Confidence interval

Question

Let x1,x2...x15 be the values of the items of a random sample of size 15 from N(null,16).If xbar is the sample mean,write a 90% confidence interval for the unknown mean

Answer 1

x̄=(x1+x2+x3+...+x15)/15.

In the normal distribution (N-distribution) the mean divides the distribution into two symmetrical halves. Each half (50%) corresponds to data either above or below the mean. For a CI of 90%, the half is reduced to 45% above and below the mean. The total width is the interval 45+45=90%. So the critical p-value is 5% or 0.05. Z=(μ-x̄)/σ, where μ is the unknown mean and σ=16 (given standard deviation). The Z-score corresponding to the p-value of 0.05 is ±1.645. This means that the absolute difference between x̄ and the population mean should be no more than 1.645 standard deviations if it is to be within the required confidence interval. So |μ-x̄|<26.32 (=16×1.645) or x̄-26.32<μ<x̄+26.32.