Edexcel A-Level Mathematics A2 S3.4 Goodness of Fit and Contingency Tables Questions

Practise chi-squared tests for distribution fit and independence, including hypotheses, expected frequencies, degrees of freedom and decisions.

Exam points

test whether a Poisson, binomial or uniform model fits observed frequency data
use contingency tables to test association between categories
choose degrees of freedom and a χ² critical value before making the conclusion

Covered syllabus points

S3.4.1 - null and alternative hypotheses S3.4.2 - Degrees of freedom

[Maximum number: 12]

A research student is investigating the number of children who are girls in families with 4 children.

The table below shows her results for 200 such families.

Table for Question 1 — Edexcel A-Level Mathematics A2

The research student suggests that a binomial distribution with $p=\frac{1}{2}$ could be a suitable model for the number of children who are girls in a family of 4 children.

(a)

Using her results and a 5% significance level, test the research student's claim. You should state your hypotheses, expected frequencies, test statistic and the critical value used.

The research student decides to refine the model and retains the idea of using a binomial distribution but does not specify the probability that the child is a girl.

[ 8 ]

$H_0$ : B(4,0.5) is a suitable model.
$H_1$ : B(4,0.5) is not a suitable model.
Expected frequencies: 12.5, 50, 75, 50, 12.5.
$\sum \frac{(O-E)^2}{E}=\frac{(15-12.5)^2}{12.5}+\cdots+\frac{(10-12.5)^2}{12.5}$
or
$\sum \frac{O^2}{E}-N=\frac{15^2}{12.5}+\cdots+\frac{10^2}{12.5}-200$ v=4 $\chi^2_4(0.05)=9.488$ , critical region >9.488.
Sufficient evidence to say that the research student's claim is not supported.
A1 ft

Figure for Answer to Question (a) — Edexcel A-Level Mathematics A2

(b)

Test, at the 5% significance level, whether using a binomial distribution is suitable to model the number of children who are girls in a family of 4 children.
You should state your hypotheses and the critical value used.

[ 4 ]

B1
v=3 $\chi^2_3(0.05)=7.815$ , critical region >7.815.
B1 ft
No significant evidence to say that the binomial is not a reasonable model.

[Maximum number: 11]

Luka wants to carry out a survey of students at his school.

He obtains a list of all 280 students.

(a)

Determine the conclusion of this test.

The digits generated by Luka are taken two at a time to form two-digit numbers.

Figure 2 shows the frequency of two-digit numbers in his table.

Figure 2

[ 3 ]

$v=10-1=9 \quad \chi_{9}^{2}(10 \%)=14.684$

B1 B1ft

No evidence (at 10\% level of significance) that the digits generated do not
follow a uniform distribution.

dB1

(3)

B1 B1ft B1

(b)

Test, at the 10% level of significance, whether the two-digit numbers generated by Luka follow a uniform distribution. You should state the hypotheses, the degrees of freedom and the critical value used for this test.

There are 70 students in Year 12 at his school.

[ 8 ]

$\mathrm{H}_{0}$ : Uniform distribution is a good fit (for the two-digit numbers generated)

$\mathrm{H}_{1}$ : Uniform distribution is not a good fit (for the two-digit numbers generated)
O & E & $\frac{(O-E)^{2}}{E}$ & $\frac{O^{2}}{E}$

31 & 40 & 2.025 & 24.025

49 & 40 & 2.025 & 60.025

30 & 40 & 2.5 & 22.5

42 & 40 & 0.1 & 44.1

48 & 40 & 1.6 & 57.6

200 & 200 & 8.25 & 208.25
$\sum \frac{(O-E)^{2}}{E}=8.25 \quad$ or $\sum \frac{O^{2}}{E}-200=8.25$ $\chi_{4}^{2}(10 \%)=7.779$
[8.25 > 7.779] Reject $\mathrm{H}_{0}$ / Significant
There is evidence to suggest the two-digit numbers generated may not follow a
uniform distribution.
(8)

B1 B1 M1 A1 A1 B1 M1 A1

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question (a)

Question (b)