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Date November 2021 Marks available 2 Reference code 21N.2.SL.TZ0.1
Level Standard Level Paper Paper 2 Time zone Time zone 0
Command term Calculate Question number 1 Adapted from N/A

Question

A group of 1280 students were asked which electronic device they preferred. The results per age group are given in the following table.

A student from the group is chosen at random. Calculate the probability that the student

A χ2 test for independence was performed on the collected data at the 1% significance level. The critical value for the test is 13.277.

prefers a tablet.

[2]
a.i.

is 1113 years old and prefers a mobile phone.

[2]
a.ii.

prefers a laptop given that they are 1718 years old.

[2]
a.iii.

prefers a tablet or is 1416 years old.

[3]
a.iv.

State the null and alternative hypotheses.

[1]
b.

Write down the number of degrees of freedom.

[1]
c.

Write down the χ2 test statistic.

[2]
d.i.

Write down the p-value.

[1]
d.ii.

State the conclusion for the test in context. Give a reason for your answer.

[2]
d.iii.

Markscheme

5601280  716, 0.4375              A1A1


Note: Award A1 for correct numerator, A1 for correct denominator.

 

[2 marks]

a.i.

721280  9160, 0.05625              A1A1


Note: Award A1 for correct numerator, A1 for correct denominator.

 

[2 marks]

a.ii.

153348  51116, 0.439655              A1A1


Note: Award A1 for correct numerator, A1 for correct denominator.

 

[2 marks]

a.iii.

160+224+128+205+131  OR  560+512-224                   (M1)

8481280  5380, 0.6625              A1A1


Note: Award A1 for correct denominator (1280) seen, (M1) for correct calculation of the numerator, A1 for the correct answer.

 

[3 marks]

a.iv.

H0: the variables are independent

H1: the variables are dependent              A1


Note: Award A1 for for both hypotheses correct. Do not accept “not correlated” or “not related” in place of “independent”.

 

[1 mark]

b.

4              A1

 

[1 mark]

c.

χ2=  23.3  23.3258             A2

 

[2 marks]

d.i.

0.000109  0.000108991  OR  1.09×10-4             A1

 

[1 mark]

d.ii.

EITHER

23.3>13.277             R1

OR

0.000109<0.01             R1


THEN

(there is sufficient evidence to accept H1 that) preferred device and age group are not independent             A1


Note: For the final A1 the answer must be in context. Do not award A1R0.

 

[2 marks]

d.iii.

Examiners report

A common error was to treat these as compound probability events. Candidates who attempted to use the conditional probability formula in part (a)(iii) often accrued at least one mark for a correct method. Some lost one mark for not reducing the total sample space. In part (a)(iv) many found PA+PB, failing to subtract the intersection of the two events. When stating test hypotheses, terminology such as "related", "correlated", "data is independent" are not acceptable. Few candidates attempted to frame this as a Goodness of Fit test. Candidates correctly wrote down the degrees of freedom. It was pleasing to see most found the correct χ2-test statistic and associated p-value. Relative to past examination sessions, fewer candidates appeared to compare the critical value with the p-value or the χ2-test statistic with the level of significance. A conclusion consistent with their hypotheses was usually seen, though this was not always expressed in context.

a.i.

A common error was to treat these as compound probability events. Candidates who attempted to use the conditional probability formula in part (a)(iii) often accrued at least one mark for a correct method. Some lost one mark for not reducing the total sample space. In part (a)(iv) many found PA+PB, failing to subtract the intersection of the two events. When stating test hypotheses, terminology such as "related", "correlated", "data is independent" are not acceptable. Few candidates attempted to frame this as a Goodness of Fit test. Candidates correctly wrote down the degrees of freedom. It was pleasing to see most found the correct χ2-test statistic and associated p-value. Relative to past examination sessions, fewer candidates appeared to compare the critical value with the p-value or the χ2-test statistic with the level of significance. A conclusion consistent with their hypotheses was usually seen, though this was not always expressed in context.

a.ii.

A common error was to treat these as compound probability events. Candidates who attempted to use the conditional probability formula in part (a)(iii) often accrued at least one mark for a correct method. Some lost one mark for not reducing the total sample space. In part (a)(iv) many found PA+PB, failing to subtract the intersection of the two events. When stating test hypotheses, terminology such as "related", "correlated", "data is independent" are not acceptable. Few candidates attempted to frame this as a Goodness of Fit test. Candidates correctly wrote down the degrees of freedom. It was pleasing to see most found the correct χ2-test statistic and associated p-value. Relative to past examination sessions, fewer candidates appeared to compare the critical value with the p-value or the χ2-test statistic with the level of significance. A conclusion consistent with their hypotheses was usually seen, though this was not always expressed in context.

a.iii.

A common error was to treat these as compound probability events. Candidates who attempted to use the conditional probability formula in part (a)(iii) often accrued at least one mark for a correct method. Some lost one mark for not reducing the total sample space. In part (a)(iv) many found PA+PB, failing to subtract the intersection of the two events. When stating test hypotheses, terminology such as "related", "correlated", "data is independent" are not acceptable. Few candidates attempted to frame this as a Goodness of Fit test. Candidates correctly wrote down the degrees of freedom. It was pleasing to see most found the correct χ2-test statistic and associated p-value. Relative to past examination sessions, fewer candidates appeared to compare the critical value with the p-value or the χ2-test statistic with the level of significance. A conclusion consistent with their hypotheses was usually seen, though this was not always expressed in context.

a.iv.

A common error was to treat these as compound probability events. Candidates who attempted to use the conditional probability formula in part (a)(iii) often accrued at least one mark for a correct method. Some lost one mark for not reducing the total sample space. In part (a)(iv) many found PA+PB, failing to subtract the intersection of the two events. When stating test hypotheses, terminology such as "related", "correlated", "data is independent" are not acceptable. Few candidates attempted to frame this as a Goodness of Fit test. Candidates correctly wrote down the degrees of freedom. It was pleasing to see most found the correct χ2-test statistic and associated p-value. Relative to past examination sessions, fewer candidates appeared to compare the critical value with the p-value or the χ2-test statistic with the level of significance. A conclusion consistent with their hypotheses was usually seen, though this was not always expressed in context.

b.

A common error was to treat these as compound probability events. Candidates who attempted to use the conditional probability formula in part (a)(iii) often accrued at least one mark for a correct method. Some lost one mark for not reducing the total sample space. In part (a)(iv) many found PA+PB, failing to subtract the intersection of the two events. When stating test hypotheses, terminology such as "related", "correlated", "data is independent" are not acceptable. Few candidates attempted to frame this as a Goodness of Fit test. Candidates correctly wrote down the degrees of freedom. It was pleasing to see most found the correct χ2-test statistic and associated p-value. Relative to past examination sessions, fewer candidates appeared to compare the critical value with the p-value or the χ2-test statistic with the level of significance. A conclusion consistent with their hypotheses was usually seen, though this was not always expressed in context.

c.

A common error was to treat these as compound probability events. Candidates who attempted to use the conditional probability formula in part (a)(iii) often accrued at least one mark for a correct method. Some lost one mark for not reducing the total sample space. In part (a)(iv) many found PA+PB, failing to subtract the intersection of the two events. When stating test hypotheses, terminology such as "related", "correlated", "data is independent" are not acceptable. Few candidates attempted to frame this as a Goodness of Fit test. Candidates correctly wrote down the degrees of freedom. It was pleasing to see most found the correct χ2-test statistic and associated p-value. Relative to past examination sessions, fewer candidates appeared to compare the critical value with the p-value or the χ2-test statistic with the level of significance. A conclusion consistent with their hypotheses was usually seen, though this was not always expressed in context.

d.i.

A common error was to treat these as compound probability events. Candidates who attempted to use the conditional probability formula in part (a)(iii) often accrued at least one mark for a correct method. Some lost one mark for not reducing the total sample space. In part (a)(iv) many found PA+PB, failing to subtract the intersection of the two events. When stating test hypotheses, terminology such as "related", "correlated", "data is independent" are not acceptable. Few candidates attempted to frame this as a Goodness of Fit test. Candidates correctly wrote down the degrees of freedom. It was pleasing to see most found the correct χ2-test statistic and associated p-value. Relative to past examination sessions, fewer candidates appeared to compare the critical value with the p-value or the χ2-test statistic with the level of significance. A conclusion consistent with their hypotheses was usually seen, though this was not always expressed in context.

d.ii.

A common error was to treat these as compound probability events. Candidates who attempted to use the conditional probability formula in part (a)(iii) often accrued at least one mark for a correct method. Some lost one mark for not reducing the total sample space. In part (a)(iv) many found PA+PB, failing to subtract the intersection of the two events. When stating test hypotheses, terminology such as "related", "correlated", "data is independent" are not acceptable. Few candidates attempted to frame this as a Goodness of Fit test. Candidates correctly wrote down the degrees of freedom. It was pleasing to see most found the correct χ2-test statistic and associated p-value. Relative to past examination sessions, fewer candidates appeared to compare the critical value with the p-value or the χ2-test statistic with the level of significance. A conclusion consistent with their hypotheses was usually seen, though this was not always expressed in context.

d.iii.

Syllabus sections

Topic 4—Statistics and probability » SL 4.5—Probability concepts, expected numbers
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Topic 4—Statistics and probability

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