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Date May Example question Marks available 2 Reference code EXM.3.AHL.TZ0.8
Level Additional Higher Level Paper Paper 3 Time zone Time zone 0
Command term Find Question number 8 Adapted from N/A

Question

This question explores methods to analyse the scores in an exam.

A random sample of 149 scores for a university exam are given in the table.

The university wants to know if the scores follow a normal distribution, with the mean and variance found in part (a).

The expected frequencies are given in the table.

The university assigns a pass grade to students whose scores are in the top 80%.

The university also wants to know if the exam is gender neutral. They obtain random samples of scores for male and female students. The mean, sample variance and sample size are shown in the table.

The university awards a distinction to students who achieve high scores in the exam. Typically, 15% of students achieve a distinction. A new exam is trialed with a random selection of students on the course. 5 out of 20 students achieve a distinction.

A different exam is trialed with 16 students. Let p be the percentage of students achieving a distinction. It is desired to test the hypotheses

H 0 : p = 0.15 against  H 1 : p > 0.15

It is decided to reject the null hypothesis if the number of students achieving a distinction is greater than 3.

Find unbiased estimates for the population mean.

[1]
a.i.

Find unbiased estimates for the population Variance.

[2]
a.ii.

Show that the expected frequency for 20 <  x ≤ 4 is 31.5 correct to 1 decimal place.

[3]
b.

Perform a suitable test, at the 5% significance level, to determine if the scores follow a normal distribution, with the mean and variance found in part (a). You should clearly state your hypotheses, the degrees of freedom, the p-value and your conclusion.

[8]
c.

Use the normal distribution model to find the score required to pass.

[2]
d.

Perform a suitable test, at the 5% significance level, to determine if there is a difference between the mean scores of males and females. You should clearly state your hypotheses, the p-value and your conclusion.

[6]
e.

Perform a suitable test, at the 5% significance level, to determine if it is easier to achieve a distinction on the new exam. You should clearly state your hypotheses, the critical region and your conclusion.

[6]
f.

Find the probability of making a Type I error.

[3]
g.i.

Given that p = 0.2 find the probability of making a Type II error.

[3]
g.ii.

Markscheme

52.8     A1

[1 mark]

a.i.

s n 1 2 = 23.7 2 = 562       M1A1

[2 marks]

a.ii.

P ( 20 < x 40 ) = 0.211       M1A1

0.211 × 149       M1

= 31.5       AG

[3 marks]

b.

use of a  χ 2  goodness of fit test      M1

H 0 : x N ( 52.8 , 562 ) and      A1A1

υ = 5 1 2 = 2       A1

p-value = 0.569       A2

Since 0.569 > 0.05        R1

Insufficient evidence to reject  H 0 . The scores follow a normal distribution.      A1

[8 marks]

c.

Φ 1 ( 0.2 ) = 32.8      M1A1

[2 marks]

d.

use of a t-test     M1

H 0 : μ m = μ f and  H 1 : μ m μ f       A1

p-value = 0.180       A2

Since 0.180 > 0.05       R1  

Insufficient evidence to reject H 0 . There is no difference between males and females.      A1

[6 marks]

e.

use of test for proportion using Binomial distribution    M1

H 0 : p = 0.15 and H 1 : p > 0.15       A1

P ( X 6 ) = 0.0673 and  P ( X 7 ) = 0.0219     M1

So the critical region is  X 7       A1

Since 5 < 7        R1 

Insufficient evidence to reject H 0 . It is not easier to achieve a distinction on the new exam.      A1

[6 marks]

f.

using H 0 , X B ( 16 , 0.15 )     M1

P ( X > 3 ) = 0.210     M1A1

[3 marks]

g.i.

using H 1 , X B ( 16 , 0.2 )     M1

P ( X 3 ) = 0.598    M1A1

[3 marks]

g.ii.

Examiners report

[N/A]
a.i.
[N/A]
a.ii.
[N/A]
b.
[N/A]
c.
[N/A]
d.
[N/A]
e.
[N/A]
f.
[N/A]
g.i.
[N/A]
g.ii.

Syllabus sections

Topic 4—Statistics and probability » SL 4.9—Normal distribution and calculations
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Topic 4—Statistics and probability

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