A Level H1 Mathematics Practice Paper 2

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A-Level Maths H1 Quiz - Statistics Probability

Name: ____________________ Class: __________ Date: __________ Score: _________

Duration: 90 Minutes
Total Marks: 50 Marks

Instructions:

1. Answer all questions.
2. Use of an approved Graphing Calculator (GC) is expected.
3. Show all necessary working. Mathematical notation must be used; calculator commands will not be accepted.
4. Give your answers to 3 significant figures unless specified otherwise.

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Section A: Probability & Counting (Questions 1-7)

1. A committee of 5 people is to be chosen from a group of 7 men and 6 women. Find the number of ways the committee can be formed if it must contain exactly 3 men.
   
   
   [2]

2. Seven students are to be seated in a row. Two of them, Alice and Bob, refuse to sit next to each other. Find the number of possible seating arrangements.
   
   
   [2]

3. Events $A$ and $B$ are such that $P(A) = 0.6$, $P(B) = 0.4$, and $P(A \cup B) = 0.8$. Find $P(A \cap B)$.
   
   
   [2]

4. Given that $P(X) = 0.7$ and $P(Y|X) = 0.3$, find $P(X \cap Y)$.
   
   
   [2]

5. A bag contains 4 red balls and 6 blue balls. Two balls are drawn one after another without replacement. Draw a probability tree diagram to represent all possible outcomes.
   
   
   [3]

6. Using the tree diagram from Question 5, find the probability that both balls drawn are of the same colour.
   
   
   [2]

7. Events $C$ and $D$ are independent. Given $P(C) = 0.3$ and $P(C \cup D) = 0.5$, find $P(D)$.
   
   
   [3]

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Section B: Probability Distributions (Questions 8-14)

8. A fair coin is tossed 15 times. Let $X$ be the number of heads. State the distribution of $X$ and its parameters.
   
   
   [2]

9. In a large population, 20% of people are left-handed. In a random sample of 12 people, find the probability that exactly 3 are left-handed.
   
   
   [2]

10. A manufacturer finds that 5% of the lightbulbs produced are defective. In a random sample of 20 bulbs, find the probability that at least 2 are defective.
    
    
    [3]

11. The weights of apples in an orchard are normally distributed with a mean of 150g and a standard deviation of 12g. Find the probability that a randomly chosen apple weighs less than 135g.
    
    
    [3]

12. For the normal distribution in Question 11, find the weight $w$ such that 10% of the apples weigh more than $w$.
    
    
    [3]

13. Let $Y$ be a random variable such that $Y \sim N(40, 25)$. Find $E(3Y + 5)$ and $\text{Var}(3Y + 5)$.
    
    
    [3]

14. A continuous random variable $Z$ follows a normal distribution. Given $P(Z < 10) = 0.2$ and $P(Z > 20) = 0.1$, find the mean $\mu$ and standard deviation $\sigma$ of $Z$.
    
    
    [5]

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Section C: Sampling & Hypothesis Testing (Questions 15-20)

15. A researcher wants to select a random sample of 50 residents from a housing estate of 1,000 residents. Describe a method of simple random sampling that could be used.
    
    
    [2]

16. A sample of 10 students' heights is recorded: 160, 165, 170, 158, 162, 175, 168, 161, 164, 167. Calculate the unbiased estimate of the population mean.
    
    
    [2]

17. Using the data from Question 16, calculate the unbiased estimate of the population variance.
    
    
    [3]

18. A population has a mean $\mu$ and a variance $\sigma^2 = 100$. A random sample of size $n = 36$ is taken. Find the probability that the sample mean $\bar{X}$ is within 2 units of the population mean.
    
    
    [4]

19. A company claims that the average life of its batteries is 500 hours. A consumer group suspects the average is lower. They test 40 batteries and find a sample mean of 485 hours with a sample standard deviation of 30 hours. State the null and alternative hypotheses for a one-tailed test.
    
    
    [2]

20. Using the data from Question 19, test the company's claim at a 5% level of significance. State your conclusion in context.
    
    
    [5]

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A-Level Maths H1 Quiz - Statistics Probability (Answers)

Section A: Probability & Counting

1. $\binom{7}{3} \times \binom{6}{2} = 35 \times 15 = 525$ ways. [2]
2. Total arrangements = $7! = 5040$. Arrangements where Alice and Bob are together = $2! \times 6! = 1440$. $5040 - 1440 = 3600$. [2]
3. $P(A \cap B) = P(A) + P(B) - P(A \cup B) = 0.6 + 0.4 - 0.8 = 0.2$. [2]
4. $P(X \cap Y) = P(X) \times P(Y|X) = 0.7 \times 0.3 = 0.21$. [2]
5. Tree diagram:
   • Branch 1: Red (4/10), Blue (6/10)
   • Branch 2 (after Red): Red (3/9), Blue (6/9)
   • Branch 2 (after Blue): Red (4/9), Blue (5/9) [3]
6. $P(\text{Same}) = P(RR) + P(BB) = (4/10 \times 3/9) + (6/10 \times 5/9) = 12/90 + 30/90 = 42/90 = 7/15 \approx 0.467$. [2]
7. $P(C \cup D) = P(C) + P(D) - P(C)P(D) \Rightarrow 0.5 = 0.3 + P(D) - 0.3P(D) \Rightarrow 0.2 = 0.7P(D) \Rightarrow P(D) = 2/7 \approx 0.286$. [3]

Section B: Probability Distributions

8. $X \sim B(15, 0.5)$. [2]
9. $P(X=3) = \binom{12}{3}(0.2)^3(0.8)^9 \approx 0.236$. [2]
10. $P(X \geq 2) = 1 - [P(X=0) + P(X=1)] = 1 - [\binom{20}{0}(0.05)^0(0.95)^{20} + \binom{20}{1}(0.05)^1(0.95)^{19}] \approx 1 - [0.358 + 0.377] = 0.265$. [3]
11. $Z = (135 - 150)/12 = -1.25$. $P(Z < -1.25) \approx 0.106$. [3]
12. $P(X > w) = 0.1 \Rightarrow P(Z > z) = 0.1 \Rightarrow z \approx 1.282$. $w = 150 + 1.282(12) = 165.38 \approx 165\text{g}$. [3]
13. $E(3Y+5) = 3(40) + 5 = 125$. $\text{Var}(3Y+5) = 3^2 \times 25 = 225$. [3]
14. $P(Z < 10) = 0.2 \Rightarrow \frac{10-\mu}{\sigma} = -0.842$. $P(Z > 20) = 0.1 \Rightarrow \frac{20-\mu}{\sigma} = 1.282$. Subtracting: $\frac{10}{\sigma} = 2.124 \Rightarrow \sigma \approx 4.71$. $\mu = 10 + 0.842(4.71) \approx 13.97$. [5]

Section C: Sampling & Hypothesis Testing

15. Assign each resident a number from 1 to 1,000. Use a random number generator to select 50 distinct numbers. Interview the residents corresponding to these numbers. [2]
16. $\bar{x} = (160+165+170+158+162+175+168+161+164+167)/10 = 1660/10 = 166\text{cm}$. [2]
17. $s^2 = \frac{\sum(x_i - 166)^2}{9} = \frac{(-6)^2 + (-1)^2 + 4^2 + (-8)^2 + (-4)^2 + 9^2 + 2^2 + (-5)^2 + (-2)^2 + 1^2}{9} = \frac{36+1+16+64+16+81+4+25+4+1}{9} = \frac{248}{9} \approx 27.6$. [3]
18. $\bar{X} \sim N(\mu, 100/36) = N(\mu, 2.778)$. $P(|\bar{X} - \mu| < 2) = P(-2 < \bar{X} - \mu < 2) = P(\frac{-2}{\sqrt{2.778}} < Z < \frac{2}{\sqrt{2.778}}) = P(-1.2 < Z < 1.2) \approx 0.770$. [4]
19. $H_0: \mu = 500$, $H_1: \mu < 500$. [2]
20. Test statistic $z = \frac{485 - 500}{30/\sqrt{40}} = \frac{-15}{4.743} \approx -3.16$. Critical value for $\alpha=0.05$ (one-tail) is $-1.645$. Since $-3.16 < -1.645$, we reject $H_0$. Conclusion: There is sufficient evidence at the 5% level of significance to suggest that the average life of the batteries is lower than 500 hours. [5]