final (A1) to be awarded.
[2 marks]
3 (A1)
[1 mark]
\(\chi _{calc}^2\) = 104(103.957...) (G2)
Note: Award (M1) if an attempt at using the formula is seen but incorrect answer obtained.
[2 marks]
11.345 (A1)(ft)
Notes: Follow through from their degrees of freedom.
[1 mark]
\(\chi _{calc}^2\) > \(\chi _{crit}^2\) OR p < 0.01 (R1)
Do not accept H0. (A1)(ft)
Note: Do not award (R0)(A1)(ft). Follow through from their parts (d), (e) and (f).
[2 marks]
Examiners report
Part (a) was generally well answered by most of the students, except for part (a)(iv) which called for conditional probability.
Most students correctly stated the null hypothesis in part (b), and answered parts (d), (e), (f) and (g).
In some responses to part (c) it seemed that the difference between calculation of the expected value and showing that the value is 79 was not clear to the candidates. It is important that teachers explain to their students that in a “show that” question they are expected to demonstrate the mathematical reasoning through which the given answer is obtained.
Most students correctly stated the null hypothesis in part (b), and answered parts (d), (e), (f) and (g).
Most students correctly stated the null hypothesis in part (b), and answered parts (d), (e), (f) and (g).
Most students correctly stated the null hypothesis in part (b), and answered parts (d), (e), (f) and (g).
Most students correctly stated the null hypothesis in part (b), and answered parts (d), (e), (f) and (g).
