CAIE A-Level Mathematics 6.5 Hypothesis Tests

[Maximum number: 10]

At a certain large school it was found that the proportion of students not wearing correct uniform was 0.15 . The school sent a letter to parents asking them to ensure that their children wear the correct uniform. The school now wishes to test whether the proportion not wearing correct uniform has been reduced.

(a)

State suitable null and alternative hypotheses.

[ 1 ]

$\mathrm{H}_0: \mathrm{P}( not correct uniform )=0.15 \mathrm{H}_1: P (not correct uniform) < 0.15 B1 Allow " p "$

(b)

Use a binomial distribution to find the probability of a Type I error. You must justify your answer fully.

[ 5 ]

$Any two probs attempted using B(50,0.15) M1 \mathrm{P}(X \leqslant 3)=0.85^50+50 x 0.85^49 x 0.15+ ^50 C_2 x 0.85^48 x 0.15^2+ ^50 C_3 x 0.85^47 x 0.15^3P(X \leqslant 3)=0.85^50+50 x 0.85^49 x 0.15+ ^50 C_2 x 0.85^48 x 0.15^2+ ^50 C_3 x 0.85^47 x0.15^3 M1 Attempt the tail probability \mathrm{P}(0,1,2,3) with B(50,0.15) must be added. \mathrm{P}(X \leqslant 4)=0.04605+ ^50 C_4 x 0.85^46 x 0.15^4 M1 OE. Their \mathrm{P}(X \leqslant 3)+\mathrm{P}(X=4) or \mathrm{P}(0,1,2,3,4) with B(50,0.15) must be added. \mathrm{P}(X \leqslant 3)=0.0460 or 0.0461 [<0.05] \mathrm{P}(X \leqslant 4)=0.112 or [>0.05] A1 Both correct. OR if \mathrm{P}(X \leqslant 4) not seen; \mathrm{P}(4)=0.06606 and 0.06606>0.05 and \mathrm{P}(X \leqslant 3)=0.0460 scores M1 A1 \mathrm{P}( Type I )=0.0460 or 0.0461(3 sf) A1 Dependent on second M1. SC If M1M1M1A0 scored allow A1FT for incorrect \mathrm{P}(X \leqslant 3) as long as <0.05$

(c)

In fact 4 students out of the 50 are not wearing correct uniform.

State the conclusion of the test, explaining your answer.

[ 2 ]

$4 is outside critical region (\leqslant 3) OE or \mathrm{P}(X \leqslant 4)=0.112 which is >0.05 M1 FT working from (c). No evidence that proportion not wearing the correct uniform has decreased (Accept Ho) A1 In context not definite, e.g. not 'Proportion has not decreased'. No contradiction.$

(d)

State, with a reason, which of the errors, Type I or Type II, may have been made.

[ 2 ]

$Not rejected \mathrm{H}_0 *B1 FT FT If Reject \mathrm{H}_0 in (d) Type II DB1 FT FT Type I$

[Maximum number: 5]

The score on one spin of a 5 -sided spinner is denoted by the random variable X with probability distribution as shown in the table.

Table for Question 6.5 Hypothesis tests question 2 — CAIE A-Level Mathematics A2

Johann has another, similar, spinner. He suspects that it is biased so that the mean score is less than 2 . He spins his spinner 200 times and finds that the mean of the 200 scores is 1.86 .

Given that the variance of the score on one spin of this spinner is also 1.2 , test Johann's suspicion at the 5\% significance level.

$H_0:$ pop mean =2
$H_1:$ pop mean < 2
B1
1.86-21.2 / 200
M1
1.807
A1
comp z=1.645
M1
There is evidence that the spinner is biased so that mean is less than 2
A1

CAIE A-Level Mathematics 6.5 Hypothesis Tests Question Bank

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6.5 Hypothesis tests question 1

Question (a)

Question (b)

Question (c)

Question (d)

6.5 Hypothesis tests question 2