Question 1

If the standard deviation of a dataset is 6, what is the standard deviation when each value is increased by 10?

Accepted Answer

Step 1: Adding a constant to all values shifts the mean but does not change the spread of data. Step 2: Standard Deviation measures spread, not location. Step 3: Therefore, SD remains 6.

Question 2

The median of the dataset {14, 22, 18, 26, 30, 16, 20} is:

Accepted Answer

Step 1: Arrange the data in ascending order: 14, 16, 18, 20, 22, 26, 30. Step 2: Count the values: there are 7 values (odd count). Step 3: The median is at position (7+1)/2 = 4. Step 4: The 4th value is 20. Therefore, the median is 20.

Question 3

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

Accepted Answer

Step 1: Calculate mean = (2 + 4 + 6 + 8 + 10) ÷ 5 = 30 ÷ 5 = 6. Step 2: Find squared deviations: (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 4

The variance of {1, 2, 3, 4, 5} is 2. What is the variance of {2, 4, 6, 8, 10}?

Accepted Answer

Step 1: The second dataset is exactly 2 times the first dataset. Step 2: When all values are multiplied by a constant k, variance is multiplied by k². Step 3: New Variance = 2 × 2² = 2 × 4 = 8.

Question 5

The ages of 6 employees are: 24, 28, 24, 32, 24, and 30 years. What is the mode of their ages?

Accepted Answer

Step 1: List all values: 24, 28, 24, 32, 24, 30. Step 2: Count the frequency of each age. The age 24 appears 3 times; 28, 32, and 30 each appear once. Step 3: Mode is the value with the highest frequency. Therefore, the mode is 24 years.

Question 6

A coin is tossed three times. What is the probability of getting at least one head?

Accepted Answer

Step 1: Total possible outcomes when tossing a coin 3 times = 2³ = 8. Step 2: Outcomes with at least one head = Total outcomes - Outcomes with no heads (all tails). Step 3: Outcomes with all tails = 1 (TTT). Step 4: Outcomes with at least one head = 8 - 1 = 7. Step 5: Probability = 7/8. Therefore, the answer is 7/8.

Question 7

The mean of four numbers is 20. If three of the numbers are 16, 18, and 22, what is the fourth number?

Accepted Answer

Step 1: Use the formula: Mean = (Sum of all values) / (Number of values). Step 2: Given Mean = 20 and count = 4, so Sum = 20 × 4 = 80. Step 3: Sum of three known numbers = 16 + 18 + 22 = 56. Step 4: Fourth number = 80 - 56 = 24. Therefore, the fourth number is 24.

Question 8

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 9

The mean of five numbers is 12. If the sum of squared deviations from the mean is 80, what is the standard deviation?

Accepted Answer

Step 1: Variance = Sum of squared deviations ÷ Number of observations = 80 ÷ 5 = 16. Step 2: Standard Deviation = √Variance = √16 = 4.

Question 10

For the dataset {3, 3, 3, 3, 3}, the standard deviation is:

Accepted Answer

Step 1: All values are identical, so mean = 3. Step 2: Each deviation from mean = 3 - 3 = 0. Step 3: Sum of squared deviations = 0. Step 4: Variance = 0 ÷ 5 = 0. Step 5: Standard Deviation = √0 = 0.

Question 11

The marks obtained by 5 students in a test are: 12, 18, 12, 15, and 23. What is the mode of their marks?

Accepted Answer

Step 1: Arrange the data: 12, 12, 15, 18, 23. Step 2: Identify the frequency of each value. The value 12 appears twice, while 15, 18, and 23 each appear once. Step 3: Mode is the value with the highest frequency. Therefore, the mode is 12.

Question 12

The median of the dataset {7, 9, 5, 11, 8} is:

Accepted Answer

Step 1: Arrange the data in ascending order: 5, 7, 8, 9, 11. Step 2: Since there are 5 values (odd count), the median is the middle value at position (5+1)/2 = 3. Step 3: The 3rd value is 8. Therefore, the median is 8.

Question 13

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, 5! = 5 × 4 × 3 × 2 × 1 = 120.

Question 14

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. 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 15

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: Use formula for permutations with repetition: n! / (n₁! × n₂! × ...). Step 3: Number of arrangements = 4! / 2! = 24 / 2 = 12.

Question 16

How many 2-digit numbers can be formed using the digits 3, 4, 5, 6 without repetition?

Accepted Answer

Step 1: We need to form 2-digit numbers from 4 available 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 or P(4,2) = 4!/(4-2)! = 4!/2! = 12.

Question 17

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

Accepted Answer

Step 1: Total balls = 4 + 3 = 7. Step 2: Red balls are identical to each other, and blue balls are identical to each other. Step 3: Use formula: 7! / (4! × 3!) = 5040 / (24 × 6) = 5040 / 144 = 35.

Question 18

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: Order of selection does not matter, so we use combination. Step 3: C(8,5) = 8! / (5! × 3!) = (8 × 7 × 6) / (3 × 2 × 1) = 336 / 6 = 56.

Question 19

A dataset has mode 15, and the frequency of 15 is 6. If all other values appear exactly 3 times each, and there are 4 other distinct values, how many total observations are in the dataset?

Accepted Answer

Step 1: Mode 15 appears 6 times. Step 2: There are 4 other distinct values, each appearing 3 times. Step 3: Total observations = 6 + (4 × 3) = 6 + 12 = 18.

Question 20

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

Accepted Answer

Step 1: Count total letters: M-I-S-S-I-S-S-I-P-P-I = 11 letters. Step 2: Identify frequencies: M appears 1 time, I appears 4 times, S appears 4 times, P appears 2 times. Step 3: Use the formula for permutations with repetition: n! / (n₁! × n₂! × ... × nₖ!) = 11! / (1! × 4! × 4! × 2!). Step 4: Calculate: 11! = 39,916,800; 4! = 24; 2! = 2. Step 5: Answer = 39,916,800 / (1 × 24 × 24 × 2) = 39,916,800 / 1,152 = 34,650.

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