A Level H1 Mathematics Practice Paper 1

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TuitionGoWhere Exam Practice (AI) — Maths H1 A-Level

Practice Paper: Statistics & Probability (Version 1 of 5)

School: TuitionGoWhere Exam Practice (AI)
Subject: Maths H1
Level: A-Level
Paper: Practice Paper — Statistics & Probability (Version 1)
Duration: 75 minutes
Total Marks: 60
Name: ________________________
Class: ________________________
Date: ________________________

Instructions:

• Answer all questions.
• Show all working clearly.
• Use an approved graphing calculator where appropriate.
• Write your answers in the spaces provided.

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Section A: Probability (Questions 1–7) [21 marks]

1. A bag contains 5 red balls and 3 green balls. Two balls are drawn at random without replacement. Draw a tree diagram to represent the possible outcomes and their probabilities. [3]

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2. In a survey, 60% of students use a laptop for note-taking. In a random sample of 10 students, find the probability that exactly 7 use a laptop. [2]

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3. Events A and B are such that $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \cap B) = 0.2$. Find $P(A \cup B)$. [2]

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4. A company finds that 8% of its products are defective. In a batch of 20 items, find the probability that more than 2 are defective. [2]

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5. Given that $P(X|Y) = 0.3$, $P(Y) = 0.6$, and $P(X \cap Y) = 0.18$, state whether events X and Y are independent. Justify your answer. [2]

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6. A committee of 4 is to be formed from 6 men and 5 women. Find the number of ways to form the committee with exactly 2 women. [2]

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7. A fair die is rolled twice. Find the probability that the sum of the two scores is at least 10. [2]

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Section B: Distributions (Questions 8–13) [18 marks]

8. The random variable $X \sim B(15, 0.3)$. Find the mean and variance of $X$. [2]

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9. The mass of a species of fish is normally distributed with mean 400 g and standard deviation 50 g. Find the probability that a randomly caught fish has mass less than 360 g. [3]

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10. Let $X \sim N(20, 4^2)$ and $Y = 3X - 5$. Find $E(Y)$ and $\text{Var}(Y)$. [3]

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11. The random variable $W \sim N(\mu, \sigma^2)$. Given $P(W < 30) = 0.05$ and $P(W > 50) = 0.10$, find the values of $\mu$ and $\sigma$. [4]

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12. A sample of size 36 is taken from a population with mean 50 and variance 144. Using the Central Limit Theorem, state the distribution of the sample mean $\bar{X}$ and find $P(\bar{X} > 53)$. [2]

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13. The number of emails received per hour by an office follows a binomial distribution with $n = 12$ and $p = 0.25$. Find $P(X \le 3)$. [2]

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Section C: Data Analysis & Sampling (Questions 14–20) [21 marks]

14. The following are the daily steps (in thousands) recorded for 8 students:
9.2, 8.1, 10.4, 7.8, 9.9, 8.6, 10.1, 9.0.
Find the unbiased estimates of the population mean and variance. [3]

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15. A random sample of 50 residents gave $\sum x = 1750$ and $\sum x^2 = 64000$. Find the unbiased estimates of the population mean and variance. [3]

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16. Describe a systematic sampling method to select 25 households from a list of 250 households. [1]

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17. The following data show study hours (x) and test scores (y) for 6 students:

x	2	3	4	5	6	7
y	55	60	65	70	78	82

Find the product moment correlation coefficient $r$ and comment on its value. [3]

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18. Using the data in Question 17, find the equation of the least squares regression line of $y$ on $x$ in the form $y = mx + c$. [3]

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19. The scatter diagram for the data in Question 17 is shown below.

Image pending generation: scatter_diagram for Q19.

Sketch the regression line from Question 18 on the scatter diagram and use it to estimate the test score for a student who studies 4.5 hours. [2]

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20. A researcher claims that the regression line can be used to predict test scores for a student studying 20 hours. State a reason why this prediction may be unreliable. [1]

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TuitionGoWhere Exam Practice (AI) — Maths H1 A-Level

Practice Paper: Statistics & Probability (Version 1) — Answer Key

Total Marks: 60

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Section A: Probability (Q1–7) — 21 marks

Q1 [3 marks]
Tree diagram:

• Stage 1: R (5/8), G (3/8)
• Stage 2 after R: R (4/7), G (3/7)
• Stage 2 after G: R (5/7), G (2/7)
  Marking: 1 mark for correct first branches, 1 mark for correct second branches (dependent probabilities), 1 mark for clear labels.
  Teaching note: Without replacement changes probabilities; total probability along paths multiplies.

Q2 [2 marks]
$X \sim B(10, 0.6)$; $P(X=7) = \binom{10}{7}(0.6)^7(0.4)^3 = 0.215$ (3 s.f.)
Marking: 1 mark for identifying binomial, 1 mark for correct value.
Common mistake: using wrong p or n.

Q3 [2 marks]
$P(A \cup B) = P(A)+P(B)-P(A\cap B) = 0.4+0.5-0.2 = 0.7$
Marking: 1 mark formula, 1 mark answer.

Q4 [2 marks]
$X \sim B(20, 0.08)$; $P(X>2) = 1 - P(X\le 2) = 1 - [P(0)+P(1)+P(2)] = 0.597$ (3 s.f.)
Marking: 1 mark setup, 1 mark final answer.

Q5 [2 marks]
Independent if $P(X|Y)=P(X)$. $P(X)=P(X\cap Y)/P(Y)=0.18/0.6=0.3 = P(X|Y)$ → independent.
Marking: 1 mark computation of P(X), 1 mark correct conclusion.

Q6 [2 marks]
$\binom{5}{2} \times \binom{6}{2} = 10 \times 15 = 150$ ways.
Marking: 1 mark combination setup, 1 mark answer.

Q7 [2 marks]
Outcomes with sum ≥10: (4,6),(5,5),(6,4),(5,6),(6,5),(6,6) → 6 of 36 → $1/6$.
Marking: 1 mark listing, 1 mark probability.

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Section B: Distributions (Q8–13) — 18 marks

Q8 [2 marks]
Mean $= np = 15 \times 0.3 = 4.5$; Variance $= np(1-p) = 15 \times 0.3 \times 0.7 = 3.15$.
Marking: 1 mark each.

Q9 [3 marks]
$Z = (360-400)/50 = -0.8$; $P(X<360)=P(Z<-0.8)=0.2119$
Marking: 1 mark standardising, 2 marks for probability (tables/GC).

Q10 [3 marks]
$E(Y)=3E(X)-5 = 3(20)-5=55$; $\text{Var}(Y)=3^2\text{Var}(X)=9\times16=144$.
Marking: 1 mark each part, 1 mark for using variance property.

Q11 [4 marks]
$P(W<30)=0.05 \Rightarrow z=-1.645 = (30-\mu)/\sigma$
$P(W>50)=0.10 \Rightarrow z=1.282 = (50-\mu)/\sigma$
Solve: $\mu = 38.6$, $\sigma = 5.24$ (3 s.f.)
Marking: 1 mark each z, 2 marks solving.

Q12 [2 marks]
By CLT, $\bar{X} \sim N(50, 144/36) = N(50,4)$. $P(\bar{X}>53)=P(Z>1.5)=0.0668$.
Marking: 1 mark distribution, 1 mark probability.

Q13 [2 marks]
$P(X\le3) = \sum_{k=0}^{3}\binom{12}{k}(0.25)^k(0.75)^{12-k} = 0.648$ (3 s.f.)
Marking: 1 mark setup, 1 mark answer.

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Section C: Data Analysis & Sampling (Q14–20) — 21 marks

Q14 [3 marks]
$\bar{x} = \frac{73.1}{8}=9.1375$
$s^2 = \frac{1}{7}[\sum x^2 - 8\bar{x}^2] = \frac{1}{7}[674.27 - 8(9.1375)^2] = 0.983$ (3 s.f.)
Marking: 1 mark mean, 2 marks variance (unbiased).

Q15 [3 marks]
$\bar{x}=1750/50=35$; $s^2=\frac{1}{49}[64000 - 50(35)^2] = \frac{1}{49}[64000-61250]=56.1$
Marking: 1 mark mean, 2 marks variance.

Q16 [1 mark]
Assign numbers 1–250, choose a random start between 1–10, then select every 10th household.
Marking: 1 mark for valid systematic method.

Q17 [3 marks]
$r = 0.992$ (from GC). Strong positive linear correlation between study hours and test score.
Marking: 2 marks calc, 1 mark comment.

Q18 [3 marks]
From GC: $y = 5.49x + 44.3$ (3 s.f.)
Marking: 3 marks for correct equation form.

Q19 [2 marks]
Line drawn through points; at x=4.5, y≈69.0.
Marking: 1 mark line, 1 mark estimate.

Q20 [1 mark]
20 hours is outside the range of data (extrapolation) so unreliable.
Marking: 1 mark reason.