1. Inference about Two Means - Independent Samples
Lesson
Inference about Two Means - Independent Samples

2. Objectives
Test claims regarding the difference of two independent means
Construct and interpret confidence intervals regarding the difference of two independent means

3. Vocabulary
Robust – minor deviations from normality will not affect results
Independent – when the individuals selected for one sample do not dictate which individuals are in the second sample
Dependent – when the individuals selected for one sample determine which individuals are in the second sample; often referred to as matched pairs samples
Welch’s approximate t – the test statistic to compare two independent means

4. Requirements
Testing a claim regarding the difference of two means using matched pairs
Sample is obtained using simple random sampling
Sample data are independent
Populations are normally distributed or the sample sizes, n1 and n2, are both large (n ≥ 30)
This procedure is robust.

5. P-Value is the area highlighted Reject null hypothesis, if
Classical and P-Value Approach – Two Means
P-Value is the area highlighted
Remember to add the areas in the two-tailed!
-tα
-tα/2
tα/2 tα t0
-|t0|
|t0| t0
Critical Region
(x1 – x2) – (μ1 – μ2 )
t0 =
s s22
n n2
Test Statistic:
Reject null hypothesis, if
P-value < α
Left-Tailed
Two-Tailed
Right-Tailed
t0 < - tα
t0 < - tα/2 or t0 > tα/2
t0 > tα

6. Confidence Interval – Difference in Two Means
Lower Bound:
Upper Bound:
tα/2 is determined using the smaller of n1 -1 or n2 -1 degrees of freedom
x1 and x2 are the means of the two samples
s1 and s2 are the standard deviations of the two samples
Note: The two populations need to be normally distributed or the sample sizes large
s s
n n2
(x1 – x2) – tα/2 ·
PE ± MOE
s s
n n2
(x1 – x2) + tα/2 ·

7. Two-sample, independent, T-Test on TI
If you have raw data:
enter data in L1 and L2
Press STAT, TESTS, select 2-SampT-Test
raw data: List1 set to L1, List2 set to L2 and freq to 1
summary data: enter as before
Set Pooled to NO
Confidence Intervals
follow hypothesis test steps, except select 2-SampTInt and input confidence level
expect slightly different answers from book

8. Example Problem Given the following data:
Test the claim that μ1 > μ2 at the α=0.05 level of significance
Construct a 95% confidence interval about μ1 - μ2
Data
Population 1
Population 2 n 23 13
x-bar
43.1
41.0 s
4.5
5.1

9. Example Problem Cont. part a
Requirements:
Hypothesis H0: H1:
Test Statistic:
Critical Value:
Conclusion:
Assumed to work the problem
μ1 = μ2 (No difference)
μ1 > μ2
x1 – x2 - 0
t0 =
(s²1/n1) + (s²2/n2)
= 1.237, p =
tc(13-1,0.05) = 1.782, α = 0.05
Fail to Reject H0

10. Example Problem Cont. part b
Confidence Interval: PE ± MOE
s s
n n2
(x1 – x2) ± tα/2 ·
tc(13-1,0.025) = 2.179
2.1 ±  (20.25/23) + (26.01/13)
2.1 ± (1.6974) = 2.1 ±
[ , ] by hand
[ , ] by calculator
It uses a different way to calculate the degrees of freedom (as shown on pg 592)

11. Summary and Homework Summary Homework
Two sets of data are independent when observations in one have no affect on observations in the other
In this case, the differences of the two means should be used in a Student’s t-test
The overall process, other than the formula for the standard error, are the general hypothesis test and confidence intervals process
Homework
pg 595 – 599: 1, 2, 7, 8, 9, 13, 19

12. Homework Answers
4 a) Reject H0 (t0 = , p = ) b) [1.1, 12.9]
6 a) Reject H0 (t0 = , p = ) b) [-30.75, ]
8 example problem in notes