Confidence Intervals

Question

 A random sample of 𝑛n measurements was selected from a population with standard deviation 𝜎=13.7 and unknown mean 𝜇. Calculate a 99% confidence interval for 𝜇 for each of the following situations: 

(a) n=55, x¯=100.4 

 ≤𝜇≤

(b)  n=75, x¯=100.4 

 ≤𝜇≤ 

(c)  n=105, x¯=100.4 

 ≤𝜇≤

Answer 1

I'll write Xbar for sample mean of the random variable X

First let me state a rule of thumb: for sample sizes ≥ 30, Xbar is well-approximated by a normal distribution. All the sample sizes are well above 30, so we'll use normal.

For a normally-distributed random variable, 99% of the values lie within 3 standard deviations of the mean.

For the random variable X, σ = 13.7. But we want the standard deviation of the sample mean. For a sample mean Xbar drawn from a sample of size n, σ_Xbar = σ/√n

(a) σ_Xbar ≈ 1.85 (this is 13.7/√55)

Confidence interval is (100.4 - 3*1.85, 100.4 + 3*1.85) = (94.85, 105.95)

(b) σ_Xbar ≈ 1.58

Confidence interval is (100.4 - 3*1.58, 100.4 + 3*1.58) = (95.66, 105.14)

(c) σ_Xbar ≈ 1.34

Confidence interval is (100.4 - 3*1.34, 100.4 + 3*1.34) = (96.38, 104.42)

This example shows you how the confidence interval narrows as sample size increases.