A Level · Mathematics 9709 · Probability & Statistics 2
Hypothesis tests: A Level Mathematics questions with answers
Free Hypothesis tests questions for A Level Mathematics (9709), part of Probability & Statistics 2. Each question has the answer and the marking points an examiner would look for.
A company claims that 25% of customers prefer its product. A manager believes the proportion is lower. In a random sample of 20 customers, 2 prefer the product.
(a)
Test at the 5% significance level whether there is evidence to support the manager’s belief.
[5]
(b)
State the critical region and the probability of a Type I error.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
P(X ≤ 2) = 0.0913 > 0.05: do not reject H0; insufficient evidence
H0: p = 0.25, H1: p < 0.25 B1 B(20, 0.25) M1 P(X ≤ 2) = 0.0913 A1 comparison with 0.05 M1 conclusion in context A1
5
(b)
X ≤ 1; 0.0243
X ≤ 1 B1 0.0243 B1
2
Question 2Structured · 9 marksA* challenge
The number of accidents on a road averaged 3.5 per week. After a speed limit was introduced, there were 15 accidents in 6 weeks.
(a)
Test at the 5% significance level whether the mean number of accidents has decreased.
[6]
(b)
State what is meant by a Type I error in this context, and find the probability of making a Type I error in this test.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
P(X ≤ 15) = 0.1111 > 0.05: do not reject H0; insufficient evidence of a decrease
H0: λ = 3.5 (per week), H1: λ < 3.5 B1 Po(21) used B1 P(X ≤ 15) attempted M1 0.1111 A1 comparison with 0.05 M1 conclusion in context, not definite A1
6
(b)
concluding that the mean has decreased when it has not; critical region X ≤ 13, so P(Type I error) = 0.0434
correct description in context B1 critical region X ≤ 13 found M1 0.0434 A1
3
Question 3Structured · 5 marks
A company claims that 20% of its customers are dissatisfied. In a random sample of 15 customers, 7 were dissatisfied.
Test at the 5% significance level whether the proportion is greater than claimed.
[5]
Show answers and mark scheme
Part
Answer and marking points
Marks
reject H₀
H₀: p = 0.2, H₁: p > 0.2 B1 B(15, 0.2) M1 P(X ≥ 7) = 0.01806 A1 compare with 0.05 M1 reject H₀: evidence proportion greater A1
5
Question 4Structured · 5 marks
The mean time for a process is believed to be 45 minutes, with standard deviation 5.3 minutes. After changes, a random sample of 61 times has mean 45.35 minutes. Test at the 5% significance level whether the mean time has increased.
[5]
Show answers and mark scheme
Part
Answer and marking points
Marks
z = 0.516; do not reject H0: insufficient evidence of an increase
H0: μ = 45, H1: μ > 45 B1 Standardise with σ/√n M1 z = 0.516 A1 Compare with 1.645 M1 Correct conclusion in context A1
5
Question 5Structured · 7 marksA* challenge
A company claims that 25% of customers prefer its product. A manager believes the proportion is lower. In a random sample of 15 customers, 1 prefers the product.
(a)
Test at the 5% significance level whether there is evidence to support the manager’s belief.
[5]
(b)
State the critical region and the probability of a Type I error.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
P(X ≤ 1) = 0.0802 > 0.05: do not reject H0; insufficient evidence
H0: p = 0.25, H1: p < 0.25 B1 B(15, 0.25) M1 P(X ≤ 1) = 0.0802 A1 comparison with 0.05 M1 conclusion in context A1
5
(b)
X ≤ 0; 0.0134
X ≤ 0 B1 0.0134 B1
2
Question 6Structured · 9 marksA* challenge
The number of accidents on a road averaged 3.5 per week. After a speed limit was introduced, there were 16 accidents in 6 weeks.
(a)
Test at the 5% significance level whether the mean number of accidents has decreased.
[6]
(b)
State what is meant by a Type I error in this context, and find the probability of making a Type I error in this test.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
P(X ≤ 16) = 0.1629 > 0.05: do not reject H0; insufficient evidence of a decrease
H0: λ = 3.5 (per week), H1: λ < 3.5 B1 Po(21) used B1 P(X ≤ 16) attempted M1 0.1629 A1 comparison with 0.05 M1 conclusion in context, not definite A1
6
(b)
concluding that the mean has decreased when it has not; critical region X ≤ 13, so P(Type I error) = 0.0434
correct description in context B1 critical region X ≤ 13 found M1 0.0434 A1
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Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in