Question 1

If each value in a dataset is multiplied by 2, how does the variance change?

Accepted Answer

Step 1: Recall the property: if all data points are multiplied by a constant k, the variance is multiplied by k². Step 2: If multiplied by 2, variance is multiplied by 2² = 4. Step 3: New variance = 4 × (Original variance).

Question 2

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

Accepted Answer

Step 1: The word 'BOOK' has 4 letters with O repeated twice. Step 2: When objects are repeated, we use: n!/(n₁! × n₂! × ...). Step 3: Number of arrangements = 4!/2! = (4 × 3 × 2 × 1)/2 = 24/2 = 12.

Question 3

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

Accepted Answer

Step 1: Recall that Variance = (Standard Deviation)². Step 2: Variance = 5² = 25.

Question 4

The standard deviation of the dataset {5, 5, 5, 5, 5} is:

Accepted Answer

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

Question 5

In 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: This is a permutation with repetition problem. Step 3: Number of arrangements = 7!/(4! × 3!) = (7 × 6 × 5 × 4!)/(4! × 3 × 2 × 1) = (7 × 6 × 5)/6 = 35.

Question 6

How many ways can a committee of 3 members 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). Step 2: This is a combination problem: C(8,3) = 8!/(3! × 5!). Step 3: C(8,3) = (8 × 7 × 6)/(3 × 2 × 1) = 336/6 = 56.

Question 7

How many 2-digit numbers can be formed using the digits 3, 5, 7, and 9 without repetition?

Accepted Answer

Step 1: We need to form 2-digit numbers using 4 different digits without repetition. Step 2: For the first position, we have 4 choices. For the second position, we have 3 remaining choices. Step 3: Total = 4 × 3 = 12. Alternatively, P(4,2) = 4!/(4-2)! = 4!/2! = 12.

Question 8

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 9

A student must answer 5 questions out of 8 questions in an exam. In how many ways can the student choose the questions?

Accepted Answer

Step 1: The student must select 5 questions from 8 available questions. Step 2: The order of selection does not matter, so this is a combination problem. Step 3: C(8,5) = 8!/(5! × 3!) = (8 × 7 × 6)/(3 × 2 × 1) = 336/6 = 56.

Question 10

The mean of five numbers is 12 and their variance is 4. If each number is increased by 3, what will be the new variance?

Accepted Answer

Step 1: Recall that adding a constant to all data points does NOT change the variance (variance measures spread, not location). Step 2: New variance = Original variance = 4.

Question 11

A coin is tossed 3 times. What is the probability of getting exactly 2 heads?

Accepted Answer

Step 1: Total possible outcomes when tossing a coin 3 times = 2³ = 8. Step 2: Favorable outcomes with exactly 2 heads are: HHT, HTH, THH = 3 outcomes. Step 3: Probability = 3/8.

Question 12

The median of the dataset {12, 15, 18, 21, 24} is:

Accepted Answer

Step 1: Arrange the data in ascending order: {12, 15, 18, 21, 24}. Step 2: The dataset has 5 values (odd count). Step 3: The median is the middle value at position (5+1)/2 = 3. Step 4: The third value is 18.

Question 13

In the dataset {3, 7, 7, 9, 11, 7, 13, 15}, what is the difference between the mode and the median?

Accepted Answer

Step 1: Find the mode by counting frequencies: 3(1×), 7(3×), 9(1×), 11(1×), 13(1×), 15(1×). Mode = 7. Step 2: Arrange in order (already sorted): {3, 7, 7, 9, 11, 7, 13, 15}. Step 3: For 8 values (even), median = (4th + 5th values)/2 = (9 + 11)/2 = 10. Step 4: Difference = 10 - 7 = 3.

Question 14

The median of {10, 14, 18, 22, 26, 30} is:

Accepted Answer

Step 1: The dataset has 6 values (even count), already in ascending order. Step 2: For even count, median = (value at position n/2 + value at position n/2+1) / 2. Step 3: Median = (value at position 3 + value at position 4) / 2 = (18 + 22) / 2 = 40 / 2 = 20.

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: This is a permutation problem where all 5 objects are used. Step 3: Number of arrangements = 5! = 5 × 4 × 3 × 2 × 1 = 120.

Question 16

In a dataset {5, 8, 8, 12, 15, 8, 20}, the mode is:

Accepted Answer

Step 1: Count the frequency of each value. Step 2: 5 appears 1 time, 8 appears 3 times, 12 appears 1 time, 15 appears 1 time, 20 appears 1 time. Step 3: The mode is the value with the highest frequency. Step 4: 8 appears most frequently (3 times), so the mode is 8.

Question 17

Seven students scored: 45, 52, 48, 52, 55, 52, 60. What is the mean score?

Accepted Answer

Step 1: Add all scores: 45 + 52 + 48 + 52 + 55 + 52 + 60 = 364. Step 2: Divide by the number of students: 364 ÷ 7 = 52. Step 3: The mean score is 52.

Question 18

A committee of 4 members is to be selected from 6 men and 5 women. In how many ways can this be done if the committee must have at least 2 women?

Accepted Answer

Step 1: 'At least 2 women' means 2 women or 3 women or 4 women. Step 2: Case 1 (2 women, 2 men): C(5,2) × C(6,2) = 10 × 15 = 150. Step 3: Case 2 (3 women, 1 man): C(5,3) × C(6,1) = 10 × 6 = 60. Step 4: Case 3 (4 women, 0 men): C(5,4) × C(6,0) = 5 × 1 = 5. Step 5: Total = 150 + 60 + 5 = 215.

Question 19

How many 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6 without repetition, such that the number is divisible by 5?

Accepted Answer

Step 1: For divisibility by 5, the last digit must be 0 or 5. Since only 5 is available, the last digit must be 5. Step 2: The first three positions must be filled with 3 digits chosen from {1, 2, 3, 4, 6} without repetition. Step 3: Number of ways = P(5,3) = 5 × 4 × 3 = 60.

Question 20

The standard deviation of five numbers is 4. If the sum of the numbers is 50, what is the sum of their squares?

Accepted Answer

Step 1: Use the formula: Variance = (Sum of squares / n) - (Mean)². Step 2: Mean = 50/5 = 10. Step 3: Standard deviation = 4, so Variance = 4² = 16. Step 4: 16 = (Sum of squares / 5) - 10². Step 5: 16 = (Sum of squares / 5) - 100. Step 6: Sum of squares / 5 = 116. Step 7: Sum of squares = 580. Therefore, the answer is 580.

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