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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: ________ / 45

Duration: 60 minutes
Total Marks: 45

Instructions:

Answer all 20 questions.
Write your answers in the spaces provided.
An approved graphing calculator is expected. Unsupported answers are generally allowed unless otherwise stated.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

Section A: Probability and Counting Principles (Questions 1–5)

1. A committee of 4 people is to be selected from a group of 6 men and 5 women. Find the number of different ways the committee can be formed if it must contain at least 2 women. [2]

2. Events $A$ and $B$ are such that $P(A) = 0.4$ , $P(B) = 0.5$ , and $P(A \cup B) = 0.7$ . (a) Find $P(A \cap B)$ . [1] (b) Determine whether events $A$ and $B$ are independent, showing your working clearly. [1]

3. In a certain factory, 5% of the light bulbs produced are defective. A random sample of 20 light bulbs is taken. Let $X$ be the number of defective bulbs in the sample. (a) State the distribution of $X$ , stating the parameters clearly. [1] (b) Find the probability that exactly 2 bulbs are defective. [1]

4. The probability that a student passes Mathematics is 0.8, and the probability that the same student passes Physics is 0.7. The probability that the student passes both subjects is 0.6. Find the probability that the student passes at least one of the subjects. [2]

5. A bag contains 4 red balls and 6 blue balls. Two balls are drawn at random without replacement. Draw a tree diagram to represent the possible outcomes and their probabilities. [2]

Section B: Discrete and Continuous Distributions (Questions 6–12)

6. The random variable $X$ follows a binomial distribution $B(15, 0.3)$ . Find $P(X \le 4)$ . [2]

7. The heights of adult males in a certain population are normally distributed with mean 175 cm and standard deviation 8 cm. Find the probability that a randomly selected male is taller than 185 cm. [2]

8. The weights of bags of rice produced by a machine are normally distributed with mean $\mu$ kg and standard deviation 0.5 kg. It is known that 10% of the bags weigh less than 4.8 kg. Find the value of $\mu$ . [3]

9. The random variable $Y$ is normally distributed with mean 50 and variance 16. Find the value of $k$ such that $P(Y < k) = 0.95$ . [2]

10. A discrete random variable $X$ has the following probability distribution:

$x$	1	2	3	4
$P(X=x)$	0.1	0.3	0.4	0.2

(a) Calculate $E(X)$ . [1] (b) Calculate $Var(X)$ . [2]

11. The time taken by students to complete a test is normally distributed with mean 45 minutes and standard deviation 10 minutes. (a) Find the probability that a student takes more than 60 minutes. [2] (b) If 200 students take the test, estimate the number of students who take more than 60 minutes. [1]

12. Let $X \sim N(20, 9)$ and $Y \sim N(15, 4)$ be independent random variables. Find $P(X - Y > 8)$ . [3]

Section C: Sampling and Estimation (Questions 13–16)

13. A random sample of 50 observations from a normal population with unknown mean $\mu$ and known variance 25 has a sample mean of 102. Find a 95% confidence interval for $\mu$ . [3]

14. The following summary statistics were obtained from a sample of 100 students' test scores: $\sum x = 7500$ and $\sum x^2 = 570,000$ . (a) Calculate the unbiased estimate of the population mean. [1] (b) Calculate the unbiased estimate of the population variance. [2]

15. A surveyor wants to estimate the mean height of plants in a large field. She takes a random sample of 64 plants. The sample mean is 25 cm and the sample standard deviation is 4 cm. State the distribution of the sample mean $\bar{X}$ , justifying your answer. [2]

16. Which of the following is an unbiased estimator for the population variance $\sigma^2$ ? A. $\frac{1}{n} \sum (x_i - \bar{x})^2$ B. $\frac{1}{n-1} \sum (x_i - \bar{x})^2$ C. $\frac{1}{n} \sum x_i^2 - \bar{x}^2$ D. $\sqrt{\frac{1}{n-1} \sum (x_i - \bar{x})^2}$

Circle the correct option. [1]

Section D: Hypothesis Testing and Regression (Questions 17–20)

17. A manufacturer claims that the mean lifetime of their batteries is 50 hours. A consumer group suspects the mean lifetime is less than 50 hours. State the null and alternative hypotheses for a test to investigate this claim. [2]

18. In a hypothesis test, the p-value obtained is 0.03. The test is conducted at the 5% significance level. State the conclusion of the test, explaining your reasoning. [2]

19. The scatter diagram below shows the relationship between the amount spent on advertising ( $x$ , in $'000) and sales ($ y $, in$ '000).

(Imagine a scatter plot with positive correlation)

The equation of the regression line of $y$ on $x$ is $y = 2.5x + 10$ . (a) Interpret the value of the gradient 2.5 in the context of the question. [1] (b) Estimate the sales when $20,000 is spent on advertising. [1]

20. The product moment correlation coefficient between two variables $x$ and $y$ is found to be $r = -0.92$ . Describe the strength and direction of the linear relationship between $x$ and $y$ . [2]

End of Quiz

Answers

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

1. [2 marks] Total ways to choose 4 from 11 is $\binom{11}{4} = 330$ . Cases for at least 2 women:

2 women, 2 men: $\binom{5}{2}\binom{6}{2} = 10 \times 15 = 150$
3 women, 1 man: $\binom{5}{3}\binom{6}{1} = 10 \times 6 = 60$
4 women, 0 men: $\binom{5}{4}\binom{6}{0} = 5 \times 1 = 5$ Total = $150 + 60 + 5 = 215$ . Answer: 215

2. [2 marks] (a) $P(A \cap B) = P(A) + P(B) - P(A \cup B) = 0.4 + 0.5 - 0.7 = 0.2$ . [1] (b) Check independence: $P(A)P(B) = 0.4 \times 0.5 = 0.2$ . Since $P(A \cap B) = P(A)P(B)$ , events A and B are independent. [1]

3. [2 marks] (a) $X \sim B(20, 0.05)$ . [1] (b) $P(X=2) = \binom{20}{2}(0.05)^2(0.95)^{18} \approx 0.1887$ . Answer: 0.189 (3 s.f.) [1]

4. [2 marks] Let M = Pass Math, P = Pass Physics. $P(M \cup P) = P(M) + P(P) - P(M \cap P)$ $P(M \cup P) = 0.8 + 0.7 - 0.6 = 0.9$ . Answer: 0.9

5. [2 marks] Tree Diagram:

First branch: Red (4/10), Blue (6/10)
If Red first: Second branch Red (3/9), Blue (6/9)
If Blue first: Second branch Red (4/9), Blue (5/9) (Award marks for correct structure and probabilities)

6. [2 marks] $X \sim B(15, 0.3)$ . Using GC: binomcdf(15, 0.3, 4) $P(X \le 4) \approx 0.5155$ . Answer: 0.516 (3 s.f.)

7. [2 marks] $H \sim N(175, 8^2)$ . $P(H > 185) = P(Z > \frac{185-175}{8}) = P(Z > 1.25)$ . Using GC or tables: $1 - 0.8944 = 0.1056$ . Answer: 0.106 (3 s.f.)

8. [3 marks] $W \sim N(\mu, 0.5^2)$ . $P(W < 4.8) = 0.10$ . From inverse normal, $Z_{0.10} \approx -1.2816$ . $\frac{4.8 - \mu}{0.5} = -1.2816$ $4.8 - \mu = -0.6408$ $\mu = 4.8 + 0.6408 = 5.4408$ . Answer: 5.44 (3 s.f.)

9. [2 marks] $Y \sim N(50, 16)$ . SD = 4. $P(Y < k) = 0.95 \implies Z = 1.6449$ . $\frac{k - 50}{4} = 1.6449$ $k = 50 + 4(1.6449) = 56.5796$ . Answer: 56.6 (3 s.f.)

10. [3 marks] (a) $E(X) = 1(0.1) + 2(0.3) + 3(0.4) + 4(0.2) = 0.1 + 0.6 + 1.2 + 0.8 = 2.7$ . [1] (b) $E(X^2) = 1^2(0.1) + 2^2(0.3) + 3^2(0.4) + 4^2(0.2) = 0.1 + 1.2 + 3.6 + 3.2 = 8.1$ . $Var(X) = E(X^2) - [E(X)]^2 = 8.1 - (2.7)^2 = 8.1 - 7.29 = 0.81$ . [2]

11. [3 marks] (a) $T \sim N(45, 10^2)$ . $P(T > 60) = P(Z > \frac{60-45}{10}) = P(Z > 1.5) = 1 - 0.9332 = 0.0668$ . [2] (b) Expected number = $200 \times 0.0668 = 13.36$ . Answer: 13 students. [1]

12. [3 marks] Let $W = X - Y$ . $E(W) = E(X) - E(Y) = 20 - 15 = 5$ . $Var(W) = Var(X) + Var(Y) = 9 + 4 = 13$ (Independent). $W \sim N(5, 13)$ . $P(W > 8) = P(Z > \frac{8-5}{\sqrt{13}}) = P(Z > \frac{3}{3.6056}) = P(Z > 0.832)$ . $P(Z > 0.832) \approx 0.2026$ . Answer: 0.203 (3 s.f.)

13. [3 marks] $n=50, \bar{x}=102, \sigma^2=25 \implies \sigma=5$ . 95% CI: $\bar{x} \pm z_{0.025} \frac{\sigma}{\sqrt{n}}$ . $102 \pm 1.96 \times \frac{5}{\sqrt{50}}$ . $102 \pm 1.96 \times 0.7071$ . $102 \pm 1.386$ . Interval: $(100.61, 103.39)$ . Answer: $100.6 < \mu < 103.4$

14. [3 marks] (a) Unbiased estimate of mean = $\bar{x} = \frac{7500}{100} = 75$ . [1] (b) Unbiased estimate of variance $s^2 = \frac{1}{n-1} (\sum x^2 - \frac{(\sum x)^2}{n})$ . $s^2 = \frac{1}{99} (570000 - \frac{7500^2}{100}) = \frac{1}{99} (570000 - 562500) = \frac{7500}{99}$ . $s^2 \approx 75.76$ . Answer: 75.8 (3 s.f.) [2]

15. [2 marks] Since $n=64$ is large ( $>30$ ), by the Central Limit Theorem, the sample mean $\bar{X}$ is approximately normally distributed. $\bar{X} \sim N(25, \frac{4^2}{64}) = N(25, 0.25)$ . (Must mention CLT or large sample size)

16. [1 mark] Answer: B

17. [2 marks] $H_0: \mu = 50$ $H_1: \mu < 50$ (Where $\mu$ is the mean lifetime of the batteries)

18. [2 marks] Since p-value (0.03) < significance level (0.05), we reject $H_0$ . There is sufficient evidence at the 5% level to support the claim that the mean lifetime is less than 50 hours.

19. [2 marks] (a) For every additional $1,000 spent on advertising, sales increase by$ 2,500 on average. [1] (b) $x=20$ . $y = 2.5(20) + 10 = 50 + 10 = 60$ . Sales = $60,000. [1]

20. [2 marks] Strong negative linear relationship. (1 mark for "Strong", 1 mark for "Negative")