What is the test statistic and p-value?

Question

The mean number of English courses taken in a two–year time period by American and international college students is believed to be different. An experiment is conducted and data are collected and summarized below.

Test the claim using a 10% level of significance. Assume the population variances are unequal and that number of English courses taken is normally distributed.

American	Students (�1)International	Students (�2)
3	4
3	2
0	2
4	5
0	5
3	3
0	1
3	2
1	4
4	2
0	3
3	1
3	5
0	2
1	1
2	2
2	5
1	4
2	3
4	4
1	2
0	4
1	4
0	5
2	2
3	1
0	2
2

Calculate your test statistic. Round to 2 decimal places. 

Calculate your p-value. Round to 4 decimal places.

Answer 1

Hi Kaitlyn,

Let me warn you--the unequal variances t-test is complicated, not due to the test statistic, but due to the degrees of freedom. Formula for test statistic is:

t=(x-bar₁ - x-bar₂)/sqrt[(s₁²/n₁) + (s₂²/n₂)

x-bar1=first sample mean

x-bar2=second sample mean

n1=first sample size

n2=second sample size

s1=first sample standard deviation

s2=second sample standard deviation

For our problem, we need either statistical software, a graphics calculator, or an online calculator to compute the sample means and standard deviations. From online calculator:

x-bar₁=1.714

s₁=1.410

n₁=28

x-bar₂=2.963

s₂=1.400

n₂=27

Computing from formula:

t=(1.714 - 2.963)/sqrt[(1.410²/28) + (1.400²/27)]

t= -3.30

That's our t-statistic, but we need degrees of freedom for p-value. Get ready. Formula is:

df=[(1/n1) + (u/n2)]²/[(1/n1²(n1-1)) + u²/(n₂²(n₂-1))]

u= s₂²/s₁²

All other variables are the same as above.

First, compute u:

u=(1.410)²/(1.400)²

u=1.015

df=[(1/n1) + (u/n2)]²/[(1/n1²(n1-1)) + u²/(n₂²(n₂-1))]

df=[(1/28) + (1.015/27)]²/[(1/(28²(28-1)) + 1.015²/27²(27-1)]

Sorry, but I don't have a calculator on me at the moment that can do this. So, get that value, then round it down to the nearest whole number. Then, you can go to the t-table, get the degrees of freedom, and search for the absolute value of the t-statistic -3.30. Whichever alpha levels it falls between--ex. 0.01 < p < 0.02--are the bounds for your p-value. Only statistical software gives an exact p-value. I hope this helps.