Inference about Two Means - Independent Samples

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
• Welchs 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
P-Value is the area highlighted
Remember to add the areas in the two-tailed!
t0
-t0
t0
t0
Critical Region
Test Statistic
Reject null hypothesis, if Reject null hypothesis, if Reject null hypothesis, if
P-value lt a P-value lt a P-value lt a
Left-Tailed Two-Tailed Right-Tailed
t0 lt - ta t0 lt - ta/2ort0 gt ta/2 t0 gt ta
6
Confidence Interval Difference in Two Means
Lower Bound
Upper Bound ta/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
s12 s22 ----- ----- n1
n2
(x1 x2) ta/2
PE MOE
s12 s22 ----- ----- n1
n2
(x1 x2) ta/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 gt µ2 at the a0.05 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
• HypothesisH0 H1
• Test Statistic
• Critical Value
• Conclusion

Assumed to work the problem
µ1 µ2 (No difference)
µ1 gt µ2
1.237, p 0.1144
tc(13-1,0.05) 1.782, a 0.05
Fail to Reject H0
10
Example Problem Cont. part b

• Confidence Interval PE MOE

s12 s22 ----- ----- n1
n2
(x1 x2) ta/2
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
-1.5986, 5.7986 by hand
-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 Students 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