1. Independent Samples: Comparing Means
Lecture 37
Sections 11.1 – 11.2, 11.4
Fri, Apr 6, 2007

2. The t Distribution
Whenever we do not know , we must use s1 and s2 to estimate .
In this case, we will have to use the t distribution instead of the standard normal distribution, unless the sample sizes are large.

3. Estimating  Individually, s1 and s2 estimate .
However, we can get a better estimate than either one if we “pool” them together.
The pooled estimate is

4. x1 –x2 and the t Distribution
If we use sp instead of , and the sample sizes are small, then we should use t instead of Z.
The number of degrees of freedom is
df = df1 + df2 = n1 + n2 – 2.
That is

5. Hypothesis Testing
See Example 11.4, p. 699 – Comparing Two Headache Treatments.
State the hypotheses.
H0: 1 = 2
H1: 1 > 2
State the level of significance.
 = 0.05.

6. The t Statistic Compute the value of the test statistic.
The test statistic is
with df = n1 + n2 – 2.

7. Computations

8. Hypothesis Testing Calculate the p-value.
The number of degrees of freedom is
df = df1 + df2 = 18.
p-value = P(t > 1.416)
= tcdf(1.416, E99, 18) =

9. Hypothesis Testing State the decision. State the conclusion.
Accept H0.
State the conclusion.
Treatment 1 is more effective than Treatment 2.

10. The TI-83 and Means of Independent Samples – Stats
Press STAT > TESTS.
Choose 2-SampTTest.
Choose Stats.

11. The TI-83 and Means of Independent Samples – Stats
Provide the information that is called for.
x1, s1, n1.
x2, s2, n2.
Alternative hypothesis.
Whether to use a pooled estimate of . Answer “yes.”

12. The TI-83 and Means of Independent Samples – Stats
Select Calculate and press ENTER.
The display shows, among other things, the value of the test statistic and the p-value.

13. The TI-83 and Means of Independent Samples – Data
Enter the data from the first sample into L1.
Enter the data from the second sample into L2.
Press STAT > TESTS.
Choose 2-SampTTest.
Choose Data.

14. The TI-83 and Means of Independent Samples – Data
Provide the information that is called for.
List 1: L1.
List 2: L2.
Freq 1: 1.
Freq 2: 1.
Alternative hypothesis.
Whether to use a pooled estimate of . Answer “yes.”

15. The TI-83 and Means of Independent Samples – Data
Select Calculate and press ENTER.
The display shows, among other things, the value of the test statistic and the p-value.

16. Paired vs. Independent Samples
The following data represent students’ calculus test scores before and after taking an algebra refresher course.
Student 1 2 3 4 5 6 7 8
Before 85 63 94 78 75 82 45 58
After 92 68 98 83 80 88 53 62

17. Paired vs. Independent Samples
Perform a test of the hypotheses
H0: 2 – 1 = 0
H1: 2 – 1 > 0
treating the samples as independent.

18. Paired vs. Independent Samples
Had we performed a test of the “same” hypotheses
H0: D = 0
H1: D > 0
treating the samples as paired, then the p-value would have been
Why so small?

19. Paired Samples Why is there a difference? 1 2 3 5 4 6 8 7 40 50 60 8090
100 70
Paired

20. Independent Samples Why is there a difference? 1 2 3 5 4 6 8 7 40 5060 80 90
100 70
Independent

21. Confidence Intervals
Confidence intervals for 1 – 2 use the same theory.
The point estimate isx1 –x2.
The standard deviation ofx1 –x2 is approximately

22. Confidence Intervals The confidence interval is or
( known, large samples)
( unknown, large samples)
( unknown, normal pops.,
small samples)

23. Confidence Intervals The choice depends on Whether  is known.
Whether the populations are normal.
Whether the sample sizes are large.

24. Example
Find a 95% confidence interval for 1 – 2 in Example 11.4, p. 699.
x1 –x2 = 3.2.
sp =
Use t =
The confidence interval is
3.2  (2.101)(2.259) = 3.2  4.75.

25. The TI-83 and Means of Independent Samples
To find a confidence interval for the difference between means on the TI-83,
Press STAT > TESTS.
Choose either 2-SampZInt or 2-SampTInt.
Choose Data or Stats.
Provide the information that is called for.
2-SampTTest will ask whether to use a pooled estimate of . Answer “yes.”