A Level · Mathematics 9709 · P6

A Level Mathematics Paper 6 · Probability & Statistics 2 practice papers

50 marks · 1h 15m. Each generated paper follows the 9709 Paper 6 · Probability & Statistics 2 format, with the same number and style of questions as the real exam, and comes with a Cambridge-style mark scheme.

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Sample Paper 6 · Probability & Statistics 2 questions

Question 1Structured · 9 marksA* challenge

The masses of apples are X ~ N(51, 25) grams and the masses of oranges are Y ~ N(80, 25) grams, independently.

(a)

Find the probability that the total mass of three randomly chosen apples is more than 78 g greater than the mass of one randomly chosen orange.

[4]
(b)

Find the probability that three times the mass of one apple is more than twice the mass of one orange.

[4]
(c)

Explain why the answer to (b) is not the same as the probability that the total mass of three apples is more than the total mass of two oranges.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.3085
E(X1 + X2 + X3 − Y) = 73 B1
Var = 100 B1
standardise M1
0.3085 A1
4
(b)0.3489
E(3X − 2Y) = -7 B1
Var(3X − 2Y) = 9 × 25 + 4 × 25 = 325 B1
standardise and correct tail M1
0.3489 A1
4
(c)3X uses one apple tripled (variance 9σ2); three different apples have variance 3σ2
variance of 3X is 9Var(X) but of X1 + X2 + X3 is 3Var(X) B1
1
Question 2Structured · 7 marksA* challenge

A random sample of 50 bags of sugar has masses x grams summarised by Σx = 2290 and Σx2 = 106765.6.

(a)

Calculate unbiased estimates of the population mean and variance.

[3]
(b)

Calculate a 99% confidence interval for the population mean.

[3]
(c)

Explain why the Central Limit Theorem was needed in (b), or state why it was not.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)45.8, 38.44
45.8 B1
149(Σx2 − (Σx)250) M1
38.44 A1
3
(b)(43.54, 48.06)
z = 2.576 B1
mean ± z √(variance ÷ n) M1
(43.54, 48.06) A1
3
(c)Needed: the population distribution is unknown; n is large so the sample mean is approximately normal
distribution of masses unknown; large n means sample mean approximately normal B1
1
Question 3Structured · 6 marks

X ~ N(293, 92) and Y ~ N(147, 62) are independent.

Find P(X + 2Y > 602) and explain why this is different from P(X + Y1 + Y2 > 602), where Y1, Y2 are independent observations of Y.

[6]
Show answers and mark scheme
PartAnswer and marking pointsMarks
0.159
E = 587 B1
Var = 225 B1
standardise M1
0.159 A1
Var(X + Y1 + Y2) = 153 B1
so the probabilities differ B1
6
Question 4Structured · 5 marks

The number of calls to a helpline in a 5-minute period has the distribution Po(1.5).

(a)

Find the probability of exactly 1 calls in a 5-minute period.

[2]
(b)

Find the probability of at least 2 calls in a 10-minute period.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.335
e−1.5 × 1.5 ÷ 1! M1
0.335 A1
2
(b)0.801
λ = 3 B1
1 − P(0) − P(1) M1
0.801 A1
3
Question 5Structured · 5 marks

The mean time for a process is believed to be 46 minutes, with standard deviation 4.9 minutes. After changes, a random sample of 43 times has mean 46.63 minutes. Test at the 5% significance level whether the mean time has increased.

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
z = 0.843; do not reject H0: insufficient evidence of an increase
H0: μ = 46, H1: μ > 46 B1
Standardise with σ/√n M1
z = 0.843 A1
Compare with 1.645 M1
Correct conclusion in context A1
5
Question 6Structured · 4 marks

A random sample of 80 observations of X gave x̄ = 25.4 and s2 = 24.8 (unbiased estimate).

Find a 95% confidence interval for the population mean, and explain why the Central Limit Theorem allows this even though the distribution of X is unknown.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(24.31, 26.49)
1.96 B1
x̄ ± z√(s2/n) M1
(24.31, 26.49) A1
n large so sample mean is approximately normal B1
4

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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