1. Tests of inference about 2 population means
Outline
Tests of inference about 2 population means
Independent vs. dependent groups
Large sample, independent groups
Z test
Small sample, independent groups
Test of equality of population variances
If variances are equal, t-test
Independent Samples

2. Tests of Inference about 2 population means
Sometimes, we want to know how 2 population means compare with each other.
To begin this procedure we ask 2 questions:
Are the samples large or small? (n < 30  small)
Are the samples independent of each other?
Independent Samples

3. Independent vs. dependent samples
Sometimes, observations in two samples are tied together in some way:
E.g., same subjects tested twice, or
subjects matched on some variable (IQ, age, education…)
In that case, we have dependent pairs – our topic two weeks from today.
When observations are not tied in any way, we have independent pairs.
Independent Samples

4. Four tests we’ll use this month
1a. Large samples, independent groups
1b. Small samples, independent groups
2a. Large samples, dependent pairs
2b. Small samples, dependent pairs
Independent Samples

5. 1a. Large samples, independent groups
We want to answer a question about the difference between two population means, 1 – 2.
As usual, we do this by
taking samples and measuring their means
comparing those sample means
Independent Samples

6. 3. Make inference back to population
Inferred
Population
3. Make inference back to population
2. Measure sample
Known
Sample
1. Draw sample
The logic of the single sample test
Independent Samples

7. Population 1 μ1
Population 2 μ2
μ1 – μ2
This time, we’re interested in the difference between two population means: μ1 – μ2.

8. The difference between two population means: μ1 – μ2:
We learn about this population difference by testing the difference between two sample means: X1 – X2
Sample 1
Sample 2
X1 – X2
Independent Samples

9. Logic of the two sample test
Inferred
Population 1
Population 2
μ1 – μ2
Sample 1
Sample 2
X1 – X2
Known
Logic of the two sample test

10. 1a. Large samples, independent groups
Our null hypothesis is typically that there is no difference between the two population means.
Sometimes we’ll hypothesize that a historical, non-zero difference still holds.
Either way, we use the sampling distribution of the difference X1 X2
( )
Independent Samples

11. 1a. Large sample, independent groupsX1 X2
( )
The sampling distribution of is approximately normal for large n, by C.L.T.
This result is directly analogous to result for tests of hypothesis respecting a single sample mean.
Independent Samples

12. 1a. Large samples, independent groups
The mean of the sampling distribution of the difference is
Because the samples are independent,
(X1–X2) = 12 22
n n2 X1 X2
( )
(1 – 2)
Independent Samples

13. The C.I. for the difference between two population means is:
Confidence Interval:
The C.I. for the difference between two population means is:
± Zα/2 (x1 – x2)
= ± Zα/2 12 22
n n2 X1 X2
( ) X1 X2
( )
Independent Samples

14. Z test
H0: 1 – 2 = D0 H0: 1 – 2 = D0
HA: 1 – 2 > D0 HA: 1 – 2 ≠ D0
or: 1 – 2 < D0
Test statistic: Z = – D0 X1 X2
( )
D0 is the historical value of the difference between population means, typically but not always 0.
12
22 n1 n2
Independent Samples

15. One-tailed: Two-tailed: Z > Zα │Z│ > Zα/2 or Z < -Zα
Z test
Rejection region:
One-tailed: Two-tailed:
Z > Zα │Z│ > Zα/2
or Z < -Zα
Independent Samples

16. 1b. Small samples, independent groups
We now turn to the case of comparing means for two independent, small samples (ns < 30).
There are 2 ways to do this – depending upon whether the two population variances are equal or different.
In order to know which method we should use, we have to test the hypothesis H0: 12 = 22
So for small, independent samples, there are always 2 steps– test the variances, then test the means.
Independent Samples

17. 1b. Small samples, independent groups
VERY IMPORTANT POINT:
We can only use the independent groups t-test when the two population variances are equal.
We must not assume that 12 = 22.
We must test H0: 12 = 22.
The test of hypothesis about the two population variances uses the ratio F= (12 / 22).
Independent Samples

18. 1b. Small samples, independent groups
On an exam, you must test the hypothesis of equal variances before doing the independent groups t-test!
If H0: 12 = 22 is rejected, we use the Wilcoxon Rank Sum test instead of the t-test (our subject next week).
Note: before t-test only; not before Z test.
Independent Samples

19. Test of hypothesis of equal variances
Notes:
Next slide shows a formal statement of test of hypothesis about two population variances.
Both one-tailed and two-tailed tests are shown.
When you test equality of variances before doing small sample, independent groups t-test, always do a two-tailed test.
One-tailed test of equality of variances has other uses.
Independent Samples

20. Test of hypothesis of equal variances
H0: 12 = 22 H0: 12 = 22
HA: 12 < 22 HA: 12 ≠ 22
or 12 > 22
Test statistic: F = S12
S22
Independent Samples

21. Test of hypothesis of equal variances
Rejection region for the one tailed test:
Fobt > F ((n1 – 1), (n2 – 1,)
Fcritical is obtained from the table, with numerator d.f., denominator d.f., and 
Note: For the one tailed-test, you can put either variance in the numerator, but if you put the smaller variance in the numerator then you have to use the lower tail critical value of F, which is computed as on slide 23. If you put larger variance in the numerator then use F with (num d.f., denom. d.f., )
Independent Samples

22. An example of an F distributionα
Problem: how do we obtain the lower-tail critical value of ? The  given in the F table is the value for the upper tail. Since the F distribution is not symmetric, we have to compute critical F for lower tail.
Independent Samples

23. Computing critical F values for lower tail
Critical F for upper tail of distribution is found in Table VII, using α/2 and d.f.
Critical F for lower tail of distribution: 1
Fα/2, n2-1, n1-1
Note that d.f. are inverted!
Independent Samples

24. Test of hypothesis of equal variances
Rejection region for the two tailed test:
Fobt < 1
F(n2-1,n1-2,/2)
or Fobt > F(n1-1, n2-1,/2)
Note the reversed d.f.
For the two sided test, you reject H0 if either Fobt is smaller than the lower-tail critical F (the reciprocal) or larger than the upper tail critical F.
Independent Samples

25. 1b. Small samples, independent groups
Now – back to our t-test.
If you do NOT reject H0 in the test of equality of variances, then you can pool the two sample variances:
Sp2 = (n1-1)s12 + (n2-1)s22
n1 + n2 - 2
Pooling the sample variances makes no sense if you have just demonstrated that they are unequal (e.g., you have just rejected the null hypothesis in your test of the equality of the variances).
Independent Samples

26. 1b. Small samples, independent groups
H0: 1 – 2 = D0 H0: 1 – 2 = D0
HA: 1 – 2 > D0 HA: 1 – 2 ≠ D0
or: 1 – 2 < D0
Test statistic: t = – D0 X1 X2
( ) Sp
n1 n2 ( )
Independent Samples

27. 1b. Small samples, independent groups
Rejection region:
One-tailed: Two-tailed:
t < -tα │t│>tα/2
or t > tα
With tα and tα/2 based on df = n1 + n2 – 2
Independent Samples

28. Example 1a
This is a question about the sampling distribution of the difference, The question is, how likely is a value of
less than 194.0?
What is 1 – 2? To get 1 – 2, we subtract population means: – = 204.4 X1 X2
( ) X1 X2
( )
Independent Samples

29. The mean of the sampling distribution of the difference between the means (X1 – X2) is the difference between the population means, – = 204.4
194
204.4
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30. 194
204.4
The question asks, what is the probability that our sample difference (X1 – X2) is < 194?
Independent Samples

31. Example 1a (X1-X2) = = 900 1225 50 70 = 5.958 12 22 n1 n2= =
12
22 n1 n2
Independent Samples

32. Example 1a
Z = 194 – =
= -1.75
P(Z < -1.75) = = .0401
Independent Samples

33. From Table
.0401
.4599
194
204.4
Independent Samples

34. Example 1b
First – compute mean # of bold-faced words in each discipline’s text:
XP = = 32
XS = = 40
Independent Samples

35. Example 1b 12 22 H0: S – P = 204.4 HA: S – P < 204.4
Rejection region: Zobt < Zcrit = (α < .001)
= =
32 40
12
22 n1 n2
Independent Samples

36. Example 1b
Zobt = ( – ) – 204.4
7.665 =
Decision: reject H0 – the difference in # of bold-faced words in P & S texts is decreasing.
Independent Samples

37. Example 1c The difference S - P = 90% of 204.4 which is 183.96.
H0: S – P = 204.4
HA: S – P < 204.4
Independent Samples

38. Fail to reject B
Reject
204.4 A
183.96
Independent Samples

39. Example 1c What is the critical value of XS – XP?
Zcrit = = (XS – XP) – 204.4
7.665
XS – XP =
For XS – XP > , do not reject H0.
Independent Samples

40. Example 1c
Z = – = 1.02
7.665
P(Z ≥ 1.02) = .5 – =
Probability you fail to reject H0 even though mean difference has decreased from to is
Independent Samples

41. Example 2
First, we have to test the hypothesis of the equality of the variances:
S21 = =
= 36.62
Independent Samples

42. Fobt < 1 or Fobt > F(6,7,.025) F(7,6,.025)
Example 2
S22 = = 192
= 27.43
Rejection region:
Fobt < or Fobt > F(6,7,.025)
F(7,6,.025)
Independent Samples

43. Example 2 Rejection region: Fobt < 1 or Fobt > 5.70 5.12
Independent Samples

44. Example 2
F = =
27.43
Do not reject H0 – we can now do the t-test of our hypothesis about the difference between customers at the two types of stores.
Independent Samples

45. ( ) Example 2 H0: 1 – 2 = 0 HA: 1 – 2 > 0
Test statistic: t = – 0 X1 X2
( ) Sp
n1 n2 ( )
Independent Samples

46. Example 2
X1 = 228/7 = 32.57
X2 = 320/8 = 40.0
S2P = (n1 – 1)*S21 + (n2 – 1)*S22
n1 + n2 – 2
Independent Samples

47. Example 2
S2P = 6 (36.62) + 7 (27.43)
7 + 8 – 2 = 13
= 31.67
Independent Samples

48. Example 2 Rejection region: tobt > tcrit = t(13, .05) = 1.771
√31.67 * (1/7 + ⅛) √8.483
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49. Example 2
t = = 2.55
2.913
Decision: Reject H0 – there is evidence that people who shop at membership-required stores spend more on average than people who shop at no-membership-required stores.
Independent Samples

50. 12
22
(1 – 2) n1 n2 X1 X2
( )
12
22 n1 n2
Independent Samples