Write down the null hypothesis, H₀, for this test.

Find the expected value of female footballers.

Write down the number of degrees of freedom.

Write down the critical value of \(\chi ^2\), at the 5 % level of significance.

Use your graphic display calculator to determine the \(\chi _{calc}^2\) value.

Determine whether H₀ should be accepted. Justify your answer.

One student is chosen at random from the 120 students.

Find the probability that this student

(i) is male;

(ii) plays tennis.

Two students are chosen at random from the 120 students.

Find the probability that

(i) both play football;

(ii) neither play basketball.

H₀ : Gender and choice of afterschool sport are independent.     (A1)

Note: Accept “not associated”, do not accept “not related”, “not correlated”, or “not linked”. Accept “the relation between  gender and sport is independent”.

[1 mark]

\(\frac{{85}}{{120}} \times \frac{{48}}{{120}} \times 120\left( {\frac{{85 \times 48}}{{120}}} \right)\)     (M1)

Note: Award (M1) for correct expression.

= 34     (A1)(G2)

[2 marks]

2     (A1)

[1 mark]

5.99 (5.991)     (A1)(ft)

Note: Follow through from part (c).

[1 mark]

2.42 (2.42094…)     (G2)

[2 marks]

Since 2.42 < 5.99 therefore accept (do not reject) H₀     (R1)(A1)(ft)

Note: The numerical values need not be seen, but must be consistent with their parts (d) and (e).

OR

p-value 0.298 > 0.05 therefore accept (do not reject) H₀     (R1)(A1)

Note: p-value comparison may not be used as part of a follow through solution. Do not award (A1)(R0). Follow through from parts (c), (d) and (e).

[2 marks]

(i) \(\frac{{35}}{{120}}\left( {\frac{7}{{24}},{\text{ }}0.292,{\text{ }}29.2\% } \right)\) (0.291666...)     (A1)

(ii) \(\frac{{25}}{{120}}\left( {\frac{5}{{24}},{\text{ }}0.208,{\text{ }}20.8\% } \right)\) (0.208333...)     (A1)

[2 marks]

(i) \(\frac{{48}}{{120}} \times \frac{{47}}{{119}}\)     (A1)(M1)

Note: Award (A1) for two correct fractions, (M1) for multiplying their two fractions.

\( = \frac{{94}}{{595}}(0.158,{\text{ }}15.8\% )\) (0.157983...)     (A1)(G2)

(ii) \(\frac{{73}}{{120}} \times \frac{{72}}{{119}}\)     (M1)

Note: Award (M1) for multiplying correct fractions. If sampling with replacement has been used in both parts (h)(i) and (h)(ii) do not penalise in part (h)(ii). Award a maximum of (M1)(A1)(ft).

\( = \frac{{219}}{{595}}(0.368,{\text{ }}36.8\% )\) (0.368067...)     (A1)(G2)

[5 marks]

This question was successfully attempted by the great majority. However, the test is for the mathematical independence of the two variables; it does not address “correlation” or whether there is “no relation” between them. Further, the result of the test should be determined by the comparison of the numerical values of either the chi-squared calculated and critical values or the associated p-value and the significance level of the test. The creeping use of k as the critical value is the notation used in one text book; it is not standard notation and its use is not accepted. Comments were made on the G2 forms as to whether the the null hypothesis should be “accepted” or not rejected; both forms are acceptable.

In the compound probability questions, the lack of an explicit tree diagram determined that many candidates were not able to proceed. Determining an appropriate technique is a skill that should be taught.

This question was successfully attempted by the great majority. However, the test is for the mathematical independence of the two variables; it does not address “correlation” or whether there is “no relation” between them. Further, the result of the test should be determined by the comparison of the numerical values of either the chi-squared calculated and critical values or the associated p-value and the significance level of the test. The creeping use of k as the critical value is the notation used in one text book; it is not standard notation and its use is not accepted. Comments were made on the G2 forms as to whether the the null hypothesis should be “accepted” or not rejected; both forms are acceptable.

In the compound probability questions, the lack of an explicit tree diagram determined that many candidates were not able to proceed. Determining an appropriate technique is a skill that should be taught.

This question was successfully attempted by the great majority. However, the test is for the mathematical independence of the two variables; it does not address “correlation” or whether there is “no relation” between them. Further, the result of the test should be determined by the comparison of the numerical values of either the chi-squared calculated and critical values or the associated p-value and the significance level of the test. The creeping use of k as the critical value is the notation used in one text book; it is not standard notation and its use is not accepted. Comments were made on the G2 forms as to whether the the null hypothesis should be “accepted” or not rejected; both forms are acceptable.

In the compound probability questions, the lack of an explicit tree diagram determined that many candidates were not able to proceed. Determining an appropriate technique is a skill that should be taught.

This question was successfully attempted by the great majority. However, the test is for the mathematical independence of the two variables; it does not address “correlation” or whether there is “no relation” between them. Further, the result of the test should be determined by the comparison of the numerical values of either the chi-squared calculated and critical values or the associated p-value and the significance level of the test. The creeping use of k as the critical value is the notation used in one text book; it is not standard notation and its use is not accepted. Comments were made on the G2 forms as to whether the the null hypothesis should be “accepted” or not rejected; both forms are acceptable.

In the compound probability questions, the lack of an explicit tree diagram determined that many candidates were not able to proceed. Determining an appropriate technique is a skill that should be taught.

This question was successfully attempted by the great majority. However, the test is for the mathematical independence of the two variables; it does not address “correlation” or whether there is “no relation” between them. Further, the result of the test should be determined by the comparison of the numerical values of either the chi-squared calculated and critical values or the associated p-value and the significance level of the test. The creeping use of k as the critical value is the notation used in one text book; it is not standard notation and its use is not accepted. Comments were made on the G2 forms as to whether the the null hypothesis should be “accepted” or not rejected; both forms are acceptable.

In the compound probability questions, the lack of an explicit tree diagram determined that many candidates were not able to proceed. Determining an appropriate technique is a skill that should be taught.

This question was successfully attempted by the great majority. However, the test is for the mathematical independence of the two variables; it does not address “correlation” or whether there is “no relation” between them. Further, the result of the test should be determined by the comparison of the numerical values of either the chi-squared calculated and critical values or the associated p-value and the significance level of the test. The creeping use of k as the critical value is the notation used in one text book; it is not standard notation and its use is not accepted. Comments were made on the G2 forms as to whether the the null hypothesis should be “accepted” or not rejected; both forms are acceptable.

In the compound probability questions, the lack of an explicit tree diagram determined that many candidates were not able to proceed. Determining an appropriate technique is a skill that should be taught.

This question was successfully attempted by the great majority. However, the test is for the mathematical independence of the two variables; it does not address “correlation” or whether there is “no relation” between them. Further, the result of the test should be determined by the comparison of the numerical values of either the chi-squared calculated and critical values or the associated p-value and the significance level of the test. The creeping use of k as the critical value is the notation used in one text book; it is not standard notation and its use is not accepted. Comments were made on the G2 forms as to whether the the null hypothesis should be “accepted” or not rejected; both forms are acceptable.

In the compound probability questions, the lack of an explicit tree diagram determined that many candidates were not able to proceed. Determining an appropriate technique is a skill that should be taught.

This question was successfully attempted by the great majority. However, the test is for the mathematical independence of the two variables; it does not address “correlation” or whether there is “no relation” between them. Further, the result of the test should be determined by the comparison of the numerical values of either the chi-squared calculated and critical values or the associated p-value and the significance level of the test. The creeping use of k as the critical value is the notation used in one text book; it is not standard notation and its use is not accepted. Comments were made on the G2 forms as to whether the the null hypothesis should be “accepted” or not rejected; both forms are acceptable.

In the compound probability questions, the lack of an explicit tree diagram determined that many candidates were not able to proceed. Determining an appropriate technique is a skill that should be taught.