IB Questionbank

User interface language: English | Español

Date	May 2021	Marks available	2	Reference code	21M.3.AHL.TZ1.2
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 1
Command term	Show that	Question number	2	Adapted from	N/A

Question

A firm wishes to review its recruitment processes. This question considers the validity and reliability of the methods used.

Every year an accountancy firm recruits new employees for a trial period of one year from a large group of applicants.

At the start, all applicants are interviewed and given a rating. Those with a rating of either Excellent, Very good or Good are recruited for the trial period. At the end of this period, some of the new employees will stay with the firm.

It is decided to test how valid the interview rating is as a way of predicting which of the new employees will stay with the firm.

Data is collected and recorded in a contingency table.

The next year’s group of applicants are asked to complete a written assessment which is then analysed. From those recruited as new employees, a random sample of size $18$ is selected.

The sample is stratified by department. Of the $91$ new employees recruited that year, $55$ were placed in the national department and $36$ in the international department.

At the end of their first year, the level of performance of each of the $18$ employees in the sample is assessed by their department manager. They are awarded a score between $1$ (low performance) and $10$ (high performance).

The marks in the written assessment and the scores given by the managers are shown in both the table and the scatter diagram.

The firm decides to find a Spearman’s rank correlation coefficient, $r_{s}$ , for this data.

The same seven employees are given the written assessment a second time, at the end of the first year, to measure its reliability. Their marks are shown in the table below.

The written assessment is in five sections, numbered $1$ to $5$ . At the end of the year, the employees are also given a score for each of five professional attributes: $V, W, X, Y$ and $Z$ .

The firm decides to test the hypothesis that there is a correlation between the mark in a section and the score for an attribute.

They compare marks in each of the sections with scores for each of the attributes.

Use an appropriate test, at the $5 %$ significance level, to determine whether a new employee staying with the firm is independent of their interview rating. State the null and alternative hypotheses, the $p$ -value and the conclusion of the test.

[6]

Show that $11$ employees are selected for the sample from the national department.

[2]

Without calculation, explain why it might not be appropriate to calculate a correlation coefficient for the whole sample of $18$ employees.

[2]

c.i.

Find $r_{s}$ for the seven employees working in the international department.

[4]

c.ii.

Hence comment on the validity of the written assessment as a measure of the level of performance of employees in this department. Justify your answer.

[2]

c.iii.

State the name of this type of test for reliability.

[1]

d.i.

For the data in this table, test the null hypothesis, $H_{0} : ρ = 0$ , against the alternative hypothesis, $H_{1} : ρ > 0$ , at the $5 %$ significance level. You may assume that all the requirements for carrying out the test have been met.

[4]

d.ii.

Hence comment on the reliability of the written assessment.

[1]

d.iii.

Write down the number of tests they carry out.

[1]

e.i.

The tests are performed at the $5 %$ significance level.

Assuming that:

there is no correlation between the marks in any of the sections and scores in any of the attributes,
the outcome of each hypothesis test is independent of the outcome of the other hypothesis tests,

find the probability that at least one of the tests will be significant.

[4]

e.ii.

The firm obtains a significant result when comparing section $2$ of the written assessment and attribute $X$ . Interpret this result.

[1]

e.iii.

Markscheme

Use of $χ^{2}$ test for independence (M1)

$H_{0} :$ Staying (or leaving) the firm and interview rating are independent.
$H_{1} :$ Staying (or leaving) the firm and interview rating are not independent A1

Note: For $H_{1}$ accept ‘…are dependent’ in place of ‘…not independent’.

$p$ -value $= 0.487 (0.487221 \dots)$ A2

Note: Award A1 for $χ^{2} = 1.438 \dots$ if $p$ -value is omitted or incorrect.

$0.487 > 0.05$ R1

(the result is not significant at the $5 %$ level)

insufficient evidence to reject the $H_{0}$ (or “accept $H_{0}$ ”) A1

Note: Do not award R0A1. The final R1A1 can follow through from their incorrect $p$ -value

[6 marks]

$\frac{55}{91} \times 18 = 10.9 (10.8791 \dots)$ M1A1

Note: Award A1 for anything that rounds to $10.9$ .

$\approx 11$ AG

[2 marks]

there seems to be a difference between the two departments (A1)

the international department manager seems to be less generous than the national department manager R1

Note: The A1 is for commenting there is a difference between the two departments and the R1 is for correctly commenting on the direction of the difference

[2 marks]

c.i.

(M1)(A1)

Note: Award (M1) for an attempt to rank the data, and (A1) for correct ranks for both variables. Accept either set of rankings in reverse.

$r_{s} = 0.909 (0.909241 \dots)$ (M1)(A1)

Note: The (M1) is for calculating the PMCC for their ranks.

Note: If a final answer of $0.9107$ is seen, from use of $1 - \frac{6 Σ d^{2}}{n (n^{2} - 1)}$ , award (M1)(A1)A1.
Accept $- 0.909$ if one set of ranks has been ordered in reverse.

[4 marks]

c.ii.

EITHER

there is a (strong) association between the written assessment mark and the manager scores. A1

there is a (strong) agreement in the rank order of the written assessment marks and the rank order of the manager scores. A1

there is a (strong linear) correlation between the rank order of the written assessment marks and the rank order of the manager scores. A1

Note: Follow through on a value for their value of $r_{s}$ in c(ii).

THEN

the written assessment is likely to be a valid measure (of the level of employee performance) R1

[2 marks]

c.iii.

test-retest A1

[1 mark]

d.i.

$p$ -value $= 0.00209 (0.0020939 \dots)$ A2

$0.00209 < 0.05$ R1

(the result is significant at the $5 %$ level)
(there is sufficient evidence to) reject $H_{0}$ A1

Note: Do not award R0A1. Accept “accept $H_{1}$ ”. The final R1A1 can follow through from their incorrect $p$ -value.

[4 marks]

d.ii.

the test seems reliable A1

Note: Follow through from their answer in part (d)(ii). Do not award if there is no conclusion in d(ii).

[1 mark]

d.iii.

$25$ A1

[1 mark]

e.i.

probability of significant result given no correlation is $0.05$ (M1)

probability of at least one significant result in $25$ tests is

$1 - 0 . 95^{25}$ (M1)(A1)

Note: Award (M1) for use of $1 - P (0)$ or the binomial distribution with any value of $p$ .

$= 0.723 (0.722610 \dots)$ A1

[4 marks]

e.ii.

(though the result is significant) it is very likely that one significant result would be achieved by chance, so it should be disregarded or further evidence sought R1

[1 mark]

e.iii.