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 be the percentage of students achieving a distinction. It is desired to test the hypotheses
against
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.
Find unbiased estimates for the population Variance.
Show that the expected frequency for 20 < ≤ 4 is 31.5 correct to 1 decimal place.
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.
Use the normal distribution model to find the score required to pass.
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.
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.
Find the probability of making a Type I error.
Given that find the probability of making a Type II error.
Markscheme
52.8 A1
[1 mark]
M1A1
[2 marks]
M1A1
M1
= 31.5 AG
[3 marks]
use of a goodness of fit test M1
and A1A1
A1
p-value = 0.569 A2
Since 0.569 > 0.05 R1
Insufficient evidence to reject . The scores follow a normal distribution. A1
[8 marks]
M1A1
[2 marks]
use of a t-test M1
and A1
p-value = 0.180 A2
Since 0.180 > 0.05 R1
Insufficient evidence to reject . There is no difference between males and females. A1
[6 marks]
use of test for proportion using Binomial distribution M1
and A1
and M1
So the critical region is A1
Since 5 < 7 R1
Insufficient evidence to reject . It is not easier to achieve a distinction on the new exam. A1
[6 marks]
using M1
M1A1
[3 marks]
using M1
M1A1
[3 marks]