Question

Complete parts ​(a) through ​(c) below.

​(a) Determine the critical​ value(s) for a​ right-tailed test of a population mean at the alpha = 0.10 level of significance with 20 degrees of freedom. ​
(b) Determine the critical​ value(s) for a​ left-tailed test of a population mean at the alpha = 0.05 level of significance based on a sample size of n=10. ​
(c) Determine the critical​ value(s) for a​ two-tailed test of a population mean at the alpha=0.01 level of significance based on a sample size of n=11.

Answer 1

a)
`df = n-1= 20-1=19`

For this case the value of
`\alpha=0.1` , since is a right tailed test we need a critical value on the t distribution with 19 degrees of freedom that accumulates 0.1 of the area on the right and we got:

`t_(cric)= 1.328`

b)
`df = n-1= 10-1=9`

For this case the value of
`\alpha=0.05` , since is a left tailed test we need a critical value on the t distribution with 9 degrees of freedom that accumulates 0.05 of the area on the right and we got:

`t_(cric)=-1.833`

c)
`df = n-1= 11-1=10`

For this case the value of
`\alpha=0.01` and
`\alpha/2=0.005` , since is a two tailed tailed test we need a critical value on the t distribution with 10 degrees of freedom that accumulates 0.005 of the area on both tails and we got:

`t_(cric)=\pm 3.169`

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

Part a

We can calculate the degrees of freedom first:

`df = n-1= 20-1=19`

For this case the value of
`\alpha=0.1` , since is a right tailed test we need a critical value on the t distribution with 19 degrees of freedom that accumulates 0.1 of the area on the right and we got:

`t_(cric)= 1.328`

Part b

We can calculate the degrees of freedom first:

`df = n-1= 10-1=9`

For this case the value of
`\alpha=0.05` , since is a left tailed test we need a critical value on the t distribution with 9 degrees of freedom that accumulates 0.05 of the area on the right and we got:

`t_(cric)=-1.833`

Part c

We can calculate the degrees of freedom first:

`df = n-1= 11-1=10`

For this case the value of
`\alpha=0.01` and
`\alpha/2=0.005` , since is a two tailed tailed test we need a critical value on the t distribution with 10 degrees of freedom that accumulates 0.005 of the area on both tails and we got:

`t_(cric)=\pm 3.169`