Tuesday, October 22

•Interval estimation.•Independent samples t-test

for the difference between two means.•Matched samples t-test

Tuesday, October 23

•Interval estimation.•Independent samples t-test

for the difference between two means.•Matched samples t-test

Interval Estimation (a.k.a. confidence interval)

Is there a range of possible values for that you can specify, onto which you can attach a statistical probability?

Interval Estimation (a.k.a. confidence interval)

Is there a range of possible values for that you can specify, onto which you can attach a statistical probability?

Confidence Interval

X - tsX X + tsX _ _

Where

t = critical value of t for df = N - 1, two-tailed

X = observed value of the sample _

Tuesday, October 23

•Interval estimation.•Independent samples t-test

for the difference between two means.•Matched samples t-test

Tuesday, October 22

•Interval estimation.•Independent samples t-test

for the difference between two means.•Matched samples t-test

H0 : 1 - 2 = 0

H1 : 1 - 2 0

1 2

30

40

50

60

70

80

SEX

RDG

1.0 1.5 2.0

30

40

50

60

70

80

SEX

RDG

Xboys=53.75_

Xgirls=51.16_

How do we know if the difference between these means,of 53.75 - 51.16 = 2.59, is reliably different from zero?

Xboys=53.75_

Xgirls=51.16_

95CI: 52.07 boys 55.43

95CI: 49.64 girls 52.68

We could find confidence intervals around each mean...

H0 : 1 - 2 = 0

H1 : 1 - 2 0

But we can directly test this hypothesis...

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

- -

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.

X1-X2 = 2X1 +2

X2

The sampling distribution of the difference between means.

This reflects the fact that two independent variancescontribute to the variance in the difference betweenthe means.

- - - -

X1-X2 = 2X1 +2

X2

The sampling distribution of the difference between means.

This reflects the fact that two independent variancescontribute to the variance in the difference betweenthe means.

- - - -

Your intuition should tell you that the variance in thedifferences between two means is larger than the variancein either of the means separately.

The sampling distribution of the difference between means,at n = , would be:

z =

(X1 - X2)

X1-X2

- -

- -

The sampling distribution of the difference between means.

Since we don’t know , we must estimate it with the sample statistic s.

X1-X2 = 21 2

2

n1 n2

+- -

The sampling distribution of the difference between means.

Rather than using s21 to estimate 2

1 and s22 to estimate 2

2 , we pool the twosample estimates to create a more stable estimate of 2

1 and 22 by assuming

that the variances in the two samples are equal, that is, 21 = 2

2 .

X1-X2 = 21 2

2

n1 n2

+- -

sX1-X2 =

sp2 sp

2

N1 N2

+

sX1-X2 =

sp2 sp

2

N1 N2

+

sX1-X2 =

sp2 sp

2

N1 N2

+

sp2 =

SSw SS1 + SS2

N-2 N-2=

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)

Assumptions

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

Get district data.