Question

find the t values that form the boundaries of the critical region for a two-tailed test with a=.05 for n=6 n=12

Answer 1

n=6

"=T.INV(1-0.025,5)", and we got the critical values given by:
`t_(critical)=\pm 2.5706`

n=12

"=T.INV(1-0.025,11)", and we got the critical values given by:
`t_(critical)=\pm 2.201`

Explanation:

Previous concepts

The t distribution (Student’s t-distribution) is a "probability distribution that is used to estimate population parameters when the sample size is small (n<30) or when the population variance is unknown".

The shape of the t distribution is determined by its degrees of freedom and when the degrees of freedom increase the t distirbution becomes a normal distribution approximately.

The degrees of freedom represent "the number of independent observations in a set of data. For example if we estimate a mean score from a single sample, the number of independent observations would be equal to the sample size minus one."

Solution to the problem

For n=6

In order to find the critical value we need to take in count that we are conducting a two tailed test, so we are looking on the t distribution with df=n-1=6-1=5 degrees of freedom, a value that accumulates
`0.05/2=0.025` of the area on each tail. We can use excel or a table to find it, for example the code in Excel is:

"=T.INV(1-0.025,5)", and we got the critical values given by:
`t_(critical)=\pm 2.5706`

For n=12

In order to find the critical value we need to take in count that we are conducting a two tailed test, so we are looking on the t distribution with df=n-1=12-1=11 degrees of freedom, a value that accumulates
`0.05/2=0.025` of the area on each tail. We can use excel or a table to find it, for example the code in Excel is:

"=T.INV(1-0.025,11)", and we got the critical values given by:
`t_(critical)=\pm 2.201`