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A Level H1 Mathematics Statistics Probability Quiz

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Questions

A-Level Maths H1 Quiz - Statistics Probability

Name: ________________________
Class: ________________________
Date: ________________________
Score: ________________________

Duration: 70 minutes
Total Marks: 40

Instructions:

Answer all 20 questions.
Show your working clearly where required.
Use your graphing calculator where appropriate.
Write your answers in the spaces provided.

Section A: Probability (Questions 1–7)

1. [2 marks] A bag contains 5 red balls and 3 blue balls. Two balls are drawn at random without replacement. Draw a tree diagram to represent the possible outcomes and their probabilities.

2. [1 mark] Using the tree diagram from Question 1, write down the probability of drawing a red ball followed by a blue ball.

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

4. [2 marks] A fair die is rolled. Let E be the event "an even number is obtained" and F be the event "a number greater than 4 is obtained". Determine whether E and F are independent.

5. [2 marks] In a survey of 200 residents, 120 use public transport and 80 do not. Of those using public transport, 30 are late at least once a week. Given that a randomly chosen resident is late at least once a week, the probability that they use public transport is 0.6. Find the total number of residents who are late at least once a week.

6. [2 marks] A box contains 4 green and 6 yellow cards. Three cards are selected at random without replacement. Find the probability that exactly two are green.

7. [2 marks] The probability that a student passes a test is 0.7. Two students are selected at random. Find the probability that at least one passes.

Section B: Distributions (Questions 8–13)

8. [2 marks] The random variable $X \sim B(20, 0.3)$ . Find $P(X > 8)$ .

9. [2 marks] The random variable $Y \sim B(15, 0.4)$ . State the mean and variance of $Y$ .

10. [2 marks] The mass of a certain fruit is normally distributed with mean 120 g and standard deviation 15 g. Find the probability that a randomly chosen fruit has mass less than 100 g.

11. [2 marks] Let $X \sim N(50, 8^2)$ . Find $P(40 < X < 60)$ .

12. [2 marks] The random variable $X \sim N(\mu, \sigma^2)$ . Given that $P(X < 30) = 0.1$ and $P(X < 50) = 0.9$ , find the value of $\mu$ and $\sigma$ .

13. [2 marks] Given $X \sim N(10, 4)$ and $Y \sim N(20, 9)$ are independent, find $E(2X - Y + 3)$ and $\text{Var}(2X - Y + 3)$ .

Section C: Sampling and Statistics (Questions 14–20)

14. [1 mark] Describe a method to select a simple random sample of 25 students from a school of 300 students.

15. [2 marks] A sample of 40 observations is taken from a normal population with variance 16. State the distribution of the sample mean $\bar{X}$ and give its mean and variance.

16. [2 marks] The following data are the number of books read by 8 students in a month: 3, 5, 2, 7, 4, 6, 3, 5. Find the unbiased estimates of the population mean and variance.

17. [2 marks] A summary of a sample gives $\sum x = 240$ and $\sum x^2 = 5800$ for $n = 10$ . Find the unbiased estimates of the population mean and variance.

18. [2 marks] The following data show the number of hours studied ( $x$ ) and test score ( $y$ ) for 6 students:

$x$	2	3	5	6	8	9
$y$	50	55	65	70	80	85

Find the product moment correlation coefficient $r$ . Comment briefly on its value.

19. [2 marks] Using the data in Question 18, find the equation of the least squares regression line of $y$ on $x$ in the form $y = ax + b$ .

20. [2 marks] Sketch the scatter diagram for the data in Question 18, labelling axes clearly.

Scatter Diagram for Q20 (ALEVEL Maths H1)

Generated scatter_diagram for Q20.

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Answers

A-Level Maths H1 Quiz - Statistics Probability (Answer Key)

Total Marks: 40

Section A: Probability

Q1 [2 marks]
Tree diagram:

Stage 1: Red (5/8), Blue (3/8)
Stage 2 after Red: Red (4/7), Blue (3/7)
Stage 2 after Blue: Red (5/7), Blue (2/7)
Marking: 1 mark for correct first branches, 1 mark for correct second branches with updated probabilities.

Q2 [1 mark]
$P(\text{Red then Blue}) = \frac{5}{8} \times \frac{3}{7} = \frac{15}{56} \approx 0.268$
Accept 15/56 or 0.268.

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

Q4 [2 marks]
$E = \{2,4,6\}, P(E) = 3/6 = 0.5$
$F = \{5,6\}, P(F) = 2/6 = 1/3$
$E \cap F = \{6\}, P(E \cap F) = 1/6$
Check: $P(E)P(F) = 0.5 \times 1/3 = 1/6 = P(E \cap F)$ → independent.
1 mark for probabilities, 1 mark for conclusion.

Q5 [2 marks]
Let L = late at least once a week.
$P(\text{PT} | L) = 0.6 = \frac{30}{\text{Total late}}$ → Total late = 30 / 0.6 = 50.
1 mark for setting up equation, 1 mark for answer 50.

Q6 [2 marks]
$P(\text{exactly 2 green}) = \frac{\binom{4}{2}\binom{6}{1}}{\binom{10}{3}} = \frac{6 \times 6}{120} = \frac{36}{120} = 0.3$
1 mark for combinations, 1 mark for final answer.

Q7 [2 marks]
$P(\text{at least one passes}) = 1 - P(\text{both fail}) = 1 - (0.3)^2 = 1 - 0.09 = 0.91$
1 mark for complement method, 1 mark for answer.

Section B: Distributions

Q8 [2 marks]
$X \sim B(20, 0.3)$ , $P(X > 8) = 1 - P(X \leq 8) \approx 1 - 0.8867 = 0.1133$ (GC).
1 mark for correct setup, 1 mark for value.

Q9 [2 marks]
Mean = $np = 15 \times 0.4 = 6$
Variance = $np(1-p) = 15 \times 0.4 \times 0.6 = 3.6$
1 mark each.

Q10 [2 marks]
$Z = \frac{100 - 120}{15} = -1.333$ , $P(Z < -1.333) \approx 0.0912$
1 mark for z, 1 mark for probability.

Q11 [2 marks]
$Z_1 = (40-50)/8 = -1.25$ , $Z_2 = (60-50)/8 = 1.25$
$P(-1.25 < Z < 1.25) = 0.8944 - 0.1056 = 0.7888$
1 mark each.

Q12 [2 marks]
Symmetry: $\mu = (30+50)/2 = 40$ .
$P(X<30)=0.1 \Rightarrow Z = -1.2816 = (30-40)/\sigma \Rightarrow \sigma = 10/1.2816 \approx 7.80$
1 mark for $\mu$ , 1 mark for $\sigma$ .

Q13 [2 marks]
$E(2X - Y + 3) = 2(10) - 20 + 3 = 3$
$\text{Var} = 2^2(4) + (-1)^2(9) = 16 + 9 = 25$
1 mark each.

Section C: Sampling and Statistics

Q14 [1 mark]
Assign numbers 1–300 to students. Use a random number generator to pick 25 distinct numbers; select corresponding students. (1 mark for valid random method)

Q15 [2 marks]
$\bar{X} \sim N(50, 16/40) = N(50, 0.4)$ . Mean = 50, Variance = 0.4.
1 mark for distribution, 1 mark for parameters.

Q16 [2 marks]
$\bar{x} = (3+5+2+7+4+6+3+5)/8 = 35/8 = 4.375$
$s^2 = \frac{1}{7}[(3-4.375)^2 + \dots + (5-4.375)^2] = \frac{1}{7}(21.875) = 3.125$
1 mark mean, 1 mark variance.

Q17 [2 marks]
$\bar{x} = 240/10 = 24$
$s^2 = \frac{1}{9}[5800 - 10(24)^2] = \frac{1}{9}(5800-5760) = 40/9 \approx 4.44$
1 mark each.

Q18 [2 marks]
Using GC: $r \approx 0.997$ . Strong positive linear correlation between hours studied and score.
1 mark calc, 1 mark comment.

Q19 [2 marks]
From GC: $y = 4.41x + 41.2$ (approx).
1 mark gradient, 1 mark intercept.

Q20 [2 marks]
Scatter diagram: x-axis 0–10, y-axis 40–90, points plotted as per placeholder. Upward trend.
1 mark axes/labels, 1 mark points/trend.