Sample z test and Sample t test

Question

A random sample X₁, X₂, …, X_n is from a normal population with mean µ and variance σ². If σ is unknown, give a 95% confidence interval of the population mean, and interpret it. Discuss the major difference between one sample z test and one sample t test.

Answer 1

Suppose X₁, X₂, …, X_n be a random sample from a normal population with mean µ and variance σ².

Assume that σ is unknown.

Define:

X‾: Sample mean
S: sample standard deviation
n: Sample Size

In this case the population variance is unknown, consequently we use t test for estimation of confidence interval.

A 95 % confidence interval for the population mean is given by:

here α = 0.05.

Note that if there population variance is known, we have to use normal distribution i.e. Z test instead of t distribution.