O Level Additional Mathematics Statistics Probability Quiz

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O-Level Additional Mathematics Quiz - Statistics Probability

Name: ____________________
Class: ____________________
Date: ____________________
Score: ________ / 65

Duration: 90 Minutes
Total Marks: 65

Instructions:

1. Answer all questions.
2. Give your answers to 3 significant figures, unless otherwise stated.
3. Show all essential working.
4. Use of an approved scientific calculator is allowed.

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Section A: Probability and Discrete Distributions (Questions 1–10)

1. A bag contains 5 red balls and 3 blue balls. Two balls are drawn at random without replacement. Find the probability that both balls are of the same colour. [3]
   
   
   
   Answer: ____________________

2. The probability that a student passes a Mathematics test is 0.7 and the probability that they pass an English test is 0.6. Given that the events are independent, find the probability that the student passes at least one of the tests. [3]
   
   
   
   Answer: ____________________

3. In a group of 100 students, 60 study Chemistry, 50 study Physics, and 30 study both. A student is chosen at random. Find the probability that the student studies neither Chemistry nor Physics. [3]
   
   
   
   Answer: ____________________

4. A fair coin is tossed 5 times. Find the probability of getting exactly 3 heads. [3]
   
   
   
   Answer: ____________________

5. The probability of a machine producing a defective part is 0.05. If 10 parts are produced independently, find the probability that at most 1 part is defective. [4]
   
   
   
   Answer: ____________________

6. Two events $A$ and $B$ are such that $P(A) = 0.4$, $P(B) = 0.5$ and $P(A \cup B) = 0.7$. Determine whether $A$ and $B$ are independent. Justify your answer. [4]
   
   
   
   Answer: ____________________

7. A box contains 8 light bulbs, 3 of which are defective. If 3 bulbs are selected at random without replacement, find the probability that exactly 2 are defective. [4]
   
   
   
   Answer: ____________________

8. The probability that a certain seed germinates is 0.8. If 6 seeds are planted, find the probability that more than 4 seeds germinate. [4]
   
   
   
   Answer: ____________________

9. A random variable $X$ follows a binomial distribution $B(n, p)$. Given that $E(X) = 4$ and $Var(X) = 3$, find the values of $n$ and $p$. [4]
   
   
   
   Answer: ____________________

10. A card is drawn from a standard deck of 52. Find the probability that the card is either a Heart or a King. [3]
    
    
    
    Answer: ____________________

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Section B: Statistics and Linear Regression (Questions 11–20)

11. The mean of 5 numbers is 12. When a 6th number is added, the new mean becomes 15. Find the value of the 6th number. [3]
    
    
    
    Answer: ____________________

12. A set of data consists of the values: 4, 7, 7, 8, 12, 15, 20. Calculate the mean and the standard deviation of the data. [4]
    
    
    
    Answer: ____________________

13. The variance of a set of 10 observations is 25. If every observation is multiplied by 3, find the new variance. [3]
    
    
    
    Answer: ____________________

14. For a set of data, $\sum x = 120$, $\sum x^2 = 3000$, and $n = 10$. Calculate the standard deviation. [4]
    
    
    
    Answer: ____________________

15. The mean of a distribution is 50 and the standard deviation is 10. If each value is increased by 5, find the new mean and new standard deviation. [3]
    
    
    
    Answer: ____________________

16. The table below shows the distance $x$ (in km) from a city centre and the house price $y$ (in thousand dollars) for 5 houses.

    $x$	2	4	6	8	10
    $y$	500	420	350	280	210
    Calculate the mean of $x$ and $y$. [3]
    \
    \
    \
    \
    Answer: ____________________

17. Using the data from Question 16, calculate the product-moment correlation coefficient $r$. [5]
    
    
    
    Answer: ____________________

18. Using the data from Question 16, find the equation of the regression line $y = a + bx$. [5]
    
    
    
    Answer: ____________________

19. Using the regression line from Question 18, estimate the house price for a house located 7 km from the city centre. [3]
    
    
    
    Answer: ____________________

20. Explain whether the correlation coefficient found in Question 17 suggests a strong or weak relationship, and whether it is positive or negative. [3]
    
    
    
    Answer: ____________________

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O-Level Additional Mathematics Quiz Answers - Statistics Probability

Section A: Probability and Discrete Distributions

1. Working: $P(\text{Same}) = P(RR) + P(BB) = (\frac{5}{8} \times \frac{4}{7}) + (\frac{3}{8} \times \frac{2}{7}) = \frac{20}{56} + \frac{6}{56} = \frac{26}{56} = \frac{13}{28}$ Answer: 0.464 (or 13/28) [3 marks]

2. Working: $P(\text{At least one}) = 1 - P(\text{None}) = 1 - (1-0.7)(1-0.6) = 1 - (0.3 \times 0.4) = 1 - 0.12 = 0.88$ Answer: 0.88 [3 marks]

3. Working: $P(C \cup P) = P(C) + P(P) - P(C \cap P) = 0.6 + 0.5 - 0.3 = 0.8$. $P(\text{Neither}) = 1 - 0.8 = 0.2$ Answer: 0.2 [3 marks]

4. Working: $\binom{5}{3} (0.5)^3 (0.5)^2 = 10 \times 0.125 \times 0.25 = 0.3125$ Answer: 0.313 [3 marks]

5. Working: $P(X \le 1) = P(X=0) + P(X=1) = \binom{10}{0}(0.05)^0(0.95)^{10} + \binom{10}{1}(0.05)^1(0.95)^9 \approx 0.5987 + 0.3151 = 0.9138$ Answer: 0.914 [4 marks]

6. Working: $P(A \cap B) = P(A) + P(B) - P(A \cup B) = 0.4 + 0.5 - 0.7 = 0.2$. Check independence: $P(A) \times P(B) = 0.4 \times 0.5 = 0.2$. Since $P(A \cap B) = P(A)P(B)$, they are independent. Answer: Independent [4 marks]

7. Working: $\frac{\binom{3}{2} \times \binom{5}{1}}{\binom{8}{3}} = \frac{3 \times 5}{56} = \frac{15}{56}$ Answer: 0.268 [4 marks]

8. Working: $P(X > 4) = P(X=5) + P(X=6) = \binom{6}{5}(0.8)^5(0.2)^1 + \binom{6}{6}(0.8)^6(0.2)^0 = 0.3932 + 0.2621 = 0.6553$ Answer: 0.655 [4 marks]

9. Working: $np = 4$ and $np(1-p) = 3$. Divide: $(1-p) = 3/4 \implies p = 0.25$. $n(0.25) = 4 \implies n = 16$. Answer: $n=16, p=0.25$ [4 marks]

10. Working: $P(H \cup K) = P(H) + P(K) - P(H \cap K) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$ Answer: 0.308 [3 marks]

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Section B: Statistics and Linear Regression

11. Working: Total of 5 = $5 \times 12 = 60$. Total of 6 = $6 \times 15 = 90$. 6th number = $90 - 60 = 30$. Answer: 30 [3 marks]

12. Working: Mean $\bar{x} = \frac{4+7+7+8+12+15+20}{7} = \frac{73}{7} \approx 10.4$. $\sigma^2 = \frac{\sum(x-\bar{x})^2}{n} = \frac{(4-10.4)^2 + \dots + (20-10.4)^2}{7} \approx \frac{211.7}{7} \approx 30.2$. $\sigma \approx 5.5$. Answer: Mean = 10.4, SD = 5.5 [4 marks]

13. Working: New Variance = $k^2 \times \text{Old Variance} = 3^2 \times 25 = 9 \times 25 = 225$. Answer: 225 [3 marks]

14. Working: $\bar{x} = 120/10 = 12$. $\text{Var} = \frac{\sum x^2}{n} - \bar{x}^2 = \frac{3000}{10} - 12^2 = 300 - 144 = 156$. $\sigma = \sqrt{156} \approx 12.5$. Answer: 12.5 [4 marks]

15. Working: New Mean = $50 + 5 = 55$. Standard deviation is invariant under translation. New SD = 10. Answer: Mean = 55, SD = 10 [3 marks]

16. Working: $\bar{x} = \frac{2+4+6+8+10}{5} = 6$. $\bar{y} = \frac{500+420+350+280+210}{5} = \frac{1760}{5} = 352$. Answer: $\bar{x} = 6, \bar{y} = 352$ [3 marks]

17. Working: $\sum(x-\bar{x})^2 = 16+4+0+4+16 = 40$. $\sum(x-\bar{x})(y-\bar{y}) = (-4)(148) + (-2)(68) + (0)(-2) + (2)(-72) + (4)(-142) = -592 - 136 + 0 - 144 - 568 = -1440$. $r = \frac{-1440}{\sqrt{40 \times \sum(y-\bar{y})^2}}$. $\sum(y-\bar{y})^2 = 148^2 + 68^2 + (-2)^2 + (-72)^2 + (-142)^2 = 21904 + 4624 + 4 + 5184 + 20164 = 51880$. $r = \frac{-1440}{\sqrt{40 \times 51880}} = \frac{-1440}{1444} \approx -0.997$. Answer: -0.997 [5 marks]

18. Working: $b = \frac{\sum(x-\bar{x})(y-\bar{y})}{\sum(x-\bar{x})^2} = \frac{-1440}{40} = -36$. $a = \bar{y} - b\bar{x} = 352 - (-36)(6) = 352 + 216 = 568$. Answer: $y = 568 - 36x$ [5 marks]

19. Working: $y = 568 - 36(7) = 568 - 252 = 316$. Answer: 316 thousand dollars [3 marks]

20. Working: $r \approx -0.997$. Since $|r|$ is very close to 1, it is a strong relationship. Since $r$ is negative, it is a negative correlation. Answer: Strong negative correlation [3 marks]