AS Level · Mathematics 9709 · Probability & Statistics 1
The normal distribution: AS Level Mathematics questions with answers
Free The normal distribution questions for AS Level Mathematics (9709), part of Probability & Statistics 1. Each question has the answer and the marking points an examiner would look for.
The lifetimes of a type of battery are normally distributed with mean 178 hours and standard deviation 14 hours.
(a)
Find the lifetime exceeded by 20% of batteries.
[3]
(b)
6 batteries are chosen at random. Find the probability that exactly 4 of them have lifetimes less than 173 hours.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
190 hours
z = 0.842 B1 μ + zσ M1 190 A1
3
(b)
0.1036
p = 0.3605 B1 binomial with this p M1 15 × p4(1 − p)2M1 0.1036 A1
4
Question 2Structured · 10 marksA* challenge
The lengths of a type of fish are normally distributed with mean μ cm and standard deviation σ cm. 27.43% of the fish are shorter than 34 cm and 6.68% are longer than 65.5 cm.
(a)
Find the values of μ and σ.
[5]
(b)
35% of fish in another lake are diseased. A random sample of 40 fish is taken. Use a suitable approximation to find the probability that at least 17 fish are diseased.
[5]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
μ = 43, σ = 15
z = -0.600 or 1.500 seen B1 34 − μσ = -0.600 M1 65.5 − μσ = 1.500 M1 solve simultaneously M1 μ = 43, σ = 15 A1
The lengths of rods are normally distributed with mean 63 cm. 10% of rods are longer than 70 cm. Find the standard deviation.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
5.46 cm
z = 1.282 B1 70 − 63σ = 1.282 M1 5.46 A1
3
Question 4Structured · 5 marks
The masses, X g, of apples are normally distributed with mean 50 and standard deviation 8.
(a)
Find P(X < 57.2).
[3]
(b)
Find P(57.2 < X < 63.6).
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
0.8159
Standardise M1 z = 0.900 A1 0.8159 A1
3
(b)
0.1395
Φ(z2) − Φ(z1) M1 0.1395 A1
2
Question 5Structured · 7 marksA* challenge
The lifetimes of a type of battery are normally distributed with mean 202 hours and standard deviation 13 hours.
(a)
Find the lifetime exceeded by 10% of batteries.
[3]
(b)
6 batteries are chosen at random. Find the probability that exactly 4 of them have lifetimes more than 207 hours.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
219 hours
z = 1.282 B1 μ + zσ M1 219 A1
3
(b)
0.0953
p = 0.3503 B1 binomial with this p M1 15 × p4(1 − p)2M1 0.0953 A1
4
Question 6Structured · 10 marksA* challenge
The lengths of a type of fish are normally distributed with mean μ cm and standard deviation σ cm. 13.57% of the fish are shorter than 51.7 cm and 18.41% are longer than 77.7 cm.
(a)
Find the values of μ and σ.
[5]
(b)
35% of fish in another lake are diseased. A random sample of 60 fish is taken. Use a suitable approximation to find the probability that at least 25 fish are diseased.
[5]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
μ = 66, σ = 13
z = -1.100 or 0.900 seen B1 51.7 − μσ = -1.100 M1 77.7 − μσ = 0.900 M1 solve simultaneously M1 μ = 66, σ = 13 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