Free Hypothesis testing & errors questions for IB Mathematics: Applications & Interpretation (SL & HL), HL only, part of Statistics & probability. Each question has the answer and the marking points an examiner would look for.
A coin is tossed 25 times and lands heads 16 times. A test is carried out at the 5% significance level to decide whether the coin is biased towards heads.
(a)
State the hypotheses.
[1]
(b)
Find the p-value.
[2]
(c)
State the conclusion of the test.
[1]
(d)
State what is meant by a Type I error in this context.
[1]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
H0: p = 0.5, H1: p > 0.5
H0: p = 0.5; H1: p > 0.5 A1
1
(b)
0.115
P(X ≥ 16), X ~ B(25, 0.5) (M1) 0.115 A1
2
(c)
do not reject H0
0.115 > 0.05, do not reject H0: insufficient evidence of bias R1
1
(d)
conclude biased when fair
concluding the coin is biased when it is actually fair R1
1
Question 2Structured · 6 marksA* challenge
A manufacturer claims that 20% of its seeds fail to germinate. A gardener believes the true proportion is higher. She plants 20 seeds and will reject the claim if X, the number that fail, is at least c. The test is at the 5% significance level.
(a)
Find the least value of c.
[3]
(b)
Write down the probability of a type I error.
[1]
(c)
The true proportion is 0.4. Find the probability of a type II error.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
c = 8
X ~ B(20, 0.2) under H0(M1) P(X ≥ 8) = 0.0321 ≤ 0.05, P(X ≥ 7) = 0.0867 > 0.05 (A1) c = 8 A1
3
(b)
0.0321
0.0321 A1
1
(c)
0.416
P(X ≤ 7) with X ~ B(20, 0.4) (M1) 0.416 A1
2
Question 3Structured · 4 marks
A coin is tested for bias by tossing it 20 times. H₀: p = 0.5; the critical region is X ≤ 5 or X ≥ 15.
(a)
Find the probability of a type I error.
[2]
(b)
The coin actually has p = 0.7. Find the probability of a type II error.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
0.0414
P(X ≤ 5) + P(X ≥ 15) (M1) 0.0414 A1
2
(b)
0.584
P(6 ≤ X ≤ 14) with p = 0.7 (M1) 0.584 A1
2
Question 4Structured · 5 marksA* challenge
A coin is tossed 20 times and lands heads 14 times. A test is carried out at the 5% significance level to decide whether the coin is biased towards heads.
(a)
State the hypotheses.
[1]
(b)
Find the p-value.
[2]
(c)
State the conclusion of the test.
[1]
(d)
State what is meant by a Type I error in this context.
[1]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
H0: p = 0.5, H1: p > 0.5
H0: p = 0.5; H1: p > 0.5 A1
1
(b)
0.0577
P(X ≥ 14), X ~ B(20, 0.5) (M1) 0.0577 A1
2
(c)
do not reject H0
0.0577 > 0.05, do not reject H0: insufficient evidence of bias R1
1
(d)
conclude biased when fair
concluding the coin is biased when it is actually fair R1
1
Question 5Structured · 6 marksA* challenge
A manufacturer claims that 30% of its seeds fail to germinate. A gardener believes the true proportion is higher. She plants 25 seeds and will reject the claim if X, the number that fail, is at least c. The test is at the 5% significance level.
(a)
Find the least value of c.
[3]
(b)
Write down the probability of a type I error.
[1]
(c)
The true proportion is 0.5. Find the probability of a type II error.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
c = 12
X ~ B(25, 0.3) under H0(M1) P(X ≥ 12) = 0.0442 ≤ 0.05, P(X ≥ 11) = 0.0978 > 0.05 (A1) c = 12 A1
3
(b)
0.0442
0.0442 A1
1
(c)
0.345
P(X ≤ 11) with X ~ B(25, 0.5) (M1) 0.345 A1
2
Question 6Structured · 5 marksA* challenge
A coin is tossed 20 times and lands heads 13 times. A test is carried out at the 5% significance level to decide whether the coin is biased towards heads.
(a)
State the hypotheses.
[1]
(b)
Find the p-value.
[2]
(c)
State the conclusion of the test.
[1]
(d)
State what is meant by a Type I error in this context.
[1]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
H0: p = 0.5, H1: p > 0.5
H0: p = 0.5; H1: p > 0.5 A1
1
(b)
0.132
P(X ≥ 13), X ~ B(20, 0.5) (M1) 0.132 A1
2
(c)
do not reject H0
0.132 > 0.05, do not reject H0: insufficient evidence of bias R1
1
(d)
conclude biased when fair
concluding the coin is biased when it is actually fair R1
Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
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