Chapter 9, Problem 1CR

Statistical Literacy When testing $\mu$ how do we decide whether to use the standard normal distribution or a Student’s t distribution?

To determine

While testing $\mu$ how we decide whether to use the standard normal distribution or a Student’s t distribution.

Explanation of Solution

When we perform hypothesis test, we must first decide we will use a normal distribution or the Student’s t distribution. The conditions for each are,

If $\sigma$ is known and the original x distribution was normal, we use the normal distribution.

If $\sigma$ is known and $n\ge 30,$ we use the normal distribution, even if the original x distribution was not normal.

If we do not know $\sigma$, we use s as its estimate, and if the x distribution is roughly normal (mound-shaped), then we use Student’s t distribution with $d.f=(n-1)$.

If we do not know $\sigma$, we use s as its estimate, and if $n\ge 30,$ we use Student’s t distribution even if the original x distribution is not normal. We use $d.f=(n-1)$.