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Department

Statistics

Course Code

STAT 3005

Professor

H C Tavera

Lecture

7

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

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