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A Level H1 Mathematics Practice Paper 2

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A Level H1 Mathematics From Real Exams Generated by Owl Alpha Updated 2026-06-07

Questions

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TuitionGoWhere Practice Paper - Maths H1 A-Level

TuitionGoWhere Secondary School (AI)


Subject: Mathematics
Level: A-Level H1
Paper: Practice Paper (Statistics & Probability)
Version: 2 of 5
Duration: 1 hour 30 minutes
Total Marks: 60

Name: ___________________________
Class: ___________________________
Date: ___________________________


Instructions

  • Write your answers in the spaces provided.
  • Show all working clearly. Marks may be awarded for correct working even if the final answer is wrong.
  • Give non-exact answers correct to 3 significant figures unless otherwise stated.
  • The use of a graphing calculator is expected.
  • The total mark for this paper is 60.
  • You are advised to spend no more than 1 hour 30 minutes on this paper.

Section A: Short Answer Questions [25 marks]

Answer ALL questions in this section.


Question 1 [2 marks]

A random sample of 6 students recorded the number of hours they spent on revision in a week:

12, 15, 10, 14, 11, 1312,\ 15,\ 10,\ 14,\ 11,\ 13

Calculate the unbiased estimate of the population mean.

xˉ=xin\bar{x} = \frac{\sum x_i}{n}

Answer: ______________ hours


Question 2 [2 marks]

Using the same data from Question 1, calculate the unbiased estimate of the population variance.

s2=(xixˉ)2n1s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}

Answer: ______________ hours²


Question 3 [2 marks]

The random variable XB(12,0.35)X \sim \mathrm{B}(12, 0.35). Find P(X=5)\mathrm{P}(X = 5).

Answer: ______________


Question 4 [2 marks]

A fair six-sided die is rolled 8 times. Using a binomial model, find the probability of obtaining exactly 3 sixes.

Answer: ______________


Question 5 [3 marks]

The heights of a certain species of plant are normally distributed with mean 42 cm and standard deviation 5 cm.

(a) Find the probability that a randomly selected plant has a height between 38 cm and 47 cm. [2 marks]

Answer: ______________

(b) A plant is classified as "tall" if its height exceeds 50 cm. In a batch of 200 plants, how many would you expect to be classified as "tall"? [1 mark]

Answer: ______________


Question 6 [3 marks]

A continuous random variable XX has probability density function given by

f(x)={kx(4x)0x40otherwisef(x) = \begin{cases} kx(4-x) & 0 \le x \le 4 \\ 0 & \text{otherwise} \end{cases}

(a) Show that k=332k = \frac{3}{32}. [2 marks]

(b) Find E(X)\mathrm{E}(X). [1 mark]

Answer: ______________


Question 7 [3 marks]

The following grouped data shows the daily commute times (in minutes) of 50 office workers.

Commute time (min)Frequency
0t<100 \le t < 106
10t<2010 \le t < 2014
20t<3020 \le t < 3016
30t<5030 \le t < 5010
50t<8050 \le t < 804

(a) State the modal class. [1 mark]

Answer: ______________

(b) Estimate the mean commute time. [2 marks]

Answer: ______________ minutes


Question 8 [2 marks]

Two events AA and BB are such that P(A)=0.6\mathrm{P}(A) = 0.6, P(B)=0.4\mathrm{P}(B) = 0.4, and P(AB)=0.76\mathrm{P}(A \cup B) = 0.76.

Determine whether AA and BB are independent. Show your reasoning.

Answer: ______________


Question 9 [3 marks]

A bag contains 5 red balls, 4 blue balls, and 3 green balls. Three balls are drawn at random without replacement.

Find the probability that all three balls are the same colour.

Answer: ______________


Question 10 [3 marks]

The random variable YN(μ,σ2)Y \sim \mathrm{N}(\mu, \sigma^2). It is known that P(Y<30)=0.1587\mathrm{P}(Y < 30) = 0.1587 and P(Y>50)=0.0228\mathrm{P}(Y > 50) = 0.0228.

Find μ\mu and σ\sigma.

Answer: μ=\mu = ______________ σ=\sigma = ______________


Section B: Structured / Application Questions [35 marks]

Answer ALL questions in this section.


Question 11 [8 marks]

A market researcher collected data on the monthly spending (in dollars) of a random sample of 8 young professionals on dining out.

PersonABCDEFGH
Spending (xx)12018095210150165130195

(a) Calculate the unbiased estimates of the population mean and population variance of monthly dining spending. [4 marks]

Answer: Mean = ______________ Variance = ______________

(b) The researcher claims that the population mean spending exceeds $140. Using your answer to part (a), comment on whether the sample data supports this claim. [2 marks]

(c) Explain why the formula for the unbiased estimate of the population variance uses n1n-1 in the denominator rather than nn. [2 marks]


Question 12 [9 marks]

A call centre receives an average of 4.2 calls per minute. The number of calls received in a given minute follows a Poisson distribution.

(a) Find the probability that exactly 5 calls are received in a randomly chosen minute. [2 marks]

Answer: ______________

(b) Find the probability that at least 3 calls are received in a randomly chosen minute. [3 marks]

Answer: ______________

(c) Using a suitable approximation, find the probability that fewer than 35 calls are received in a 10-minute period. State any assumptions you make. [4 marks]

Answer: ______________


Question 13 [9 marks]

A company manufactures light bulbs. The lifetime of a light bulb (in hours) is normally distributed with mean 800 hours and standard deviation 100 hours.

(a) Find the probability that a randomly selected light bulb lasts between 700 and 950 hours. [3 marks]

Answer: ______________

(b) The company offers a warranty that replaces any bulb that fails within the first kk hours. If the company wants to replace no more than 5% of bulbs, find the value of kk. [3 marks]

Answer: ______________ hours

(c) A quality control inspector takes a random sample of 16 bulbs. Find the probability that the sample mean lifetime exceeds 820 hours. [3 marks]

Answer: ______________


Question 14 [9 marks]

A survey was conducted on 120 university students regarding their preferred mode of transport to campus. The results are summarised below.

WalkBusMRTCarTotal
Male1520251070
Female201215350
Total35324013120

(a) Find the probability that a randomly selected student is female and prefers the MRT. [2 marks]

Answer: ______________

(b) Given that a student prefers to walk, find the probability that the student is male. [2 marks]

Answer: ______________

(c) A student is selected at random. Are the events "the student is male" and "the student prefers the bus" independent? Show your working clearly. [3 marks]

(d) Two students are selected at random without replacement. Find the probability that both prefer the MRT. [2 marks]

Answer: ______________


Answers

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TuitionGoWhere Practice Paper - Maths H1 A-Level

Answer Key — Version 2 of 5


Section A


Question 1 [2 marks]

xˉ=12+15+10+14+11+136=756=12.5\bar{x} = \frac{12 + 15 + 10 + 14 + 11 + 13}{6} = \frac{75}{6} = 12.5

Answer: 12.5 hours

Marking: M1 for correct substitution into the formula. A1 for correct answer.

Teaching note: The sample mean xˉ\bar{x} is the best point estimate of the population mean μ\mu. We sum all observations and divide by the number of observations nn. This is the standard unbiased estimator for the population mean.


Question 2 [2 marks]

First compute each deviation from the mean xˉ=12.5\bar{x} = 12.5:

xix_ixixˉx_i - \bar{x}(xixˉ)2(x_i - \bar{x})^2
120.5-0.50.25
152.56.25
102.5-2.56.25
141.52.25
111.5-1.52.25
130.50.25

(xixˉ)2=0.25+6.25+6.25+2.25+2.25+0.25=17.5\sum(x_i - \bar{x})^2 = 0.25 + 6.25 + 6.25 + 2.25 + 2.25 + 0.25 = 17.5

s2=17.561=17.55=3.50s^2 = \frac{17.5}{6-1} = \frac{17.5}{5} = 3.50

Answer: 3.50 hours²

Marking: M1 for correct deviations/squared deviations. A1 for correct answer with n1n-1 denominator.

Common mistake: Using n=6n = 6 in the denominator gives 17.56=2.92\frac{17.5}{6} = 2.92, which is the biased sample variance. The unbiased estimator uses n1=5n-1 = 5 because one degree of freedom is "used up" when estimating the sample mean. This correction (Bessel's correction) ensures that on average, s2s^2 equals the true population variance σ2\sigma^2.


Question 3 [2 marks]

XB(12,0.35)X \sim \mathrm{B}(12, 0.35)

P(X=5)=(125)(0.35)5(0.65)7\mathrm{P}(X = 5) = \binom{12}{5}(0.35)^5(0.65)^7

=792×0.005252×0.049022= 792 \times 0.005252 \times 0.049022

=792×0.0002575=0.2039= 792 \times 0.0002575 = 0.2039

Answer: 0.204 (3 s.f.)

Marking: M1 for correct binomial probability formula with (125)\binom{12}{5}, p=0.35p = 0.35, q=0.65q = 0.65. A1 for correct answer to 3 s.f.

Teaching note: For XB(n,p)X \sim \mathrm{B}(n, p), the probability of exactly rr successes is P(X=r)=(nr)pr(1p)nr\mathrm{P}(X = r) = \binom{n}{r}p^r(1-p)^{n-r}. The graphing calculator's binomialPdf function can be used directly.


Question 4 [2 marks]

Let XX be the number of sixes in 8 rolls. Then XB(8,16)X \sim \mathrm{B}\left(8, \frac{1}{6}\right).

P(X=3)=(83)(16)3(56)5\mathrm{P}(X = 3) = \binom{8}{3}\left(\frac{1}{6}\right)^3\left(\frac{5}{6}\right)^5

=56×1216×31257776= 56 \times \frac{1}{216} \times \frac{3125}{7776}

=56×0.0046296×0.40188= 56 \times 0.0046296 \times 0.40188

=0.1042= 0.1042

Answer: 0.104 (3 s.f.)

Marking: M1 for identifying n=8n = 8, p=16p = \frac{1}{6} and using the binomial formula. A1 for correct answer.


Question 5 [3 marks]

Let XN(42,52)X \sim \mathrm{N}(42, 5^2).

(a) [2 marks]

P(38<X<47)=P(38425<Z<47425)=P(0.8<Z<1.0)\mathrm{P}(38 < X < 47) = \mathrm{P}\left(\frac{38-42}{5} < Z < \frac{47-42}{5}\right) = \mathrm{P}(-0.8 < Z < 1.0)

=Φ(1.0)Φ(0.8)=Φ(1.0)[1Φ(0.8)]= \Phi(1.0) - \Phi(-0.8) = \Phi(1.0) - [1 - \Phi(0.8)]

=0.8413(10.7881)=0.84130.2119=0.6294= 0.8413 - (1 - 0.7881) = 0.8413 - 0.2119 = 0.6294

Answer: 0.629 (3 s.f.)

Marking: M1 for standardising and using normal tables/calculator. A1 for correct answer.

(b) [1 mark]

P(X>50)=P(Z>50425)=P(Z>1.6)=1Φ(1.6)=10.9452=0.0548\mathrm{P}(X > 50) = \mathrm{P}\left(Z > \frac{50-42}{5}\right) = \mathrm{P}(Z > 1.6) = 1 - \Phi(1.6) = 1 - 0.9452 = 0.0548

Expected number =200×0.0548=10.96= 200 \times 0.0548 = 10.96

Answer: 11 plants (approximately)

Marking: A1 for correct calculation and rounding.


Question 6 [3 marks]

(a) [2 marks]

Since f(x)f(x) is a PDF, 04f(x)dx=1\int_0^4 f(x)\,dx = 1:

04kx(4x)dx=k04(4xx2)dx=k[2x2x33]04\int_0^4 kx(4-x)\,dx = k\int_0^4 (4x - x^2)\,dx = k\left[2x^2 - \frac{x^3}{3}\right]_0^4

=k[(2(16)643)0]=k[32643]=k[96643]=k323= k\left[\left(2(16) - \frac{64}{3}\right) - 0\right] = k\left[32 - \frac{64}{3}\right] = k\left[\frac{96-64}{3}\right] = k \cdot \frac{32}{3}

Setting equal to 1: k323=1k \cdot \frac{32}{3} = 1, so k=332k = \frac{3}{32}. \quad \blacksquare

(b) [1 mark]

E(X)=04x332x(4x)dx=33204(4x2x3)dx\mathrm{E}(X) = \int_0^4 x \cdot \frac{3}{32}x(4-x)\,dx = \frac{3}{32}\int_0^4 (4x^2 - x^3)\,dx

=332[4x33x44]04=332[256364]=332[2561923]=332×643=2= \frac{3}{32}\left[\frac{4x^3}{3} - \frac{x^4}{4}\right]_0^4 = \frac{3}{32}\left[\frac{256}{3} - 64\right] = \frac{3}{32}\left[\frac{256 - 192}{3}\right] = \frac{3}{32} \times \frac{64}{3} = 2

Answer: 2

Marking: M1 for setting up the expectation integral. A1 for correct answer.

Teaching note: For a continuous random variable with PDF f(x)f(x), the expected value is E(X)=xf(x)dx\mathrm{E}(X) = \int x\,f(x)\,dx over the domain. The symmetry of f(x)=332x(4x)f(x) = \frac{3}{32}x(4-x) about x=2x = 2 confirms E(X)=2\mathrm{E}(X) = 2.


Question 7 [3 marks]

(a) [1 mark]

The class with the highest frequency is 20t<3020 \le t < 30 (frequency 16).

Answer: 20t<3020 \le t < 30

(b) [2 marks]

Using midpoints: 5, 15, 25, 40, 65.

xˉ=6(5)+14(15)+16(25)+10(40)+4(65)50=30+210+400+400+26050=130050=26.0\bar{x} = \frac{6(5) + 14(15) + 16(25) + 10(40) + 4(65)}{50} = \frac{30 + 210 + 400 + 400 + 260}{50} = \frac{1300}{50} = 26.0

Answer: 26.0 minutes

Marking: M1 for using midpoints correctly. A1 for correct answer.

Teaching note: For grouped data, we use the midpoint of each class as the representative value. The last class 50t<8050 \le t < 80 has midpoint 50+802=65\frac{50+80}{2} = 65.


Question 8 [2 marks]

Check independence: AA and BB are independent if and only if P(AB)=P(A)×P(B)\mathrm{P}(A \cap B) = \mathrm{P}(A) \times \mathrm{P}(B).

First find P(AB)\mathrm{P}(A \cap B) using the addition rule:

P(AB)=P(A)+P(B)P(AB)\mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cap B)

0.76=0.6+0.4P(AB)0.76 = 0.6 + 0.4 - \mathrm{P}(A \cap B)

P(AB)=1.00.76=0.24\mathrm{P}(A \cap B) = 1.0 - 0.76 = 0.24

Now check:

P(A)×P(B)=0.6×0.4=0.24\mathrm{P}(A) \times \mathrm{P}(B) = 0.6 \times 0.4 = 0.24

Since P(AB)=P(A)×P(B)=0.24\mathrm{P}(A \cap B) = \mathrm{P}(A) \times \mathrm{P}(B) = 0.24, the events are independent.

Answer: Yes, A and B are independent because P(A ∩ B) = P(A) × P(B) = 0.24.

Marking: M1 for finding P(A ∩ B) using the addition formula. A1 for correct conclusion with justification.


Question 9 [3 marks]

Total balls = 5 + 4 + 3 = 12. Number of ways to draw 3 balls from 12: (123)=220\binom{12}{3} = 220.

All red: (53)=10\binom{5}{3} = 10
All blue: (43)=4\binom{4}{3} = 4
All green: (33)=1\binom{3}{3} = 1

P(all same colour)=10+4+1220=15220=344=0.0682\mathrm{P}(\text{all same colour}) = \frac{10 + 4 + 1}{220} = \frac{15}{220} = \frac{3}{44} = 0.0682

Answer: 344\frac{3}{44} or 0.0682 (3 s.f.)

Marking: M1 for using combinations for each colour. M1 for summing and dividing by total. A1 for correct answer.

Teaching note: Since the balls are drawn without replacement, we use combinations (not permutations) because the order of selection does not matter. The events "all red", "all blue", and "all green" are mutually exclusive, so we add their probabilities.


Question 10 [3 marks]

From standard normal tables:

  • P(Y<30)=0.1587\mathrm{P}(Y < 30) = 0.1587 means P(Z<30μσ)=0.1587\mathrm{P}\left(Z < \frac{30-\mu}{\sigma}\right) = 0.1587, so 30μσ=1.0\frac{30-\mu}{\sigma} = -1.0 (since Φ(1)=0.1587\Phi(-1) = 0.1587)

  • P(Y>50)=0.0228\mathrm{P}(Y > 50) = 0.0228 means P(Z>50μσ)=0.0228\mathrm{P}\left(Z > \frac{50-\mu}{\sigma}\right) = 0.0228, so 50μσ=2.0\frac{50-\mu}{\sigma} = 2.0 (since Φ(2)=0.9772\Phi(2) = 0.9772)

Solving the simultaneous equations:

30μ=σμ=30+σ...(i)30 - \mu = -\sigma \quad \Rightarrow \quad \mu = 30 + \sigma \quad \text{...(i)}

50μ=2σ...(ii)50 - \mu = 2\sigma \quad \text{...(ii)}

Substitute (i) into (ii):

50(30+σ)=2σ50 - (30 + \sigma) = 2\sigma

20σ=2σ20 - \sigma = 2\sigma

20=3σσ=203=6.6720 = 3\sigma \quad \Rightarrow \quad \sigma = \frac{20}{3} = 6.67

μ=30+203=1103=36.67\mu = 30 + \frac{20}{3} = \frac{110}{3} = 36.67

Answer: μ=3623\mu = 36\frac{2}{3} (or 36.7), σ=623\sigma = 6\frac{2}{3} (or 6.67)

Marking: M1 for setting up both equations from the given probabilities. M1 for solving the simultaneous equations. A1 for both correct values.


Section B


Question 11 [8 marks]

(a) [4 marks]

xˉ=120+180+95+210+150+165+130+1958=12458=155.625\bar{x} = \frac{120 + 180 + 95 + 210 + 150 + 165 + 130 + 195}{8} = \frac{1245}{8} = 155.625

For the unbiased variance, compute (xixˉ)2\sum(x_i - \bar{x})^2:

xix_ixixˉx_i - \bar{x}(xixˉ)2(x_i - \bar{x})^2
12035.625-35.6251269.14
18024.375594.14
9560.625-60.6253675.39
21054.3752956.64
1505.625-5.62531.64
1659.37587.89
13025.625-25.625656.64
19539.3751550.39

(xixˉ)2=10821.875\sum(x_i - \bar{x})^2 = 10821.875

s2=10821.87581=10821.8757=1545.98s^2 = \frac{10821.875}{8-1} = \frac{10821.875}{7} = 1545.98

Answer: Mean = $155.6 (or $155.63), Variance = 1550 (3 s.f.)

Marking: M1 for correct mean. M1 for correct squared deviations (at least 3 correct). M1 for using n1=7n-1 = 7 in the denominator. A1 for correct variance to 3 s.f.

(b) [2 marks]

The sample mean is \bar{x} = \155.63, which is greater than \140. This provides some support for the researcher's claim that the population mean exceeds $140. However, with only a sample of 8, there is considerable sampling variability, and a formal hypothesis test would be needed to draw a statistically significant conclusion.

Marking: B1 for noting the sample mean exceeds $140. B1 for acknowledging the limitation of the small sample size.

(c) [2 marks]

Using nn in the denominator gives the average of the squared deviations from the sample mean. However, the sample mean xˉ\bar{x} is itself estimated from the same data, so the deviations (xixˉ)(x_i - \bar{x}) tend to be slightly smaller than the true deviations (xiμ)(x_i - \mu) from the population mean. Dividing by n1n-1 instead of nn corrects this downward bias, making s2s^2 an unbiased estimator of σ2\sigma^2. This correction is known as Bessel's correction.

Marking: B1 for explaining that the sample mean causes deviations to be smaller. B1 for stating that n1n-1 corrects the bias (Bessel's correction).


Question 12 [9 marks]

Let XX be the number of calls per minute. XPo(4.2)X \sim \mathrm{Po}(4.2).

(a) [2 marks]

P(X=5)=e4.2(4.2)55!=e4.2×1306.91120=1306.91120×0.0150=0.1634\mathrm{P}(X = 5) = \frac{e^{-4.2}(4.2)^5}{5!} = \frac{e^{-4.2} \times 1306.91}{120} = \frac{1306.91}{120} \times 0.0150 = 0.1634

Answer: 0.163 (3 s.f.)

Marking: M1 for correct Poisson formula with λ=4.2\lambda = 4.2. A1 for correct answer.

(b) [3 marks]

P(X3)=1P(X2)=1[P(X=0)+P(X=1)+P(X=2)]\mathrm{P}(X \ge 3) = 1 - \mathrm{P}(X \le 2) = 1 - [\mathrm{P}(X=0) + \mathrm{P}(X=1) + \mathrm{P}(X=2)]

P(X=0)=e4.2=0.0150\mathrm{P}(X = 0) = e^{-4.2} = 0.0150

P(X=1)=4.2×e4.2=4.2×0.0150=0.0630\mathrm{P}(X = 1) = 4.2 \times e^{-4.2} = 4.2 \times 0.0150 = 0.0630

P(X=2)=4.222×e4.2=8.82×0.0150=0.1323\mathrm{P}(X = 2) = \frac{4.2^2}{2} \times e^{-4.2} = 8.82 \times 0.0150 = 0.1323

P(X3)=1(0.0150+0.0630+0.1323)=10.2103=0.7897\mathrm{P}(X \ge 3) = 1 - (0.0150 + 0.0630 + 0.1323) = 1 - 0.2103 = 0.7897

Answer: 0.790 (3 s.f.)

Marking: M1 for using the complement method. M1 for calculating individual probabilities correctly. A1 for correct final answer.

(c) [4 marks]

Over 10 minutes, λ=4.2×10=42\lambda = 4.2 \times 10 = 42. Let YPo(42)Y \sim \mathrm{Po}(42).

Since λ=42\lambda = 42 is large, use the normal approximation: YN(42,42)Y \approx \mathrm{N}(42, 42).

Assumption: The Poisson rate is constant over time, and calls in different minutes are independent.

With continuity correction:

P(Y<35)=P(Y34)P(Z<34.54242)=P(Z<7.56.481)=P(Z<1.157)\mathrm{P}(Y < 35) = \mathrm{P}(Y \le 34) \approx \mathrm{P}\left(Z < \frac{34.5 - 42}{\sqrt{42}}\right) = \mathrm{P}\left(Z < \frac{-7.5}{6.481}\right) = \mathrm{P}(Z < -1.157)

=1Φ(1.157)=10.8764=0.1236= 1 - \Phi(1.157) = 1 - 0.8764 = 0.1236

Answer: 0.124 (3 s.f.)

Marking: M1 for λ=42\lambda = 42 and stating the normal approximation. M1 for stating assumptions (constant rate, independence). M1 for continuity correction. A1 for correct answer.


Question 13 [9 marks]

Let XN(800,1002)X \sim \mathrm{N}(800, 100^2).

(a) [3 marks]

P(700<X<950)=P(700800100<Z<950800100)=P(1.0<Z<1.5)\mathrm{P}(700 < X < 950) = \mathrm{P}\left(\frac{700-800}{100} < Z < \frac{950-800}{100}\right) = \mathrm{P}(-1.0 < Z < 1.5)

=Φ(1.5)Φ(1.0)=Φ(1.5)[1Φ(1.0)]= \Phi(1.5) - \Phi(-1.0) = \Phi(1.5) - [1 - \Phi(1.0)]

=0.9332(10.8413)=0.93320.1587=0.7745= 0.9332 - (1 - 0.8413) = 0.9332 - 0.1587 = 0.7745

Answer: 0.775 (3 s.f.)

Marking: M1 for standardising both bounds. M1 for correct use of Φ\Phi values. A1 for correct answer.

(b) [3 marks]

We need P(X<k)=0.05\mathrm{P}(X < k) = 0.05.

From tables, Φ(1.645)=0.05\Phi(-1.645) = 0.05, so:

k800100=1.645\frac{k - 800}{100} = -1.645

k=800164.5=635.5k = 800 - 164.5 = 635.5

Answer: 636 hours (or 635 hours)

Marking: M1 for setting up P(X<k)=0.05\mathrm{P}(X < k) = 0.05. M1 for using the correct zz-value (1.645-1.645). A1 for correct answer.

(c) [3 marks]

By the Central Limit Theorem, the sample mean XˉN(μ,σ2n)=N(800,100216)=N(800,625)\bar{X} \sim \mathrm{N}\left(\mu, \frac{\sigma^2}{n}\right) = \mathrm{N}\left(800, \frac{100^2}{16}\right) = \mathrm{N}(800, 625), so σXˉ=25\sigma_{\bar{X}} = 25.

P(Xˉ>820)=P(Z>82080025)=P(Z>0.8)=1Φ(0.8)=10.7881=0.2119\mathrm{P}(\bar{X} > 820) = \mathrm{P}\left(Z > \frac{820 - 800}{25}\right) = \mathrm{P}(Z > 0.8) = 1 - \Phi(0.8) = 1 - 0.7881 = 0.2119

Answer: 0.212 (3 s.f.)

Marking: M1 for correct distribution of sample mean with σ/n=25\sigma/\sqrt{n} = 25. M1 for standardising. A1 for correct answer.

Teaching note: The distribution of the sample mean has the same mean μ\mu but a reduced standard deviation σ/n\sigma/\sqrt{n} (the standard error). This reflects the fact that sample means vary less than individual observations.


Question 14 [9 marks]

(a) [2 marks]

P(Female and MRT)=15120=18\mathrm{P}(\text{Female and MRT}) = \frac{15}{120} = \frac{1}{8}

Answer: 18\frac{1}{8} or 0.125

Marking: M1 for identifying the correct cell (Female, MRT = 15). A1 for correct probability.

(b) [2 marks]

P(MaleWalk)=P(Male and Walk)P(Walk)=15/12035/120=1535=37\mathrm{P}(\text{Male} \mid \text{Walk}) = \frac{\mathrm{P}(\text{Male and Walk})}{\mathrm{P}(\text{Walk})} = \frac{15/120}{35/120} = \frac{15}{35} = \frac{3}{7}

Answer: 37\frac{3}{7} or 0.429 (3 s.f.)

Marking: M1 for using conditional probability formula. A1 for correct answer.

(c) [3 marks]

Check if P(MaleBus)=P(Male)×P(Bus)\mathrm{P}(\text{Male} \cap \text{Bus}) = \mathrm{P}(\text{Male}) \times \mathrm{P}(\text{Bus}).

P(MaleBus)=20120=16=0.1667\mathrm{P}(\text{Male} \cap \text{Bus}) = \frac{20}{120} = \frac{1}{6} = 0.1667

P(Male)×P(Bus)=70120×32120=70×3214400=224014400=0.1556\mathrm{P}(\text{Male}) \times \mathrm{P}(\text{Bus}) = \frac{70}{120} \times \frac{32}{120} = \frac{70 \times 32}{14400} = \frac{2240}{14400} = 0.1556

Since 0.16670.15560.1667 \ne 0.1556, the events are not independent.

Answer: The events "male" and "prefers bus" are NOT independent because P(Male ∩ Bus) ≠ P(Male) × P(Bus).

Marking: M1 for calculating both sides. M1 for comparing. A1 for correct conclusion.

(d) [2 marks]

P(both MRT)=40120×39119=13×39119=39357=13119=0.1092\mathrm{P}(\text{both MRT}) = \frac{40}{120} \times \frac{39}{119} = \frac{1}{3} \times \frac{39}{119} = \frac{39}{357} = \frac{13}{119} = 0.1092

Answer: 13119\frac{13}{119} or 0.109 (3 s.f.)

Marking: M1 for multiplying probabilities without replacement. A1 for correct answer.


Mark Summary

QuestionMarks
12
22
32
42
53
63
73
82
93
103
Section A Total25
118
129
139
149
Section B Total35
Grand Total60