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Date	May 2022	Marks available	1	Reference code	22M.1.AHL.TZ2.9
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 2
Command term	Find	Question number	9	Adapted from	N/A

Question

A psychologist records the number of digits ( $d$ $d$ ) of $π$ $π$ that a sample of IB Mathematics higher level candidates could recall.

The psychologist has read that in the general population people can remember an average of $4.4$ $4.4$ digits of $π$ $π$ . The psychologist wants to perform a statistical test to see if IB Mathematics higher level candidates can remember more digits than the general population.

$H_{0} : μ = 4.4$ $H_{0} : μ = 4.4$ is the null hypothesis for this test.

Find an unbiased estimate of the population mean of $d$ $d$ .

[1]

Find an unbiased estimate of the population variance of $d$ $d$ .

[2]

State the alternative hypothesis.

[1]

c.i.

Given that all assumptions for this test are satisfied, carry out an appropriate hypothesis test. State and justify your conclusion. Use a $5 %$ $5 %$ significance level.

[4]

c.ii.

Markscheme

$¯ x = 4.63 (4.62686 \dots)$ $\bar{x} = 4.63 (4.62686 \dots)$ A1

[1 mark]

$s_{n - 1} = 1.098702$ $s_{n - 1} = 1.098702$ (A1)

$s_{n - 1}^{2} = 1.21 (1.207146 \dots)$ $s_{n - 1}^{2} = 1.21 (1.207146 \dots)$ A1

Note: Award A0A0 for an answer of $1.19$ $1.19$ from biased estimate.

[2 marks]

$H_{1} : μ > 4.4$ $H_{1} : μ > 4.4$ A1

[1 mark]

c.i.

METHOD 1

using a $z$ $z$ -test (M1)

$p = 0.0454992 \dots$ $p = 0.0454992 \dots$ A1

$p < 0.05$ $p < 0.05$ R1

reject null hypothesis A1

(therefore there is significant evidence that the IB HL math students know more digits of $π$ $π$ than the population in general)

Note: Do not award R0A1. Allow R1A1 for consistent conclusion following on from their $p$ $p$ -value.

METHOD 2

using a $t$ $t$ -test (M1)

$p = 0.0478584 \dots$ $p = 0.0478584 \dots$ A1

$p < 0.05$ $p < 0.05$ R1

reject null hypothesis A1

(therefore there is significant evidence that the IB HL math students know more digits of $π$ $π$ than the population in general)

Note: Do not award R0A1. Allow R1A1 for consistent conclusion following on from their $p$ $p$ -value.

[4 marks]

c.ii.

Examiners report

In parts (a) and (b), candidates used the 1-Var Stats facility to find the estimates of mean and variance although some forgot to include the frequency list so that they just found the mean and variance of the numbers 2, 3, …6, 7. Candidates who looked ahead realized that the answers to parts (a) and (b) would be included in the output from using their test. In part (c), the question was intended to use the t-test (as the population variance was unknown), however since the population could not be assumed to be normally distributed, the Principal Examiner condoned the use of the z-test (with the estimated variance from part (b)). As both methods could only produce an approximate p-value, either method (and the associated p-value) was awarded full marks.

[N/A]

c.i.

[N/A]

c.ii.

Syllabus sections

Topic 4—Statistics and probability » AHL 4.14—Linear transformation of a single RV, E(X) and VAR(X), unbiased estimators

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Topic 4—Statistics and probability

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