What is the p-value and positive significance level?

Question

Test the claim that the mean GPA of night students is significantly different than 3.1 at the 0.1 significance level.

Based on a sample of 37 people, the sample mean GPA was 3.13 with a standard deviation of 0.14

The p-value is:

The positive significance level is: 

Answer 1

Hi Kaitlyn,

You do not know population standard deviation or shape, so we have to use a t-test. First, compute the test statistic:

t=(x-bar-mu)/SE

x-bar=sample mean

mu=population mean

SE=standard error

Recall that standard error is computed by:

SE=sigma/sqrt(n)

sigma=standard deviation

n=sample size

For our problem:

sigma=0.14

n=37

SE=0.14/sqrt(37)=0.023

Returning to our original formula:

t=(x-bar-mu)/SE

For our problem:

x-bar=3.13

mu=3.1

SE=0.023

So:

t=(3.13-3.1)/0.023

t=1.303

Now, let's talk about degrees of freedom (df). Sample size was n=37. df=n-1

df=37-1

df=36

Now, my t-table does not have a row for 36 degrees of freedom and we cannot exceed 36, so we have to go with 30.

Looking down the row for t=1.303, closest values are 1.055 and 1.310. Your question asked "significantly different," not greater than or less than, so we go with the two-sided p-values. Our bound is:

0.20<p<0.30

True p-value is likely on the low end of this, since t is closer to 1.310, but we can't be any more precise without statistical software. Regardless, these both exceed:

0.1, which is the significance level in your problem. You weren't directly asked this, but we can conclude that true mean GPA does not differ significantly from 3.1 for night students. Not sure why they asked for significance level and gave it to you already, but positive significance level is also known as alpha, which is 0.1. I hope this helps.