Using T to Compare 2 Means

Two Sample Problems

Two Sample ProblemsCompare the responses of two

treatments or compare the characteristics of 2 populations

Separate samples from each population*Different from Matched pairs

May have 2 different sample sizes

No matching of the units

That means you could test results

from a group of real men like me with a group of geeks like

that guy?!!

Comparing 2 Population Means

Conditions1. Two SRS’s; Independent samples (no

matching); measuring same variable2. Both populations are normally distributed w/

unknown parameters.Needs

1. Sample Sizes (n1, n2)

2. Sample Statistics (x-bar1, x-bar2, s1, s2)

PurposeCompare 2 means to look for a SIGNIFICANT

difference

Degrees of Freedom for 2 Sample Tests

Conservative Estimate when using the t-table

Calculate degrees of freedom for each sample and use the smaller of the two

n1 - 1

n2 – 1

Use the smaller of the two…

If you use the calculator or statistical software, they

will use a formula to calculate the t-statistic &

degrees of freedom….

Hypothesis Testing for 2 Means We are testing the Ho:µ1=µ2

Write the Hypotheses (context)

Check the Conditions (show)

Calculate the T Statistic

Find the P-Value for the appropriate df

Make statistical decision and interpret results in context

Quick Look at the Hypotheses

The hypotheses for this test will be as follows, depending on the situation and context of the problem:Ho: µ1 = µ2

Ha: µ1 > µ2, µ1 < µ2, µ1 ≠ µ2

2 Sample T Statistic

2

22

1

21

2

__

1

__

)(

ns

ns

xxt

Standard Error Using the smaller of the n-1

degrees of freedom

2-Sample T-Statistic formula:

Fantastic Fishy vs. Nibbles n’ Bits Fantastic Fishy Food advertises the best fish growing

formula on the market, but so does Nibbles n’ Bits, however. As a research project, Ronnie has decided to study the growth rates of fish given these two foods over a set period of time. After carefully setting up the experiment, Ronnie measured a SRS of 48 fish from the FFF group, finding an average growth of 15 g with a standard deviation of 2.32g. The Nibbles n’ Bits group of fish grew an average of 16.7 g with a standard deviation of 1.87g out of a SRS of 52 measurements. Help Ron decide if there’s a significant difference between the two food types and the growth they produce in the fish at a 5% level.

NnB

FFF

Fish Food Formulas Ho: The average mass increase of fish is the same

between Fantastic Fishy Food and Nibbles n’ Bits. µFFF = µNnB

Ha: The average mass increase of fish is significantly different between Fantastic Fishy Food and Nibbles n’ Bits. µFFF ≠ µNnB

2 2

15 16.7

(2.32) (1.87)48 52

t

t = -4.0138 w/ df = 49

p < .0005 < a = .05 so we reject that the 2 foods cause the same growth.

2 Sample T Tests with the Calculator

Using the calculator gives even more accurate results using exact degrees of freedom (not our limited chart)

Stat – Tests – 4:2-SampTTest

Fantastic Fishy vs. Nibbles n’ Bits Fantastic Fishy Food advertises the best fish growing

formula on the market, but so does Nibbles n’ Bits, however. As a research project, Ronnie has decided to study the growth rates of fish given these two foods over a set period of time. After carefully setting up the experiment, Ronnie measured a SRS of 48 fish from the FFF group, finding an average growth of 15 g with a standard deviation of 2.32g. The Nibbles n’ Bits group of fish grew an average of 16.7 g with a standard deviation of 1.87g out of a SRS of 52 measurements. Help Ron decide if there’s a significant difference between the two food types and the growth they produce in the fish.

Try using the

Calculator this time…

Pooled 2 Sample TestsThere is an option on your

calculator to pool your degrees of freedom.

This option can only be used if:

the sample sizes are exactly the same

But also, only if the variances of the two population are known to be the same.

This basically means we WON’T

be using the Pooled option, Sucka!