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Date	May 2018	Marks available	2	Reference code	18M.2.sl.TZ2.3
Level	SL only	Paper	2	Time zone	TZ2
Command term	State and Justify	Question number	3	Adapted from	N/A

Question

The weight, W, of basketball players in a tournament is found to be normally distributed with a mean of 65 kg and a standard deviation of 5 kg.

The probability that a basketball player has a weight that is within 1.5 standard deviations of the mean is q.

A basketball coach observed 60 of her players to determine whether their performance and their weight were independent of each other. Her observations were recorded as shown in the table.

She decided to conduct a χ ² test for independence at the 5% significance level.

Find the probability that a basketball player has a weight that is less than 61 kg.

[2]

a.i.

In a training session there are 40 basketball players.

Find the expected number of players with a weight less than 61 kg in this training session.

[2]

a.ii.

Sketch a normal curve to represent this probability.

[2]

b.i.

Find the value of q.

[1]

b.ii.

Given that P(W > k) = 0.225 , find the value of k.

[2]

c.

For this test state the null hypothesis.

[1]

d.i.

For this test find the p-value.

[2]

d.ii.

State a conclusion for this test. Justify your answer.

[2]

e.

Markscheme

P(W < 61) (M1)

Note: Award (M1) for correct probability statement.

OR

(M1)

Note: Award (M1) for correct region labelled and shaded on diagram.

= 0.212 (0.21185…, 21.2%) (A1)(G2)

[2 marks]

a.i.

40 × 0.21185… (M1)

Note: Award (M1) for product of 40 and their 0.212.

= 8.47 (8.47421...) (A1)(ft)(G2)

Note: Follow through from their part (a)(i) provided their answer to part (a)(i) is less than 1.

[2 marks]

a.ii.

(A1)(M1)

Note: Award (A1) for two correctly labelled vertical lines in approximately correct positions. The values 57.5 and 72.5, or μ − 1.5σ and μ + 1.5σ are acceptable labels. Award (M1) for correctly shaded region marked by their two vertical lines.

[2 marks]

b.i.

0.866 (0.86638…, 86.6%) (A1)(ft)

Note: Follow through from their part (b)(i) shaded region if their values are clear.

[1 mark]

b.ii.

P(W < k) = 0.775 (M1)

OR

(M1)

Note: Award (A1) for correct region labelled and shaded on diagram.

(k =) 68.8 (68.7770…) (A1)(G2)

[2 marks]

c.

(H₀:) performance (of players) and (their) weight are independent. (A1)

Note: Accept “there is no association between performance (of players) and (their) weight”. Do not accept "not related" or "not correlated" or "not influenced".

[1 mark]

d.i.

0.287 (0.287436…) (G2)

[2 marks]

d.ii.

accept/ do not reject null hypothesis/H₀ (A1)(ft)

OR

performance (of players) and (their) weight are independent. (A1)(ft)

0.287 > 0.05 (R1)(ft)

Note: Accept p-value>significance level provided their p-value is seen in b(ii). Accept 28.7% > 5%. Do not award (A1)(R0). Follow through from part (d).

[2 marks]

e.

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

c.

[N/A]

d.i.

[N/A]

d.ii.

[N/A]

e.

Syllabus sections

Topic 4 - Statistical applications » 4.4 » The \({\chi ^2}\) test for independence: formulation of null and alternative hypotheses; significance levels; contingency tables; expected frequencies; degrees of freedom; \(p\)-values.

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