A Level · Mathematics 9709 · Probability & Statistics 1
The normal distribution: A Level Mathematics questions with answers
Free The normal distribution questions for A 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 225 hours and standard deviation 26 hours.
(a)
Find the lifetime exceeded by 5% of batteries.
[3]
(b)
6 batteries are chosen at random. Find the probability that exactly 4 of them have lifetimes more than 230 hours.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
268 hours
z = 1.645 B1 μ + zσ M1 268 A1
3
(b)
0.1606
p = 0.4238 B1 binomial with this p M1 15 × p4(1 − p)2M1 0.1606 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. 13.57% of the fish are shorter than 38.8 cm and 11.51% are longer than 66.4 cm.
(a)
Find the values of μ and σ.
[5]
(b)
30% of fish in another lake are diseased. A random sample of 50 fish is taken. Use a suitable approximation to find the probability that at least 18 fish are diseased.
[5]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
μ = 52, σ = 12
z = -1.100 or 1.200 seen B1 38.8 − μσ = -1.100 M1 66.4 − μσ = 1.200 M1 solve simultaneously M1 μ = 52, σ = 12 A1
The masses, X g, of apples are normally distributed with mean 78 and standard deviation 4.
(a)
Find P(X < 80.8).
[3]
(b)
Find P(80.8 < X < 81.2).
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
0.7580
Standardise M1 z = 0.700 A1 0.7580 A1
3
(b)
0.0301
Φ(z2) − Φ(z1) M1 0.0301 A1
2
Question 4Structured · 3 marks
The lengths of rods are normally distributed with mean 56 cm. 10% of rods are longer than 68 cm. Find the standard deviation.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
9.36 cm
z = 1.282 B1 68 − 56σ = 1.282 M1 9.36 A1
3
Question 5Structured · 7 marksA* challenge
The lifetimes of a type of battery are normally distributed with mean 151 hours and standard deviation 23 hours.
(a)
Find the lifetime exceeded by 5% of batteries.
[3]
(b)
5 batteries are chosen at random. Find the probability that exactly 3 of them have lifetimes less than 146 hours.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
189 hours
z = 1.645 B1 μ + zσ M1 189 A1
3
(b)
0.2436
p = 0.4140 B1 binomial with this p M1 10 × p3(1 − p)2M1 0.2436 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. 27.43% of the fish are shorter than 61.6 cm and 11.51% are longer than 77.8 cm.
(a)
Find the values of μ and σ.
[5]
(b)
40% 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 18 fish are diseased.
[5]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
μ = 67, σ = 9
z = -0.600 or 1.200 seen B1 61.6 − μσ = -0.600 M1 77.8 − μσ = 1.200 M1 solve simultaneously M1 μ = 67, σ = 9 A1
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