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Secondary 4 Additional Mathematics Statistics Probability Quiz

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Questions

Secondary 4 Additional Mathematics Quiz - Statistics Probability

Name: ___________________________
Class: ___________________________
Date: ___________________________
Score: ________ / 50

Duration: 60 minutes
Total Marks: 50

Instructions:

Answer ALL questions.
Show your working clearly. Marks will be awarded for correct reasoning and method, not only for the final answer.
Non-programmable scientific calculators may be used.
Give non-exact answers correct to 3 significant figures unless otherwise stated.
The number of marks for each question or part-question is shown in brackets [ ].

Section A: Probability (Questions 1–10)

1. A bag contains 5 red balls, 4 blue balls and 3 green balls. Two balls are drawn at random without replacement. Find the probability that both balls are red. [3]

2. Two fair six-sided dice are thrown. Find the probability that the sum of the two scores is 8. [3]

3. In a class of 30 students, 18 study Biology and 15 study Chemistry. If 5 students study neither subject, find the probability that a randomly chosen student studies both Biology and Chemistry. [3]

4. A box contains 8 cards numbered 1 to 8. A card is drawn at random, its number recorded, and then it is replaced. A second card is drawn. Find the probability that the sum of the two numbers is 10. [3]

5. The probability that it rains on any given day is 0.3. Find the probability that it rains on exactly 2 days out of 5 consecutive days. [3]

6. A bag contains 6 white balls and $k$ black balls. The probability of drawing a black ball at random is $\frac{3}{5}$ . Find the value of $k$ . [3]

7. Two events $A$ and $B$ are such that $\mathrm{P}(A) = 0.6$ , $\mathrm{P}(B) = 0.4$ and $\mathrm{P}(A \cap B) = 0.2$ . Find:

(a) $\mathrm{P}(A \cup B)$ [2]

(b) $\mathrm{P}(A' \cap B')$ [2]

8. A fair coin is tossed 4 times. Find the probability of obtaining exactly 3 heads. [3]

9. In a game, a player throws a single fair six-sided die. If the score is even, the player wins that number of dollars. If the score is odd, the player loses that number of dollars. Find the probability that the player wins more than $2 in a single throw. [3]

10. A committee of 3 is to be chosen at random from 4 boys and 5 girls. Find the probability that the committee contains at least one boy. [3]

Section B: Statistics (Questions 11–20)

11. The following table shows the marks obtained by 40 students in a test.

Mark ( $x$ )	10	20	30	40	50
Frequency ( $f$ )	4	8	12	10	6

(a) Calculate the mean mark. [2]

(b) Calculate the standard deviation of the marks. [3]

12. The heights (in cm) of 8 plants are recorded as follows:

$12,\ 15,\ 18,\ 14,\ 20,\ 16,\ 13,\ 17$

Find the mean and variance of the heights. [4]

13. The mean of five numbers is 12. When a sixth number is added, the mean becomes 14. Find the sixth number. [3]

14. The following stem-and-leaf diagram shows the ages of 15 participants in a competition.

Stem | Leaf
1    | 2 4 5 8
2    | 0 1 3 3 6 7 9
3    | 1 2 5 8

Key: $1|2$ represents 12 years

(a) Find the median age. [2]

(b) Find the interquartile range. [3]

15. A set of data has mean 25 and standard deviation 4. Each value in the data set is multiplied by 3 and then increased by 2. Find the new mean and new standard deviation. [4]

16. The cumulative frequency table below shows the time taken (in minutes) by 50 students to complete a task.

Time (min)	< 10	< 20	< 30	< 40	< 50	< 60
Cumulative frequency	5	14	28	40	47	50

(a) Draw a cumulative frequency curve (ogive) for this data. [3]

(b) Use your graph to estimate the median time. [2]

17. The mean and standard deviation of a set of 20 observations are 15 and 3 respectively. A new observation of value 21 is added to the set. Find the new mean and explain, without further calculation, whether the standard deviation will increase or decrease. [4]

18. Two groups of students sit for the same test. Group A has 30 students with a mean score of 68 and standard deviation 5. Group B has 20 students with a mean score of 75 and standard deviation 6. Find the mean score and standard deviation of the combined group of 50 students. [5]

19. The following table shows the distribution of the number of books read by 60 students in a year.

Number of books	0–2	3–5	6–8	9–11	12–14
Frequency	8	15	22	10	5

(a) State the modal class. [1]

(b) Calculate the mean number of books read. [3]

20. A set of 10 data values has a mean of 20 and a variance of 16. Two of the values are removed. The mean of the remaining 8 values is 19.5.

(a) Find the sum of the two removed values. [2]

(b) Given that one of the removed values is 24, find the other removed value. [1]

(c) The variance of the original 10 values is 16. Without detailed calculation, state whether the variance of the remaining 8 values is greater than, less than, or equal to 16. Justify your answer. [2]

End of Quiz

Answers

Secondary 4 Additional Mathematics Quiz - Statistics Probability

Answer Key

Question 1 [3]

Total balls = 5 + 4 + 3 = 12.

$\mathrm{P}(\text{both red}) = \frac{5}{12} \times \frac{4}{11} = \frac{20}{132} = \frac{5}{33}$

Answer: $\dfrac{5}{33}$

Marking notes: M1 for correct probability setup (multiplication of two fractions), M1 for correct substitution, A1 for final answer.

Question 2 [3]

Total outcomes = 36.

Favourable outcomes for sum = 8: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 outcomes.

$\mathrm{P}(\text{sum} = 8) = \frac{5}{36}$

Answer: $\dfrac{5}{36}$

Marking notes: M1 for identifying total outcomes = 36, M1 for listing/counting favourable outcomes, A1 for final answer.

Question 3 [3]

Let $x$ = number who study both.

Students studying at least one subject = 30 − 5 = 25.

$18 + 15 - x = 25$ $33 - x = 25$ $x = 8$

$\mathrm{P}(\text{studies both}) = \frac{8}{30} = \frac{4}{15}$

Answer: $\dfrac{4}{15}$

Marking notes: M1 for using inclusion-exclusion principle, M1 for solving for $x$ , A1 for probability.

Question 4 [3]

Total outcomes = $8 \times 8 = 64$ (with replacement).

Favourable outcomes for sum = 10: (2,8), (3,7), (4,6), (5,5), (6,4), (7,3), (8,2) → 7 outcomes.

$\mathrm{P}(\text{sum} = 10) = \frac{7}{64}$

Answer: $\dfrac{7}{64}$

Marking notes: M1 for total outcomes = 64, M1 for listing favourable pairs, A1 for final answer.

Question 5 [3]

This is a binomial distribution: $X \sim \mathrm{B}(5, 0.3)$ .

$\mathrm{P}(X = 2) = \binom{5}{2}(0.3)^2(0.7)^3 = 10 \times 0.09 \times 0.343 = 0.3087$

Answer: 0.309 (3 s.f.)

Marking notes: M1 for identifying binomial with $n = 5, p = 0.3$ , M1 for correct binomial formula substitution, A1 for answer to 3 s.f.

Question 6 [3]

Total balls = $6 + k$ .

$\frac{k}{6 + k} = \frac{3}{5}$

$5k = 3(6 + k)$ $5k = 18 + 3k$ $2k = 18$ $k = 9$

Answer: $k = 9$

Marking notes: M1 for setting up the equation, M1 for solving, A1 for $k = 9$ .

Question 7 [4]

(a) [2]

$\mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cap B) = 0.6 + 0.4 - 0.2 = 0.8$

Answer: 0.8

(b) [2]

$\mathrm{P}(A' \cap B') = 1 - \mathrm{P}(A \cup B) = 1 - 0.8 = 0.2$

Answer: 0.2

Marking notes: (a) M1 for addition formula, A1 for answer. (b) M1 for using complement of union, A1 for answer.

Question 8 [3]

$X \sim \mathrm{B}(4, 0.5)$

$\mathrm{P}(X = 3) = \binom{4}{3}(0.5)^3(0.5)^1 = 4 \times \frac{1}{16} = \frac{4}{16} = \frac{1}{4}$

Answer: $\dfrac{1}{4}$ or 0.25

Marking notes: M1 for binomial identification, M1 for correct substitution, A1 for answer.

Question 9 [3]

Even scores: 2, 4, 6. Winning more than $2 means winning$ 4 or $6.

Favourable outcomes: scores of 4 or 6 → 2 outcomes.

$\mathrm{P}(\text{wins more than \$2}) = \frac{2}{6} = \frac{1}{3}$

Answer: $\dfrac{1}{3}$

Marking notes: M1 for identifying even scores and the condition "more than $2", M1 for counting favourable outcomes, A1 for answer.

Question 10 [3]

Total ways to choose 3 from 9: $\binom{9}{3} = 84$ .

$\mathrm{P}(\text{at least one boy}) = 1 - \mathrm{P}(\text{no boys}) = 1 - \frac{\binom{5}{3}}{\binom{9}{3}} = 1 - \frac{10}{84} = \frac{74}{84} = \frac{37}{42}$

Answer: $\dfrac{37}{42}$

Marking notes: M1 for using complementary probability, M1 for correct calculation of $\binom{5}{3}$ and $\binom{9}{3}$ , A1 for final answer.

Question 11 [5]

(a) [2]

$\bar{x} = \frac{\sum fx}{\sum f} = \frac{10(4) + 20(8) + 30(12) + 40(10) + 50(6)}{4 + 8 + 12 + 10 + 6}$

$= \frac{40 + 160 + 360 + 400 + 300}{40} = \frac{1260}{40} = 31.5$

Answer: Mean = 31.5

(b) [3]

$\sigma = \sqrt{\frac{\sum fx^2}{\sum f} - \bar{x}^2}$

$\sum fx^2 = 100(4) + 400(8) + 900(12) + 1600(10) + 2500(6) = 400 + 3200 + 10800 + 16000 + 15000 = 45400$

$\sigma = \sqrt{\frac{45400}{40} - (31.5)^2} = \sqrt{1135 - 992.25} = \sqrt{142.75} \approx 11.9$

Answer: Standard deviation ≈ 11.9 (3 s.f.)

Marking notes: (a) M1 for correct formula, A1 for answer. (b) M1 for computing $\sum fx^2$ , M1 for correct substitution into formula, A1 for answer to 3 s.f.

Question 12 [4]

Mean:

$\bar{x} = \frac{12 + 15 + 18 + 14 + 20 + 16 + 13 + 17}{8} = \frac{125}{8} = 15.625$

Variance:

$\sigma^2 = \frac{\sum x^2}{n} - \bar{x}^2$

$\sum x^2 = 144 + 225 + 324 + 196 + 400 + 256 + 169 + 289 = 2003$

$\sigma^2 = \frac{2003}{8} - (15.625)^2 = 250.375 - 244.140625 = 6.234375 \approx 6.23$

Answer: Mean = 15.625, Variance ≈ 6.23 (3 s.f.)

Marking notes: M1 for correct mean calculation, M1 for computing $\sum x^2$ , M1 for correct variance formula, A1 for both answers.

Question 13 [3]

Sum of five numbers = $5 \times 12 = 60$ .

Sum of six numbers = $6 \times 14 = 84$ .

Sixth number = $84 - 60 = 24$ .

Answer: 24

Marking notes: M1 for finding sum of five numbers, M1 for finding sum of six numbers, A1 for answer.

Question 14 [5]

Data in order: 12, 14, 15, 18, 20, 21, 23, 23, 26, 27, 29, 31, 32, 35, 38

(a) [2]

$n = 15$ , median is the 8th value.

Answer: Median = 23

(b) [3]

Lower quartile $Q_1$ : median of first 7 values = 4th value = 18.

Upper quartile $Q_3$ : median of last 7 values = 12th value = 31.

$\mathrm{IQR} = Q_3 - Q_1 = 31 - 18 = 13$

Answer: Interquartile range = 13

Marking notes: (a) M1 for identifying position of median, A1 for answer. (b) M1 for finding $Q_1$ , M1 for finding $Q_3$ , A1 for IQR.

Question 15 [4]

New value = $3x + 2$ .

New mean = $3(25) + 2 = 77$ .

New standard deviation = $3 \times 4 = 12$ (adding a constant does not affect spread).

Answer: New mean = 77, New standard deviation = 12

Marking notes: M1 for new mean formula, A1 for new mean. M1 for understanding that SD scales by multiplication factor only, A1 for new SD.

Question 16 [5]

(a) [3]

Plot points: (10, 5), (20, 14), (30, 28), (40, 40), (50, 47), (60, 50) with time on the horizontal axis and cumulative frequency on the vertical axis. Draw a smooth curve through the points.

Marking: M1 for correct axes and labels, M1 for correct plotting of all points, A1 for smooth curve.

(b) [2]

Median position = $\frac{50}{2} = 25$ . Read from the graph at cumulative frequency = 25.

From the table, cumulative frequency 25 lies between 14 (at 20 min) and 28 (at 30 min).

$\text{Median} \approx 20 + \frac{25 - 14}{28 - 14} \times 10 = 20 + \frac{11}{14} \times 10 \approx 27.9$

Answer: Median ≈ 28 minutes (accept 27–29 from graph reading)

Marking notes: M1 for identifying median position, A1 for reading from graph (allow reasonable range).

Question 17 [4]

New mean:

$\bar{x}_{\text{new}} = \frac{20 \times 15 + 21}{21} = \frac{300 + 21}{21} = \frac{321}{21} = 15.2857... \approx 15.3$

The new observation (21) is above the original mean (15), so it increases the spread of the data. Since 21 is 2 standard deviations above the mean ( $21 = 15 + 2 \times 3$ ), it is not an extreme outlier but does add variability. The standard deviation will increase because the new value is further from the original mean than the typical deviation, pulling the spread wider.

Answer: New mean ≈ 15.3; standard deviation will increase.

Marking notes: M1 for new mean formula, A1 for correct new mean. M1 for stating SD increases with reasoning, A1 for valid justification.

Question 18 [5]

Combined mean:

$\bar{x} = \frac{30(68) + 20(75)}{50} = \frac{2040 + 1500}{50} = \frac{3540}{50} = 70.8$

Combined standard deviation:

For Group A: $\sum x_A = 30 \times 68 = 2040$ , $\sum x_A^2 = 30(5^2 + 68^2) = 30(25 + 4624) = 30 \times 4649 = 139470$

For Group B: $\sum x_B = 20 \times 75 = 1500$ , $\sum x_B^2 = 20(6^2 + 75^2) = 20(36 + 5625) = 20 \times 5661 = 113220$

$\sum x^2_{\text{total}} = 139470 + 113220 = 252690$

$\sigma = \sqrt{\frac{252690}{50} - (70.8)^2} = \sqrt{5053.8 - 5012.64} = \sqrt{41.16} \approx 6.42$

Answer: Combined mean = 70.8, Combined standard deviation ≈ 6.42 (3 s.f.)

Marking notes: M1 for combined mean formula, A1 for combined mean. M1 for computing $\sum x^2$ for each group using $\sigma^2 = \frac{\sum x^2}{n} - \bar{x}^2$ , M1 for combined variance formula, A1 for final answer.

Question 19 [7]

Midpoints: 1, 4, 7, 10, 13

(a) [1]

Answer: Modal class = 6–8 (highest frequency 22)

(b) [3]

$\bar{x} = \frac{1(8) + 4(15) + 7(22) + 10(10) + 13(5)}{60} = \frac{8 + 60 + 154 + 100 + 65}{60} = \frac{387}{60} = 6.45$

Answer: Mean = 6.45

(c) [3]

$\sum fx^2 = 1(8) + 16(15) + 49(22) + 100(10) + 169(5) = 8 + 240 + 1078 + 1000 + 845 = 3171$

$\sigma = \sqrt{\frac{3171}{60} - (6.45)^2} = \sqrt{52.85 - 41.6025} = \sqrt{11.2475} \approx 3.35$

Answer: Standard deviation ≈ 3.35 (3 s.f.)

Marking notes: (a) A1 for modal class. (b) M1 for midpoints, M1 for correct formula, A1 for answer. (c) M1 for $\sum fx^2$ , M1 for correct substitution, A1 for answer.

Question 20 [5]

(a) [2]

Sum of original 10 values = $10 \times 20 = 200$ .

Sum of remaining 8 values = $8 \times 19.5 = 156$ .

Sum of two removed values = $200 - 156 = 44$ .

Answer: 44

(b) [1]

Other removed value = $44 - 24 = 20$ .

Answer: 20

(c) [2]

The variance of the remaining 8 values is less than 16. The two removed values are 24 and 20, both of which are at or above the original mean of 20. Removing values that are at or above the mean (and in this case, 24 is one standard deviation above the mean) reduces the overall spread of the remaining data, so the variance decreases.

Answer: Less than 16, because the removed values include one that is above the mean, reducing the spread of the remaining data.

Marking notes: (a) M1 for finding total sum, A1 for answer. (b) A1 for answer. (c) M1 for correct comparison, A1 for valid justification.

End of Answer Key