CAIE A-Level Further Mathematics 4.3 Chi-Squared Tests

Practise deriving expected frequencies, combining sparse categories and carrying out chi-squared goodness-of-fit or independence tests with justified degrees of freedom.

Exam points

calculate expected frequencies from the fitted distribution or row and column totals
combine adjacent categories before testing when an expected frequency is below five
use the adjusted degrees of freedom and compare the chi-squared statistic in context

Covered syllabus points

4.3.1—Fitted distributions 4.3.2—Goodness-of-fit test 4.3.3—Independence test

[Maximum number: 9]

A random sample of 50 values of the continuous random variable X was taken. These values are summarised in the following table.

Table for Question 4.3 Chi-squared tests question 1 — CAIE A-Level Further Math A2

It is required to test the goodness of fit of the distribution with probability density function f given by

f(x)= \begin{cases}\frac{1}{24}\left(\frac{4}{x^{2}}+x^{2}\right) & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise }\end{cases}

The expected frequencies, correct to 4 decimal places, are given in the following table.

(a)

Show that a=4.6007 and find the value of b.

[ 3 ]

$\int_{1.5}^{2} \frac{1}{24}\left(\frac{4}{x^{2}}+x^{2}\right) \mathrm{d} x=\frac{1}{24}\left[-\frac{4}{x}+\frac{x^{3}}{3}\right]$

M1

Integration with correct powers and correct limits
seen.

$a=50 \times 0.092014=4.6007$

A1

AG

$\frac{53}{576}$ or $\frac{1325}{288}$ or 0.092014 or 4.60069 seen

b=11.4211

B1

Or 11.421(0)

3

(b)

Carry out a goodness of fit test, at the 10% significance level, to test whether f is a satisfactory model for the data.

[ 6 ]

Combine first two columns: 6, 9.0278

M1

Must be seen, or implied by 1.0155 .

Calculate value of chi-squared: 1.0155+0.5715+0.7661+0.2183+0.5885

M1

At least 2 correct values (at least 3sf) or
expressions seen.
Allow columns not combined or three columns
combined.

3.16

A1

CWO.

Dependent on M1M1 scored. SCB1 for 3.16 with no working

$\mathrm{H}_{0}: \mathrm{f}$ is a good fit for the data

$\mathrm{H}_{1}$ : f is not a good fit for the data

B1

Tabular value of chi-squared: 7.78

' 3.16 ' < 7.78 and accept $\mathrm{H}_{0}$

M1

Correct tabular value: allow correct FT value if
columns not combined (9.236) or three columns
combined (6.251).

Insufficient evidence to suggest that f is not a good fit for the data

Condone: sufficient evidence to suggest that f is a good fit for the data.

A1

Correct conclusion in context, following correct
work, level of uncertainty in language (not 'prove',
not 'there is no evidence'), no contradictions.
A0 if hypotheses reversed

6

[Maximum number: 7]

A town council has published its plans for redeveloping the town centre and residents are being asked whether they approve or disapprove. A random sample of 250 responses has been selected from residents in the four main streets in the town: North, East, South and West Streets. The results are shown in the table.

Table for Question 4.3 Chi-squared tests question 2 — CAIE A-Level Further Math A2

Test, at the 5% significance level, whether the opinions of the residents are independent of the streets on which they live.

$\mathrm{H}_{0}$ : opinion is independent of street $\mathrm{H}_{1}$ : opinion is not independent of street

B1

33 & 32.24 & 54 & 57.66 & 42 & 43.4 & 26 & 21.7 & 155

19 & 19.76 & 39 & 35.34 & 28 & 26.6 & 9 & 13.3 & 95

52 & & 93 & & 70 & & 35 & & 250

M1

At least 2 correct values or expressions.

A1

6 correct values or expressions.

Calculate chi-squared values:
0.0179+0.2323+0.0452+0.8521+0.0292+0.3790+0.0737+1.3902

M1

Correct to 3 decimal places.
At least 2 correct values or expressions.

3.02

A1

SC B1 3.02 following M1M0
SC B2 3.02 following M0M0

Tabular value, 3 degrees of freedom 7.815: ' 3.02 ' <7.815, accept $\mathrm{H}_{0}$

M1

Allow 'not significant'.

Insufficient evidence to suggest that opinion depends on street.

A1

CWO. Correct conclusion in context, following correct
work, level of uncertainty in language.
A0 if hypotheses wrong way round or missing.

7

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Covered syllabus points

4.3 Chi-squared tests question 1

Question (a)

Question (b)

4.3 Chi-squared tests question 2