1. Independent samples t-test for the difference between two means.
Monday, October 29
Independent samples t-test
for the difference between two means.

2. H0 : 1 - 2 = 0
H1 : 1 - 2  0

4. Xboys=53.75 _
Xgirls=51.16 _
How do we know if the difference between these means,
of = 2.59, is reliably different from zero?

5. We could find confidence intervals around each mean...
Xboys=53.75 _
95CI:   boys 
Xgirls=51.16 _
95CI:   girls 

6. H0 : 1 - 2 = 0
H1 : 1 - 2  0
But we can directly test this hypothesis...

7. H0 : 1 - 2 = 0
H1 : 1 - 2  0
To test this hypothesis, you need to know …
…the sampling distribution of the difference between means. - -
X1-X2

8. H0 : 1 - 2 = 0
H1 : 1 - 2  0
To test this hypothesis, you need to know …
…the sampling distribution of the difference between means. - -
X1-X2
…which can be used as the error term
in the test statistic.

9. The sampling distribution of the difference between means.
X1-X2 =   2X1 +2X2 - - - -
This reflects the fact that two independent variances
contribute to the variance in the difference between
the means.

10. The sampling distribution of the difference between means.
X1-X2 =   2X1 +2X2 - - - -
This reflects the fact that two independent variances
contribute to the variance in the difference between
the means.
Your intuition should tell you that the variance in the
differences between two means is larger than the variance
in either of the means separately.

11. The sampling distribution of the difference between means,
at n = , would be: - -
z =
(X1 - X2)
X1-X2 - -

12. The sampling distribution of the difference between means.- -
X1-X2 =    22 +
n n2
Since we don’t know , we must estimate it with the sample statistic s.

13. The sampling distribution of the difference between means.- -
X1-X2 =    22 +
n n2
Rather than using s21 to estimate 21 and s22 to estimate 22 , we pool the two
sample estimates to create a more stable estimate of 21 and 22 by assuming
that the variances in the two samples are equal, that is, 21 = 22 .

14. sp sp2
sX1-X2 = +
N N2

15. sp sp2
sX1-X2 = +
N N2

16. sp sp2
sX1-X2 = +
N N2
sp2 =
SSw SS1 + SS2
N N-2 =

17. Because we are making estimates that vary by degrees of freedom,
we use the t-distribution to test the hypothesis.
t =
(X1 - X2) - (1 - 2 )
sX1-X2
…at (n1 - 1) + (n2 - 1) degrees of freedom
(or N-2)

18. Assumptions
X1 and X2 are normally distributed.
Homogeneity of variance.
Samples are randomly drawn from their respective populations.
Samples are independent.