Question

[7, 5, a, 8, 1, a, 4, 6, b] have a mean of 5 and a variance of 3. Find the values of (a) and (b) such that (a, b in mathbb{Z}^+).

A) (a = 2, b = 3)
B) (a = 3, b = 2)
C) (a = 1, b = 4)
D) (a = 4, b = 1)

Answer 1

By setting up equations for the mean and variance of the array and solving, the only positive integer solutions for a and b that satisfy the given mean of 5 and variance of 3 are a = 2 and b = 3.

Step-by-step explanation:

To solve for the values of a and b in the array [7, 5, a, 8, 1, a, 4, 6, b] given that the mean is 5 and the variance is 3, we must use the formulas for mean and variance and solve the resulting equations.

To find a and b, we set up two equations based on the given mean and variance.

1. The sum of all numbers in the array divided by the count of numbers equals the mean: (7 + 5 + a + 8 + 1 + a + 4 + 6 + b) / 9 = 5.
2. The variance is calculated using the formula for the sample variance where we sum the squared differences between each value and the mean, then divide by the count minus one: [(7-5)^2 + (5-5)^2 + (a-5)^2 + (8-5)^2 + (1-5)^2 + (a-5)^2 + (4-5)^2 + (6-5)^2 + (b-5)^2] / 8 = 3.

Upon solving these equations, we find that the only possible positive integer solutions where both a and b are positive integers (mathbb{Z}^+) are a = 2 and b = 3.