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A Level H1 Mathematics Statistics Probability Quiz

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A-Level Maths H1 Quiz - Statistics Probability

Name: __________________________
Class: __________________________
Date: __________________________
Score: _______ / 50

Duration: 60 Minutes
Total Marks: 50

Instructions:

Answer all 20 questions.
An approved graphing calculator is expected. Show all necessary mathematical working; answers supported only by calculator output without working may not receive full credit.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless otherwise specified.
The total mark for each question or part question is given in brackets [ ].

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

1. A committee of 4 people is to be chosen from a group of 6 men and 5 women. (a) Find the number of different committees that can be formed if there are no restrictions. [1] (b) Find the number of different committees that can be formed if the committee must contain at least 2 women. [2]

2. Events $A$ and $B$ are defined 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, with a reason, whether events $A$ and $B$ are independent. [2]

3. In a certain school, 60% of the students study Mathematics, 40% study Physics, and 20% study both. A student is selected at random. (a) Find the probability that the student studies neither Mathematics nor Physics. [1] (b) Given that the student studies Physics, find the probability that they also study Mathematics. [2]

4. A bag contains 4 red balls and 6 blue balls. Two balls are drawn from the bag one after the other without replacement. (a) Draw a tree diagram to represent the possible outcomes and their probabilities. [2] (b) Find the probability that the two balls are of different colors. [1]

5. The probability that it rains on any given day in June is 0.3. Assume that the weather on each day is independent. (a) Find the probability that it rains on exactly 2 days in a specific 5-day period. [2] (b) Find the probability that it rains on at least one day in a specific 5-day period. [1]

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

6. A random variable $X$ follows a binomial distribution $B(12, 0.25)$ . (a) Find $P(X = 3)$ . [1] (b) Find $P(X \ge 2)$ . [2]

7. A manufacturer claims that 5% of the light bulbs produced are defective. A quality control inspector takes a random sample of 20 bulbs. (a) State two conditions required for the number of defective bulbs to be modelled by a binomial distribution. [2] (b) Find the probability that more than 2 bulbs are defective. [2]

8. The heights of adult males in a certain population are normally distributed with a mean of 175 cm and a standard deviation of 8 cm. Let $H$ be the height of a randomly selected adult male. (a) Find $P(H < 165)$ . [1] (b) Find the value of $h$ such that $P(H > h) = 0.1$ . [2]

9. 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 4.36 kg. (a) Find the value of $\mu$ . [3] (b) Find the probability that a randomly selected bag weighs between 4.5 kg and 5.5 kg. [2]

10. Let $X \sim N(50, 16)$ and $Y \sim N(30, 9)$ be independent random variables. (a) Find the distribution of the random variable $W = X - Y$ . [2] (b) Find $P(W > 22)$ . [2]

11. The daily sales of a newspaper shop follow a normal distribution with mean 200 copies and standard deviation 20 copies. (a) Find the probability that the sales on a randomly chosen day exceed 230 copies. [1] (b) The shop owner wants to ensure that he has enough stock to meet demand on 95% of days. How many copies should he stock? [2]

12. A continuous random variable $T$ represents the time (in minutes) a student spends on a homework task. $T$ is normally distributed. It is given that $P(T < 30) = 0.2$ and $P(T < 50) = 0.9$ . (a) Find the mean and standard deviation of $T$ . [4] (Note: You may use the inverse normal function on your calculator)

Section C: Sampling, Estimation, and Hypothesis Testing (Questions 13–20)

13. A random sample of 100 students is taken from a large population. The sample mean height is 168 cm, and the sample variance is 36 cm $^2$ . (a) Find the unbiased estimate of the population mean. [1] (b) Find the unbiased estimate of the population variance. [1] (c) Find the variance of the sample mean. [1]

14. The masses of apples from an orchard are normally distributed with a known standard deviation of 15 g. A random sample of 25 apples has a mean mass of 140 g. (a) Construct a 90% confidence interval for the population mean mass. [3] (b) State what is meant by "90% confidence" in this context. [1]

15. A company claims that the mean lifetime of its batteries is 500 hours. A consumer group suspects the mean lifetime is less than 500 hours. They test a random sample of 40 batteries and find a mean lifetime of 485 hours. The population standard deviation is known to be 40 hours. (a) State the null and alternative hypotheses. [2] (b) Perform a hypothesis test at the 5% significance level. State your conclusion in context. [4]

16. In a previous year, the mean score of students in a national test was 65. This year, a random sample of 50 students had a mean score of 68 with a sample standard deviation of 10. (a) Test, at the 1% significance level, whether the mean score has changed. [4] (b) Explain why the Central Limit Theorem is applicable in this test. [1]

17. The table below shows the age ( $x$ years) and the systolic blood pressure ( $y$ mmHg) of 8 individuals.

Age ( $x$ )	40	45	50	55	60	65	70	75
Pressure ( $y$ )	120	125	130	138	142	148	155	160

(a) Calculate the product moment correlation coefficient, $r$ . [1] (b) Find the equation of the regression line of $y$ on $x$ in the form $y = a + bx$ . [2] (c) Estimate the blood pressure of a 62-year-old individual. [1] (d) Comment on the reliability of estimating the blood pressure of a 90-year-old individual using this model. [1]

18. For a set of bivariate data, the regression line of $y$ on $x$ is $y = 2.5x + 10$ and the regression line of $x$ on $y$ is $x = 0.3y - 2$ . (a) Find the product moment correlation coefficient $r$ . [2] (b) Determine the sign of $r$ with a reason. [1]

19. A surveyor wants to estimate the mean income of residents in a town. He selects the first 50 people who enter a shopping mall on a Monday morning. (a) Identify the sampling method used. [1] (b) Explain one reason why this sample may be biased. [1] (c) Describe how a simple random sample of 50 residents could be obtained from the electoral register of 10,000 residents. [2]

20. The time taken for workers to assemble a component is normally distributed with mean 12 minutes and standard deviation 2 minutes. (a) Find the probability that the mean time for a random sample of 16 workers is less than 11 minutes. [3] (b) Find the probability that the total time for 16 workers to assemble their components is more than 200 minutes. [2]

*** End of Quiz ***

Answers

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

1. (a) Total people = 11. Choose 4. $\binom{11}{4} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = 330$ . [1] (b) At least 2 women means: 2W 2M, 3W 1M, or 4W 0M. 2W 2M: $\binom{5}{2}\binom{6}{2} = 10 \times 15 = 150$ 3W 1M: $\binom{5}{3}\binom{6}{1} = 10 \times 6 = 60$ 4W 0M: $\binom{5}{4}\binom{6}{0} = 5 \times 1 = 5$ Total = $150 + 60 + 5 = 215$ . [2]

2. (a) $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ $0.7 = 0.4 + 0.5 - P(A \cap B)$ $P(A \cap B) = 0.9 - 0.7 = 0.2$ . [1] (b) Check independence: Is $P(A \cap B) = P(A)P(B)$ ? $P(A)P(B) = 0.4 \times 0.5 = 0.2$ . Since $0.2 = 0.2$ , events $A$ and $B$ are independent. [2]

3. Let $M$ = Math, $P$ = Physics. $P(M) = 0.6, P(P) = 0.4, P(M \cap P) = 0.2$ . (a) $P(\text{Neither}) = 1 - P(M \cup P) = 1 - [P(M) + P(P) - P(M \cap P)]$ $= 1 - [0.6 + 0.4 - 0.2] = 1 - 0.8 = 0.2$ . [1] (b) $P(M | P) = \frac{P(M \cap P)}{P(P)} = \frac{0.2}{0.4} = 0.5$ . [2]

4. (a) Tree Diagram: Start -> Red (4/10) -> Red (3/9), Blue (6/9) Start -> Blue (6/10) -> Red (4/9), Blue (5/9) [2] (b) Different colors: (Red then Blue) or (Blue then Red). $P(RB) = \frac{4}{10} \times \frac{6}{9} = \frac{24}{90}$ $P(BR) = \frac{6}{10} \times \frac{4}{9} = \frac{24}{90}$ Total = $\frac{48}{90} = \frac{8}{15} \approx 0.533$ . [1]

5. Let $X$ be number of rainy days. $X \sim B(5, 0.3)$ . (a) $P(X=2) = \binom{5}{2}(0.3)^2(0.7)^3 = 10 \times 0.09 \times 0.343 = 0.3087$ . [2] (b) $P(X \ge 1) = 1 - P(X=0) = 1 - (0.7)^5 = 1 - 0.16807 = 0.83193 \approx 0.832$ . [1]

6. $X \sim B(12, 0.25)$ . (a) $P(X=3) = \binom{12}{3}(0.25)^3(0.75)^9 \approx 0.258$ . [1] (b) $P(X \ge 2) = 1 - P(X \le 1) = 1 - [P(X=0) + P(X=1)]$ . $P(X=0) = (0.75)^{12} \approx 0.0317$ $P(X=1) = 12(0.25)(0.75)^{11} \approx 0.1267$ $P(X \ge 2) = 1 - (0.0317 + 0.1267) = 1 - 0.1584 = 0.8416 \approx 0.842$ . [2]

7. (a) Conditions:

Fixed number of trials ( $n=20$ ).
Constant probability of success ( $p=0.05$ ).
Trials are independent.
Two outcomes (defective/not defective). (Any two). [2] (b) Let $D$ be number of defective bulbs. $D \sim B(20, 0.05)$ . $P(D > 2) = 1 - P(D \le 2) = 1 - \text{binomcdf}(20, 0.05, 2)$ . Using calculator: $P(D \le 2) \approx 0.9245$ . $P(D > 2) = 1 - 0.9245 = 0.0755$ . [2]

8. $H \sim N(175, 8^2)$ . (a) $P(H < 165) = \text{normalcdf}(-\infty, 165, 175, 8) \approx 0.1056$ . [1] (b) $P(H > h) = 0.1 \Rightarrow P(H < h) = 0.9$ . $h = \text{invNorm}(0.9, 175, 8) \approx 185.25$ cm. [2]

9. $W \sim N(\mu, 0.5^2)$ . (a) $P(W < 4.36) = 0.1$ . Standardizing: $Z = \frac{4.36 - \mu}{0.5}$ . From tables/calc, $P(Z < z) = 0.1 \Rightarrow z \approx -1.2816$ . $\frac{4.36 - \mu}{0.5} = -1.2816 \Rightarrow 4.36 - \mu = -0.6408 \Rightarrow \mu = 5.0008 \approx 5.00$ kg. [3] (b) $P(4.5 < W < 5.5)$ with $\mu=5, \sigma=0.5$ . $= \text{normalcdf}(4.5, 5.5, 5, 0.5) \approx 0.6827$ . [2]

10. (a) $W = X - Y$ . $E(W) = E(X) - E(Y) = 50 - 30 = 20$ . Since independent, $Var(W) = Var(X) + Var(Y) = 16 + 9 = 25$ . $W \sim N(20, 25)$ . [2] (b) $P(W > 22) = \text{normalcdf}(22, \infty, 20, 5)$ . $Z = \frac{22-20}{5} = 0.4$ . $P(Z > 0.4) = 1 - 0.6554 = 0.3446$ . [2]

11. $S \sim N(200, 20^2)$ . (a) $P(S > 230) = \text{normalcdf}(230, \infty, 200, 20) \approx 0.0668$ . [1] (b) Find $k$ such that $P(S < k) = 0.95$ . $k = \text{invNorm}(0.95, 200, 20) \approx 232.9$ . Stock 233 copies. [2]

12. $T \sim N(\mu, \sigma^2)$ . $P(T < 30) = 0.2 \Rightarrow \frac{30-\mu}{\sigma} = z_1 \approx -0.8416$ . $P(T < 50) = 0.9 \Rightarrow \frac{50-\mu}{\sigma} = z_2 \approx 1.2816$ . Eq 1: $30 - \mu = -0.8416\sigma$ Eq 2: $50 - \mu = 1.2816\sigma$ Subtract Eq 1 from Eq 2: $20 = 2.1232\sigma \Rightarrow \sigma \approx 9.42$ . Substitute $\sigma$ : $\mu = 30 + 0.8416(9.42) \approx 37.93$ . Mean $\approx 37.9$ , SD $\approx 9.42$ . [4]

13. (a) Unbiased estimate of mean = sample mean = 168 cm. [1] (b) Unbiased estimate of variance $s^2 = \frac{n}{n-1} \times \text{sample variance}$ ? Note: Question says "sample variance is 36". Usually, if "sample variance" is given as $s^2_{biased} = \frac{\sum(x-\bar{x})^2}{n}$ , then unbiased is $\frac{n}{n-1}s^2_{biased}$ . However, standard convention in many texts: if "variance of the sample" is stated, it often implies $s^2 = \frac{\sum(x-\bar{x})^2}{n-1}$ is already the unbiased estimator if calculated from data. Clarification for H1: If 36 is $\frac{\sum x^2}{n} - \bar{x}^2$ (biased), then Unbiased $= \frac{100}{99} \times 36 = 36.36$ . If 36 is already $s^2$ (unbiased), then answer is 36. Given "sample variance" usually refers to biased estimator in raw data contexts unless specified "unbiased", we assume biased. Unbiased Estimate $= \frac{100}{99}(36) \approx 36.4$ . [1] (c) Variance of sample mean $\bar{X} = \frac{\sigma^2}{n}$ . We estimate $\sigma^2$ with unbiased estimate 36.36. $Var(\bar{X}) = \frac{36.36}{100} = 0.3636$ . [1]

14. (a) $\sigma = 15, n=25, \bar{x}=140$ . 90% CI. $z_{0.05} = 1.645$ . CI $= \bar{x} \pm z \frac{\sigma}{\sqrt{n}} = 140 \pm 1.645 \frac{15}{5} = 140 \pm 1.645(3) = 140 \pm 4.935$ . $(135.065, 144.935)$ . [3] (b) If we were to take many random samples of size 25 and construct a 90% CI for each, 90% of these intervals would contain the true population mean. [1]

15. (a) $H_0: \mu = 500$ . $H_1: \mu < 500$ . [2] (b) Test statistic $Z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} = \frac{485 - 500}{40/\sqrt{40}} = \frac{-15}{6.3246} \approx -2.37$ . Critical value for 1-tail 5%: $-1.645$ . Since $-2.37 < -1.645$ , we reject $H_0$ . Alternatively, p-value $= P(Z < -2.37) \approx 0.0089$ . Since $0.0089 < 0.05$ , reject $H_0$ . Conclusion: There is sufficient evidence at the 5% level to suggest the mean lifetime is less than 500 hours. [4]

16. (a) $H_0: \mu = 65$ . $H_1: \mu \ne 65$ . $n=50$ (large), so use Z-test approx. $s=10$ . $Z = \frac{68 - 65}{10/\sqrt{50}} = \frac{3}{1.414} \approx 2.12$ . Critical values for 2-tail 1%: $\pm 2.576$ . Since $2.12 < 2.576$ , we do not reject $H_0$ . Conclusion: There is insufficient evidence to suggest the mean score has changed. [4] (b) The sample size $n=50$ is large ( $>30$ ), so by the Central Limit Theorem, the sampling distribution of the mean is approximately normal, regardless of the population distribution. [1]

17. (a) Using GC: $r \approx 0.998$ . [1] (b) Regression line $y$ on $x$ : $a \approx 66.57, b \approx 1.257$ . $y = 66.6 + 1.26x$ (to 3 sf). [2] (c) $x=62 \Rightarrow y = 66.57 + 1.257(62) \approx 144.5$ mmHg. [1] (d) Unreliable. 90 is outside the range of the data (40-75), so this is extrapolation. The linear relationship may not hold for older ages. [1]

18. (a) $b_{yx} = 2.5$ , $b_{xy} = 0.3$ . $r^2 = b_{yx} \times b_{xy} = 2.5 \times 0.3 = 0.75$ . $r = \sqrt{0.75} \approx 0.866$ . [2] (b) Positive, because both regression coefficients ( $2.5$ and $0.3$ ) are positive. [1]

19. (a) Convenience sampling. [1] (b) People at a mall on Monday morning may not represent the whole population (e.g., working people are excluded, elderly/retired over-represented). [1] (c) Assign each of the 10,000 residents a unique number from 1 to 10,000. Use a random number generator to select 50 distinct numbers. Select the residents corresponding to these numbers. [2]

20. $T \sim N(12, 2^2)$ . (a) Sample mean $\bar{T}$ for $n=16$ . $\bar{T} \sim N(12, \frac{2^2}{16}) = N(12, 0.25)$ . SD = 0.5. $P(\bar{T} < 11) = \text{normalcdf}(-\infty, 11, 12, 0.5)$ . $Z = \frac{11-12}{0.5} = -2$ . $P(Z < -2) \approx 0.0228$ . [3] (b) Total time $S = \sum_{i=1}^{16} T_i$ . $E(S) = 16 \times 12 = 192$ . $Var(S) = 16 \times 2^2 = 64$ . SD = 8. $S \sim N(192, 64)$ . $P(S > 200) = \text{normalcdf}(200, \infty, 192, 8)$ . $Z = \frac{200-192}{8} = 1$ . $P(Z > 1) \approx 0.1587$ . [2]