Question

Using R code solve

Here, we look at how t critical values behave as their df (degrees of freedom) increases:

a. First, what is z.05?

b. Second, if you look at t.05,df (t critical values for α = .05) with df = 20, 40, 60, etc (continuing up by 20 each time), for what df does the t critical value first fall strictly within (e.g. < ) i. .05 of z.05? ii. .02 of z.05? iii. .01 of z.05? c. What do you think the difference will be between z.05 and t.05,df as df → ∞?

Answer 1

a. First, what is z.05?

Z_0.05 = -1.644854

(by using R command > qnorm(0.05))

b. Second, if you look at t.05,df (t critical values for α = .05) with df = 20, 40, 60, etc (continuing up by 20 each time), for what df does the t critical value first fall strictly within (e.g. < ) i. .05 of z.05? ii. .02 of z.05? iii. .01 of z.05?

The t critical values by using R for different sample sizes are given as below:

> qnorm(0.05)

[1] -1.644854

> qt(0.05,20)

[1] -1.724718

> qt(0.05,40)

[1] -1.683851

> qt(0.05,60)

[1] -1.670649

> qt(0.05,80)

[1] -1.664125

> qt(0.05,100)

[1] -1.660234

> qt(0.05,200)

[1] -1.652508

> qt(0.05,300)

[1] -1.649949

> qt(0.05,500)

[1] -1.647907

> qt(0.05,1000)

[1] -1.646379

> qt(0.05,10000)

[1] -1.645006

> qt(0.05,100000)

[1] -1.644869

> qt(0.05,1000000)

[1] -1.644855

>

> qnorm(0.05)

[1] -1.644854

> qnorm(0.02)

[1] -2.053749

> qnorm(0.01)

[1] -2.326348

> qt(0.05,10000000)

[1] -1.644854

> qt(0.02,10000000)

[1] -2.053749

> qt(0.01,10000000)

[1] -2.326348

c. What do you think the difference will be between z.05 and t.05,df as df → ∞?

It is observed that as as df → ∞, the t critical value and z critical values become equal.