tuesday, october 22
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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. - PowerPoint PPT PresentationTRANSCRIPT
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Tuesday, October 22
•Interval estimation.•Independent samples t-test
for the difference between two means.•Matched samples t-test
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Tuesday, October 23
•Interval estimation.•Independent samples t-test
for the difference between two means.•Matched samples t-test
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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?
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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?
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Confidence Interval
X - tsX X + tsX _ _
Where
t = critical value of t for df = N - 1, two-tailed
X = observed value of the sample _
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Tuesday, October 23
•Interval estimation.•Independent samples t-test
for the difference between two means.•Matched samples t-test
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Tuesday, October 22
•Interval estimation.•Independent samples t-test
for the difference between two means.•Matched samples t-test
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H0 : 1 - 2 = 0
H1 : 1 - 2 0
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1 2
30
40
50
60
70
80
SEX
RDG
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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?
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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...
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H0 : 1 - 2 = 0
H1 : 1 - 2 0
But we can directly test this hypothesis...
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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
- -
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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.
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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.
- - - -
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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.
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The sampling distribution of the difference between means,at n = , would be:
z =
(X1 - X2)
X1-X2
- -
- -
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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
+- -
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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
+- -
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sX1-X2 =
sp2 sp
2
N1 N2
+
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sX1-X2 =
sp2 sp
2
N1 N2
+
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sX1-X2 =
sp2 sp
2
N1 N2
+
sp2 =
SSw SS1 + SS2
N-2 N-2=
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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)
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Assumptions
•X1 and X2 are normally distributed.•Homogeneity of variance.•Samples are randomly drawn from their respective populations.•Samples are independent.
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Get district data.