IB · Mathematics: Applications & Interpretation SL & HL · Statistics & probability

χ² tests: IB Mathematics: Applications & Interpretation questions with answers

Free χ² tests questions for IB Mathematics: Applications & Interpretation (SL & HL), part of Statistics & probability. Each question has the answer and the marking points an examiner would look for.

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χ² tests practice questions

Question 1Structured · 7 marksA* challenge

Each of 120 students answers 4 questions. It is claimed that the number of correct answers follows B(4, 0.5).

Correct answers01234
Frequency132638358
Table 1.1
(a)

Find the expected frequencies under the claim.

[2]
(b)

Explain why some categories must be combined, and state the number of degrees of freedom.

[2]
(c)

Find the p-value and state the conclusion at the 5% level.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)7.50, 30.0, 45.0, 30.0, 7.50
N × P(X = k) (M1)
7.50, 30.0, 45.0, 30.0, 7.50 A1
2
(b)4
expected frequencies less than 5 are combined R1
df = 4 A1
2
(c)0.163
χ2 = 6.52 (A1)
p = 0.163 A1
no evidence to reject the claim R1
3
Question 2Structured · 12 marks

A sweet company makes star-shaped and moon-shaped sweets. 70% are stars and the rest are moons. 10% of stars and 35% of moons are yellow.

(a)

A sweet is chosen at random.
(i) Find the probability that it is yellow.
(ii) Given that it is yellow, find the probability that it is a star.

[4]

The company claims the colours in each bag follow the distribution below. A bag of 120 sweets is tested with a χ2 goodness of fit test at the 5% level.

ColourBrownRedGreenOrangeYellowPurple
Claimed %152520201010
Observed191629221618
Table 2.1
(b)

Write down the null hypothesis.

[1]
(c)

Find the expected frequencies.

[2]
(d)

Write down the number of degrees of freedom.

[1]
(e)

Find the p-value.

[2]
(f)

State the conclusion of the test, giving a reason.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.175; 0.400
0.7 × 0.1 + 0.3 × 0.35 (M1)
0.175 A1
0.07000.175 (M1)
0.400 A1
4
(b)colours follow the claimed distribution
the colours are distributed as the company claims A1
1
(c)18.0, 30.0, 24.0, 24.0, 12.0, 12.0
percentage × 120 (M1)
18.0, 30.0, 24.0, 24.0, 12.0, 12.0 A1
2
(d)5
6 − 1 = 5 A1
1
(e)0.0330
χ2 = 12.1 (A1)
0.0330 A1
2
(f)reject H0
0.0330 < 0.05 R1
reject H0: the claim is not supported A1
2
Question 3Structured · 7 marks

A survey records the preferred type of music of 179 students. A χ2 test for independence is carried out at the 5% significance level.

PopRockClassicalTotal
Male31312587
Female28333192
Total596456179
Table 3.1
(a)

Write down the null hypothesis.

[1]
(b)

Find the expected frequency of female students who prefer classical.

[2]
(c)

Write down the number of degrees of freedom.

[1]
(d)

Find the p-value for the test, and state the conclusion, giving a reason.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)independent
preference is independent of gender A1
1
(b)28.8
92 × 56179 (M1)
28.8 A1
2
(c)2
(2 − 1)(3 − 1) = 2 A1
1
(d)p = 0.698
χ2 = 0.719 (A1)
p = 0.698 A1
0.698 > 0.05, so do not reject H0 R1
3
Question 4Structured · 7 marksA* challenge

Each of 200 students answers 4 questions. It is claimed that the number of correct answers follows B(4, 0.3).

Correct answers01234
Frequency448459103
Table 4.1
(a)

Find the expected frequencies under the claim.

[2]
(b)

Explain why some categories must be combined, and state the number of degrees of freedom.

[2]
(c)

Find the p-value and state the conclusion at the 5% level.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)48.0, 82.3, 52.9, 15.1, 1.62
N × P(X = k) (M1)
48.0, 82.3, 52.9, 15.1, 1.62 A1
2
(b)3
expected frequencies less than 5 are combined R1
df = 3 A1
2
(c)0.592
χ2 = 1.90 (A1)
p = 0.592 A1
no evidence to reject the claim R1
3
Question 5Structured · 12 marks

A sweet company makes star-shaped and moon-shaped sweets. 75% are stars and the rest are moons. 15% of stars and 40% of moons are yellow.

(a)

A sweet is chosen at random.
(i) Find the probability that it is yellow.
(ii) Given that it is yellow, find the probability that it is a star.

[4]

The company claims the colours in each bag follow the distribution below. A bag of 80 sweets is tested with a χ2 goodness of fit test at the 5% level.

ColourBrownRedGreenOrangeYellowPurple
Claimed %152520201010
Observed1817211752
Table 5.1
(b)

Write down the null hypothesis.

[1]
(c)

Find the expected frequencies.

[2]
(d)

Write down the number of degrees of freedom.

[1]
(e)

Find the p-value.

[2]
(f)

State the conclusion of the test, giving a reason.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.212; 0.529
0.75 × 0.15 + 0.25 × 0.4 (M1)
0.212 A1
0.1120.212 (M1)
0.529 A1
4
(b)colours follow the claimed distribution
the colours are distributed as the company claims A1
1
(c)12.0, 20.0, 16.0, 16.0, 8.00, 8.00
percentage × 80 (M1)
12.0, 20.0, 16.0, 16.0, 8.00, 8.00 A1
2
(d)5
6 − 1 = 5 A1
1
(e)0.0577
χ2 = 10.7 (A1)
0.0577 A1
2
(f)do not reject H0
0.0577 > 0.05 R1
do not reject H0: the data are consistent with the claim A1
2
Question 6Structured · 7 marks

A survey records the preferred type of music of 107 students. A χ2 test for independence is carried out at the 5% significance level.

PopRockClassicalTotal
Male28191562
Female11151945
Total393434107
Table 6.1
(a)

Write down the null hypothesis.

[1]
(b)

Find the expected frequency of female students who prefer classical.

[2]
(c)

Write down the number of degrees of freedom.

[1]
(d)

Find the p-value for the test, and state the conclusion, giving a reason.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)independent
preference is independent of gender A1
1
(b)14.3
45 × 34107 (M1)
14.3 A1
2
(c)2
(2 − 1)(3 − 1) = 2 A1
1
(d)p = 0.0551
χ2 = 5.80 (A1)
p = 0.0551 A1
0.0551 > 0.05, so do not reject H0 R1
3

More Statistics & probability questions

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Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in