Question 1

The mean of 8 numbers is 24. If one number is removed, the mean of the remaining 7 numbers becomes 22. What is the number that was removed?

Accepted Answer

Step 1: Sum of 8 numbers = Mean × Count = 24 × 8 = 192. Step 2: Sum of remaining 7 numbers = 22 × 7 = 154. Step 3: Removed number = 192 - 154 = 38.

Question 2

The mean of five numbers is 18. Four of the numbers are 14, 16, 20, and 22. What is the fifth number?

Accepted Answer

Step 1: Sum of five numbers = Mean × Count = 18 × 5 = 90. Step 2: Sum of four known numbers = 14 + 16 + 20 + 22 = 72. Step 3: Fifth number = 90 - 72 = 18.

Question 3

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

Accepted Answer

Step 1: Sum of all marks = 12 + 18 + 15 + 12 + 23 = 80. Step 2: Number of students = 5. Step 3: Mean = Sum ÷ Number of observations = 80 ÷ 5 = 16.

Question 4

If each value of a data set is multiplied by 3, the variance of the new data set becomes how many times the original variance?

Accepted Answer

Step 1: Recall the effect of scaling on variance. If each observation is multiplied by a constant k, the new variance becomes k² times the original variance. Step 2: Here k = 3, so the new variance = 3² × original variance = 9 × original variance. Therefore, the variance becomes 9 times the original variance. (Note: Standard deviation would become 3 times, but variance becomes 9 times.)

Question 5

A teacher records the test scores of 6 students: 45, 52, 48, 52, 55, 48. What is the median score?

Accepted Answer

Step 1: Arrange data in ascending order: 45, 48, 48, 52, 52, 55. Step 2: Count observations: n = 6 (even). Step 3: For even n, median = (value at position n/2 + value at position (n/2)+1) ÷ 2 = (value at 3rd position + value at 4th position) ÷ 2 = (48 + 52) ÷ 2 = 100 ÷ 2 = 50.

Question 6

In how many ways can the letters of the word 'GARDEN' be arranged so that the vowels always occupy odd positions?

Accepted Answer

Step 1: Identify vowels and consonants in 'GARDEN'. Vowels: A, E (2 vowels). Consonants: G, R, D, N (4 consonants). Step 2: The word has 6 letters. Odd positions are 1, 3, 5 — that's 3 odd positions. We need to place 2 vowels in 3 odd positions. Ways = P(3,2) = 3×2 = 6. Step 3: The remaining 4 consonants occupy the remaining 4 positions. Ways = 4! = 24. Step 4: Total arrangements = 6 × 24 = 144. Therefore, answer is 144.

Question 7

The mean of 10 observations is 20 and the mean of their squares is 450. A new observation of value 30 is added to the dataset. What is the variance of the 11 observations?

Accepted Answer

Step 1: For original 10 observations, Mean = 20, Mean of squares = 450. Variance₁₀ = 450 − 20² = 450 − 400 = 50. Step 2: Sum of original 10 observations = 10 × 20 = 200. Sum of squares = 10 × 450 = 4500. Step 3: After adding 30, new sum = 200 + 30 = 230, new mean = 230/11. New sum of squares = 4500 + 30² = 4500 + 900 = 5400. Step 4: New variance = (5400/11) − (230/11)² = 5400/11 − 52900/121 = (5400 × 11 − 52900)/121 = (59400 − 52900)/121 = 6500/121 ≈ 53.72. Wait — let me use cleaner numbers. New variance = 5400/11 − (230/11)² = (5400 × 121 − 230² × 11)/(11² × 11). Using direct formula: Var = [ΣX²/n − (ΣX/n)²] = 5400/11 − (230/11)² = 490.909 − 437.190 = 53.719. Rounding to clean form = 6500/121. Therefore, variance = 6500/121.

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