1. Lesson 11 - 2
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. Classical and P-Value Approach – Two Means
Test Statistic:
tα
-tα/2 tα/2
-tα
Critical Region
P-Value is the area highlighted
|t0|
-|t0|
t0 t0
Reject null hypothesis, if
P-value < α
Left-Tailed Two-Tailed Right-Tailed
t0 < - tα
t0 < - tα/2
or
t0 > tα/2
t0 > tα
Remember to add the areas in the two-tailed!
(x1 – x2) – (μ1 – μ2 )
t0 = -------------------------------
s1
2 s2
2
----- + -----
n1 n2
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
(x1 – x2) – tα/2 ·
s1
2 s2
2
----- + -----
n1 n2
(x1 – x2) + tα/2 ·
s1
2 s2
2
----- + -----
n1 n2
PE ± MOE
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:
a) Test the claim that μ1 > μ2 at the α=0.05 level of
significance
b) 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:
μ1 > μ2
μ1 = μ2 (No difference)
Assumed to work the problem
tc(13-1,0.05) = 1.782, α = 0.05
Fail to Reject H0
= 1.237, p = 0.1144
x1 – x2 - 0
t0 = ------------------------
(s²1/n1) + (s²2/n2)
10. Example Problem Cont. part b
• Confidence Interval: PE ± MOE
[ -1.5986, 5.7986] by hand
(x1 – x2) ± tα/2 ·
s1
2 s2
2
----- + -----
n1 n2
tc(13-1,0.025) = 2.179
2.1 ± 2.179 (20.25/23) + (26.01/13)
2.1 ± 2.179 (1.6974) = 2.1 ± 3.6986
[ -1.4166, 5.6156] by calculator
It uses a different way to calculate the degrees of freedom
(as shown on pg 592)
11. Summary and Homework
• Summary
– 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 = -4.393, p = 0.0000268)
b) [1.1, 12.9]
• 6 a) Reject H0 (t0 = 2.4858, p = 0.01746)
b) [-30.75, -11.25]
• 8 example problem in notes
