Question 1

In how many ways can 6 students be divided into 2 groups of 3 each?

Accepted Answer

Step 1: We need to divide 6 students into 2 indistinguishable groups of 3 each. Step 2: If groups were distinguishable, we'd use C(6,3) = 20. Step 3: Since the 2 groups are indistinguishable (Group 1 and Group 2 are not labeled), we divide by 2!. Step 4: Answer = C(6,3) / 2! = 20 / 2 = 10.

Question 2

How many ways can 4 red balls and 3 blue balls be arranged in a line?

Accepted Answer

Step 1: We have 7 balls total: 4 identical red balls and 3 identical blue balls. Step 2: Use the formula for permutations with repetition: n! / (n₁! × n₂!), where n = total objects, n₁ = red balls, n₂ = blue balls. Step 3: Arrangements = 7! / (4! × 3!) = 5040 / (24 × 6) = 5040 / 144 = 35.

Question 3

The mode of the dataset {3, 5, 7, 5, 9, 5, 11, 7, 5} is:

Accepted Answer

Step 1: Count the frequency of each number: 3 appears 1 time, 5 appears 4 times, 7 appears 2 times, 9 appears 1 time, 11 appears 1 time. Step 2: The mode is the value with the highest frequency. Step 3: 5 appears 4 times, which is the maximum frequency. Therefore, the mode is 5.

Question 4

The variance of {1, 2, 3} is 2/3. What is the standard deviation (rounded to 2 decimal places)?

Accepted Answer

Step 1: Given variance = 2/3 ≈ 0.667. Step 2: Standard Deviation = √Variance = √(2/3) = √0.667 ≈ 0.816 ≈ 0.82 (rounded to 2 decimal places).

Question 5

If every value in a dataset is increased by 5, how does the standard deviation change?

Accepted Answer

Step 1: Adding a constant to all values shifts the entire dataset but does not change the spread. Step 2: Standard deviation measures spread around the mean. Step 3: When all values increase by 5, the mean also increases by 5, so all deviations from the mean remain unchanged. Step 4: Therefore, the standard deviation remains the same.

Question 6

Two datasets have the same mean of 10. Dataset A has a standard deviation of 2, and Dataset B has a standard deviation of 4. Which statement is true?

Accepted Answer

Step 1: Both datasets have mean = 10. Step 2: Dataset A has SD = 2, so Variance_A = 2² = 4. Step 3: Dataset B has SD = 4, so Variance_B = 4² = 16. Step 4: Since Variance_B > Variance_A, Dataset B has greater spread/dispersion around the mean. Step 5: Dataset B is more dispersed than Dataset A.

Question 7

The standard deviation of the dataset {3, 3, 3, 3, 3} is:

Accepted Answer

Step 1: All values are identical (3). Step 2: Mean = 3. Step 3: All deviations from mean = 0. Step 4: Sum of squared deviations = 0. Step 5: Variance = 0 ÷ 5 = 0. Step 6: Standard Deviation = √0 = 0.

Question 8

If the standard deviation of a dataset is 5, what is the variance?

Accepted Answer

Step 1: Recall the relationship between standard deviation and variance: Variance = (Standard Deviation)². Step 2: Variance = 5² = 25.

Question 9

The variance of the dataset {2, 4, 6, 8, 10} is:

Accepted Answer

Step 1: Find the mean = (2 + 4 + 6 + 8 + 10) ÷ 5 = 30 ÷ 5 = 6. Step 2: Find squared deviations from mean: (2−6)² = 16, (4−6)² = 4, (6−6)² = 0, (8−6)² = 4, (10−6)² = 16. Step 3: Sum of squared deviations = 16 + 4 + 0 + 4 + 16 = 40. Step 4: Variance = 40 ÷ 5 = 8.

Question 10

A number is selected at random from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. What is the probability that the number is either even or a multiple of 3?

Accepted Answer

Step 1: Total numbers = 10. Step 2: Even numbers = {2, 4, 6, 8, 10} = 5 numbers. Step 3: Multiples of 3 = {3, 6, 9} = 3 numbers. Step 4: Numbers that are both even AND multiples of 3 = {6} = 1 number. Step 5: Using inclusion-exclusion: Even OR Multiple of 3 = 5 + 3 − 1 = 7 numbers. Step 6: Probability = 7/10. Therefore, the answer is 7/10.

Question 11

The median of the dataset {7, 12, 5, 18, 9, 14, 11} is:

Accepted Answer

Step 1: Arrange the numbers in ascending order: {5, 7, 9, 11, 12, 14, 18}. Step 2: Count the numbers: 7 numbers (odd count). Step 3: Median is the middle value at position (7+1)/2 = 4. Step 4: The 4th number in the ordered list is 11. Therefore, the median is 11.

Question 12

Five students scored 45, 52, 48, 55, and 50 marks. If a sixth student joins and the new mean becomes 51, what are the marks of the sixth student?

Accepted Answer

Step 1: Sum of first 5 students' marks = 45 + 52 + 48 + 55 + 50 = 250. Step 2: New mean with 6 students = 51. Step 3: Total sum with 6 students = 51 × 6 = 306. Step 4: Marks of sixth student = 306 − 250 = 56. Therefore, the sixth student scored 56 marks.

Question 13

The mean of a dataset is 25 and the median is 23. If the dataset has 9 numbers and the sum of the first 4 numbers is 88, what is the sum of the last 4 numbers?

Accepted Answer

Step 1: Mean of 9 numbers = 25, so total sum = 25 × 9 = 225. Step 2: Sum of first 4 numbers = 88 (given). Step 3: The 5th number is the median = 23 (since with 9 numbers, the median is at position 5). Step 4: Sum of last 4 numbers = Total sum − Sum of first 4 − Median = 225 − 88 − 23 = 114. Therefore, the sum of the last 4 numbers is 114.

Question 14

The mean of 5 numbers is 24. If one number is replaced by 36, the new mean becomes 28. What was the original number that was replaced?

Accepted Answer

Step 1: Original sum of 5 numbers = Mean × Count = 24 × 5 = 120. Step 2: New sum after replacement = New Mean × Count = 28 × 5 = 140. Step 3: Difference = 140 − 120 = 20. Step 4: Original number = Replaced number − Difference = 36 − 20 = 16. Therefore, the original number is 16.

Question 15

In how many ways can 5 different books be arranged on a shelf?

Accepted Answer

Step 1: We need to arrange 5 different books in a line. Step 2: The number of arrangements of n distinct objects is n!. Step 3: Therefore, arrangements = 5! = 5 × 4 × 3 × 2 × 1 = 120.

Question 16

How many ways can a committee of 3 people be selected from a group of 8 people?

Accepted Answer

Step 1: We need to select 3 people from 8, where order does not matter (committee selection). Step 2: Use combination formula: C(n,r) = n! / [r!(n-r)!]. Step 3: C(8,3) = 8! / [3! × 5!] = (8 × 7 × 6) / (3 × 2 × 1) = 336 / 6 = 56.

Question 17

In how many ways can the letters of the word 'APPLE' be arranged?

Accepted Answer

Step 1: The word 'APPLE' has 5 letters with 'P' repeated twice. Step 2: Formula for arrangements with repetition: n! / (n₁! × n₂! × ... × nₖ!), where n₁, n₂, etc. are frequencies of repeated letters. Step 3: Number of arrangements = 5! / 2! = 120 / 2 = 60.

Question 18

A dataset has the following values: 5, 8, 8, 12, 15, 8, 20. What is the difference between the mode and the median of this dataset?

Accepted Answer

Step 1: Arrange in order: 5, 8, 8, 8, 12, 15, 20. Step 2: Mode (most frequent) = 8 (appears 3 times). Step 3: Median (middle value of 7 numbers) = 4th value = 8. Step 4: Difference = 8 − 8 = 0.

Question 19

A dataset has mean 50 and standard deviation 10. The sum of squared deviations from the mean is 4000. How many observations are in the dataset?

Accepted Answer

Step 1: Variance = (Sum of squared deviations) / n. Step 2: SD² = Variance, so 10² = 100 = 4000 / n. Step 3: n = 4000 / 100 = 40.

Question 20

Two datasets have the same mean of 60. Dataset A has variance 16, and Dataset B has variance 64. If both datasets have 25 observations, what is the ratio of the standard deviation of A to the standard deviation of B?

Accepted Answer

Step 1: SD of A = √16 = 4. Step 2: SD of B = √64 = 8. Step 3: Ratio = 4 : 8 = 1 : 2.

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