1. 12.5 Differences between Means (s’s known)
Two populations: (1, 1) & (2, 2)
Two samples: one from each population
Two sample means and
sample sizes: n1 & n2
Compare two population means:
H0: 1-2= (=0 in most cases)
Alternatives: 1-2>; 1-2<; 1-2
We are testing whether the two populations are the same if delta=0.

2. Let’s go through a two sided alternative
H0: 1-2=0 vs HA: 1-2≠0
Reject H0 if is too far from zero in either direction.
How far from zero might be if 1-2=0?
Sampling distribution of is asymptotically normal with mean 0 and standard deviation
We need to know

3. Fact:
If the sample means are from independent samples, then

4. Thus under certain assumptions:
Correspondingly, a confidence interval for m1-m2 is

5. Assumptions 1 & 2 are known Normal populations or large sample sizes
Under null hypothesis
is (asymptotically) standard normal

6. Rejection Regions: 1-2> 1-2< 1-2 Rejection Regions
Alternative
Hypotheses
1-2>
1-2<
1-2
Rejection Regions
z>z
z<-z
z>z/2 or
z<-z/2

7. Example 12.4
Two labs measure the specific gravity of metal. On average do the two labs give the same answer?
 Population mean by lab1
 Population mean by lab2
H0: 1=2 vs HA: 12
1=0.02, n1=20,
2=0.03, n2=25,

8. 95% Confidence Interval
from –0.014 to 0.016

9. Two-tailed Hypotheses Test
Two sample test
Rejection region: |Z|>z0.025=1.96
Conclusion: Don’t reject H0.

10. Rejection Regions Alternative Hypotheses HA: 1>2 HA: 1<2
z>z
z<-z
z>z/2 or
z<-z/2

11. Exercise
An investigation of two kinds of photocopying equipment showed that a random sample of 60 failures of one kind of equipment took on the average 84.2 minutes to repair, while a random sample of 60 failures of another kind of equipment took on the average 91.6 minutes to repair. If, on the basis of collateral information, it can be assumed that s1=s2=19.0 minutes for such data, test at the 0.02 level of significance whether the difference between these two sample means is significant.

12. 12.6 Differences Between Means (unknown equal variances)
Large samples
n130; n230
Small samples
1. 1=2
2. 12

13. Large Samples
n130; n230
Estimate 1 and 2 by s1 and s2
Set

14. Rejection Regions Alternative Hypotheses HA: 1>2 HA: 1<2
z>z
z<-z
z>z/2 or
z<-z/2

15. Small Samples 1=2= unknown Two populations are normal
Standard error
Estimate the common variance

16. Pooled standard deviation
Using both s12 and s22 to estimate 2, we combine these estimates, weighting each by its d.f.. The combined estimate of 2 is sp2, the pooled estimate:
Estimate  by sp

17. Two-Sample T-test T-test (t distribution with df=n1+n2-2) 100(1-)% CI
Hypothesized m1-m2

18. Example 12.5 Compare blood pressures Two populations: common variance
=0.05
n1=10, s1=16.2,
n2=12, s2=14.3,

19. CI & test sp=15.2 df=10+12-2=20 Critical value t0.025=2.086
t statistic: reject H0 if |t|>2.086
Conclusion?
Don’t Reject.
CI: -122.086(6.51)=-12 13.6
-1.6 to 25.6

20. What happens when variances are not equal?
Testing: H0: m1-m2=δ.
Normal population
s1 and s2 are not necessarily equal
s1 and s2 unknown

21. Two sample t-test with unequal variances
d.f. =min(n1-1, n2-1)

22. Exercise
In a department store’s study designed to test whether or not the mean balance outstanding on 30-day charge accounts is the same in its two suburban branch stores, random samples yielded the following results:
Use the 0.05 level of significance to test the null hypothesis m1-m2=0.

23. 12.7 Paired Data 1982 study of trace metals in South Indian River.
6 random locations 5 3 1 6 2 4
T=top water zinc concentration (mg/L)
B=bottom water zinc (mg/L)
Top
Bottom

24. One of the first things to do when analyzing data is to PLOT the data
This is not a useful way to plot the data. There is not a clear distinction between bottom water and top water zinc—even though Bottom>Top at all 6 locations.
Top
Bottom

25. A better way
Top
Bottom
Connect points in the same pair.

26. A better way Bottom=Top
The plot suggests that mBottom>mTop. Is it true?

27. That is equivalent to ask: is it true that mdifference>0?
Top
Bottom
D=B-T
Ho: mD= vs HA: mD>0

28. First check the assumption that the population is normal

29. Doing a one-sided test Ho: mD=0 vs HA: mD>0
t0.05 at 5 d.f. is So anything greater than will be an evidence against H0.
We reject H0: mB-T=0 in favor of HA: mB-T>0.

30. Another example
The average weekly losses of man-hours due to accidents in 10 industrial plants before and after installation of an elaborate safety program:
Plants
Before
After
diff(B-A)
Is the safety program effective? (level=0.05)

31. Two Populations: Before and After
Normal?
Independent?
No, No

32. Normal Probability Plots
Small sizes
Skew to right somehow

33. Normal Probability Plot for Difference
Looks better

34. Consider the Differences
Paired Observations:before and after the installation of safety program are from the same plants (dependent)
Data from different plants may be independent
Diff:

35. Set up a Test—Paired T-Test
‘effective’ means the program reduces the accidents, i.e., before > after (D>0)
=difference of average accidents
H0: D=0 vs HA: D>0
The procedure is the same as the one-
sample t-test
Df=n-1

36. Rejection Regions for Paired T-test
Alternative
Hypotheses
D>
D<
D
Rejection Regions
t>t
t<-t
t>t/2 or
t<-t/2

37. Paired t-test One-tailed test Critical value: df=9, t0.05=1.833
Sample mean & standard deviation:
t-statistic:
Conclusion: reject H0 since t=4.03>1.833