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Lecture 7

STAT 3005 Lecture 7: STAT-3005 - Notes - 7-2 - Comparing Two Means
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5 Pages
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Spring 2018

Department
Statistics
Course Code
STAT 3005
Professor
H C Tavera
Lecture
7

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Daniel T. Eisert STAT-3005
1
7.2 Comparing Two Means
Chapter VII: Inference for Means
Comparing Two
Means
Recall ~ the goal given two independent samples with quantitative responses
is to compare two populations where the response is quantitative, examine the
difference between two parameters, and construct confidence intervals or
perform hypothesis tests.
Testing:
Test Statistic: 


Standard Error of :
- First, one should determine whether and are equal or not because
o If they are equal, all of the data can be pooled together to get one
good estimate of the common rather than two less accurate
estimates.
o To do a confidence interval or hypothesis test, the distribution
being dealt with (Z or t, for example), must be known and this
requires knowing whether and are equal or not.
- There are tests for equality of unknown variances, but they are quite
sensitive to departures from normality; therefore, they are NOT robust.
- Generally, consider the variances different if the ratio of larger to
smaller variance is more than 4.




- Case I: Unequal Variance
1. Assume (i.e. 

).
2. Test statistic: 
, where  represents the
degrees of freedom (df).
Two Sample T-Interval
- When the Random, Normal, and Independent conditions are met, a
level C confidence interval for comparing two means is:

where t* is the critical value for confidence level C for the t distribution
with degrees of freedom v.
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Daniel T. Eisert STAT-3005
2
Comparing Two
Means
Estimating Degrees of Freedom:
- Satterthwaite’s Approximation (Welch-Satterthwaite
Approximation): approximates the degrees of freedom from the t-test
with unequal variance.
- The floor function (L shaped brackets) denotes rounding down to the
next integer.
EXAMPLE: Consider two plant sciences graduate students working on
developing better fertilizers for a particular type of wheat. Joe fertilizes
20 plants and gets heights with  and  while Laura
fertilizes 32 plants and obtains  and . All
measurements are in centimeters.
Conduct a level 0.01 test to see if Laura’s fertilizer is better.
Find a 95% confidence interval for the difference in heights.
- Case II: Equal Variances
1. Assume (i.e. 

).
2. Use the pooled estimate for
2
21
2
22
2
11
2
2
)1()1(
pp
p
ss
nn
snsn
s
=
-+
-+-
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Description
Daniel T. Eisert STAT-3005 7.2 Comparing Two Means Chapter VII: Inference for Means Comparing Two Recall ~ the goal given two independent samples with quantitative responses is to compare two populations where the response is quantitative, examine the Means difference between two parameters, and construct confidence intervals or perform hypothesis tests. Testing: : = 0 0 1 2 1 1 2 Test Statistic:12)0 [12] Standard Error of : - First, one should determine whether and are equal or not because 1 2 o If they are equal, all of the data can be pooled together to get one good estimate of the common rather than two less accurate estimates. o To do a confidence interval or hypothesis test, the distribution being dealt with (Z or t, for example), must be known and this requires knowing whether and1 are 2qual or not. - There are tests for equality of unknown variances, but they are quite sensitive to departures from normality; therefore, they are NOT robust. - Generally, consider the variances different if the ratio of larger to smaller variance is more than 4. > 4 or > 2 - Case I: Unequal Variance 2 1. Assume (i.e. > 4). 1 2 (12)0 2. Test statistic: = ~(), where () represents the (1+ 2) 1 2 degrees of freedom (df). Two Sample T-Interval - When the Random, Normal, and Independent conditions are met, a level C confidence interval for comparing two means is: 12 1 (1 2) ()( + ) 1 2 * where t is the critical value for confidence level C for the t distribution with degrees of freedom v. 1Daniel T. Eisert STAT-3005 Comparing Two Estimating Degrees of Freedom: - Satterthwaites Approximation (Welch-Satterthwaite Means Approximation): approximates the degrees of freedom from the t-test with unequal variance. - The floor function (L shaped brackets) denotes rounding down to the next integer. 2 2 2 (1 + 1 ) 1 2 = 2 2 2 2 1 2 ( 1 ) ( 2) 1 + 1 1 2 EXAMPLE: Consider two plant sciences graduate students working on developing better fertilizers for a particular type of wheat. Joe fertilizes
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