Anthony Greene 1

Two Sample t-test: Hypothesis of Differences

Between Two Groups1. Is Group “A” Different Than Group “B”?

2. Does an Experimental Manipulation Have an Effect?

• Is an experimental group different than a control group?

• If so, then the experimental manipulation had an effect

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Use of the Two Independent Sample t-test

This is the most universally used inferential statistic

Why?

• Population parameters μ and σ are almost never known

• Most experiments require a comparison, and that requires at least two groups

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Significant Differences?

M1= 40 M2=60

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Significant Differences?

M1= 40 M2=60

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Where You’ve Been Thus Far

1. Computation of descriptives

2. Probability theory, especially standard normal distributions (z-scores)

3. Hypothesis Testing

• Using z-scores

• Single sample t-test

• Two sample t-test

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Overview of Procedure

Two independent sample t-testa) 1 and 2 are hypothesized or predicted

(not computed and generally not known): M1 and M2 are computed

b) M1 and M2 are unknown ( is unknown) :

sM1 and sM2 are computed

c) Degrees freedom (df) is computed

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The Basic Idea• A new distribution is used that is

normally distributed

• This time the parent distribution is

2121 MM

)/()/( 2221

2121

nnMM

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The Basic Idea• So the sampling distribution is

has the following mean and standard error:

21 MM

2

2

1

2

21 n

s

n

ss pp

MM

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The Basic Idea

• So the basic t-test has the form:

21

2121

soerror standard

mean edhypothesiz -mean observed

MMs

MMt

t

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Alternate Forms

)/1()/1(

)()(

)/()/(

)()(

)()(

21p

2121

22

12

2121

)(

2121

21

nns

MM

nsns

MM

MMt

pp

MMs

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The Basic Idea

21

21

MMs

MMt

Since the usual H0 is that μ1=μ2 OR μ1 - μ2= 0

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The Basic Idea

What is sp?

It’s the pooled variance and its meant to allow you to make a comparison of means even if the σs aren’t equal

21

212

dfdf

SSSSsp

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Annual salaries ($1000s) for30 faculty members in public institutions and 35 faculty members in private institutions

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Process for comparing two population means using independent samples

Compare M1 and M2

Based on the Pooled VarianceMake a Decision

Compute M1

Compute M2

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Notation for parameters and statistics when considering two populations

M M

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The t-test for two population means(Slide 1 of 3)

Step 1 The null hypothesis is H0: 1 = 2 or 1 - 2 = 0; the alternative hypothesis is one of the following:

Ha: 1 2 Ha: 1 < 2 Ha: 1 > 2 (Two Tailed) (Left Tailed) (Right Tailed)

Step 2 Decide on the significance level, Step 3 The critical values are

±t/2 -t +t(Two Tailed) (Left Tailed) (Right Tailed)

with df = n1+n2 - 2.

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The t-test for two population means (Slide 2 of 3)

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The t-test for two population means(Slide 3 of 3)

Step 4 Compute the value of the test statistic

Where

Step 5 If the value of the test statistic falls in the rejection region, reject H0, otherwise do not reject H0.

)/1()/1(

)()(

21p

2121

nns

MMt

21

21

dfdf

SSSSsp

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Summary statistics

Public Institution

Private Institution

Sample Mean

57.48 66.39

Sum of Squares

16634 1982

Sample Size 30 35

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Computing the t-value

18.224.017

91.8

351

30117

39.6648.57

1729563

18616

21

21

t

dfdf

SSSSsp

21

21

MMs

MMt

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Criterion for deciding whether or not to reject the null hypothesis

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Criterion for deciding whether or not to reject the null hypothesis

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Sample Problem:Minutes required to comprehend the self-study manual

X1 (X-μ)2 X2 (X-μ)2

139 61.73 142 663.06

118 832.73 109 52.56

164 293.88 130 189.06 M1-M2 30.61

151 17.16 107 85.56 Sp22.44

182 1235.02 155 1501.56 sM1-M211.61

140 47.02 88 798.06

134 165.31 95 451.56 t 2.64

104 150.06

M 146.86 116.25

SS 2652.86 3891.50

Sample Problem

Remember the goal is always to see if the effect is large compared to random variation.

1.In the t-test this is done by the ratio: (diff. in means)/(randomness). For a reasonable sample this ratio must exceed ~1.96.

2.In the graph this is done by comparing the mean difference to the error bars which are each 1 s.e. If the difference is greater than 2 s.e. (~1.96) then the difference is significant.

21 MM

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WithoutExperience

WithExperience

Sample Problem