Question

Suppose we have a random sample X1, X2, …, Xn such that the Xi’s follow an unknown distribution with mean and variance ଶ = 25. Assuming the sample size n > 40, what is the value of n such that P(|ത − | < 1) ≅ 0.95?

Answer 1

μ_ȳ = μ

σ_ȳ = σ/√n

Here, we want to find the sample size n such that P(|ത − μ| < 1) ≅ 0.95. Using the CLT, we can write this as:

P(|(ȳ - μ)/(σ/√n)| < 1) ≅ 0.95

We can simplify this expression by multiplying both sides by σ/√n and rearranging:

P(|ȳ - μ| < σ/√n) ≅ 0.95

Now, we can use the fact that the standard deviation σ is known to substitute σ/√n with its value of 5, and we get:

P(|ȳ - μ| < 5/√n) ≅ 0.95

The absolute value of the difference |ȳ - μ| is equivalent to the distance between ȳ and μ, which is a measure of how far the sample mean is from the population mean. We want this distance to be less than 1, so we can rewrite the inequality as:

5/√n < 1

Solving for n, we get:

n > 25

Therefore, the minimum sample size required to ensure that P(|ത − μ| < 1) ≅ 0.95 is n = 26 (rounded up to the nearest integer).