Aron chpt 9 ed f2011

Chapter 9: t-test for Chapter 9: t-test for Independent MeansIndependent Means Arthur Aron, Elaine N. Aron, Elliot CoupsArthur Aron, Elaine N. Aron, Elliot Coups

t t Tests for Independent Tests for Independent MeansMeansHypothesis-testing procedure used

for studies with two sets of scores ◦Each set of scores is from an entirely different group of people and the population variance is not known. e.g., a study that compares a treatment group to a control group

The Distribution of Differences The Distribution of Differences Between MeansBetween Means

When you have one score for each person with two different groups of people, you can compare the mean of one group to the mean of the other group.◦The t test for independent means

focuses on the difference between the means of the two groups.

Copyright © 2011 by Pearson Education, Inc. All rights reserved

The Logic of a Distribution of The Logic of a Distribution of Differences Between MeansDifferences Between Means

The null hypothesis is that Population M1 = Population M2

◦ If the null hypothesis is true, the two population means from which the samples are drawn are the same.

The population variances are estimated from the sample scores.

The variance of the distribution of differences between means is based on estimated population variances.◦ The goal of a t test for independent means is to

decide whether the difference between means of your two actual samples is a more extreme difference than the cutoff difference on this distribution of differences between means.


Mean of the Distribution of Mean of the Distribution of Differences Between MeansDifferences Between Means

With a t test for independent means, two populations are considered.◦ An experimental group is taken from one of these

populations and a control group is taken from the other population.

If the null hypothesis is true: ◦ The populations have equal means.◦ The distribution of differences between means has a

mean of 0.


Estimating the Population Estimating the Population VarianceVariance In a t test for independent means, you calculate two

estimates of the population variance.◦ Each estimate is weighted by a proportion consisting of its

sample’s degrees of freedom divided by the total degrees of freedom for both samples. The estimates are weighted to account for differences in sample

size.

◦ The weighted estimates are averaged. This is known as the pooled estimate of the population variance.

S2Pooled = df1(S2

1) + df2(S22)

df Total df Total

df Total = df1 + df2


Figuring the Variance of Each of the Figuring the Variance of Each of the Two Distributions of MeansTwo Distributions of Means The pooled estimate of the population variance is

the best estimate for both populations. Even though the two populations have the same

variance, if the samples are not the same size, the distributions of means taken from them do not have the same variance.◦ This is because the variance of a distribution of means

is the population variance divided by the sample size. S2

M1 = S2

Pooled / N1

S2M2

= S2Pooled / N2


The The Variance and Standard Variance and Standard Deviation of the Distribution of Deviation of the Distribution of Differences Between MeansDifferences Between Means

The Variance of the distribution of differences between means (S2

Difference) is the variance of Population 1’s distribution of means plus the variance of Population 2’s distribution of means.

◦ S2Difference = S2

M1 + S2

M2

The standard deviation of the distribution of difference between means (SDifference ) is the square root of the variance.

◦ SDifference = √S2Diifference


Steps to Find the Standard Deviation Steps to Find the Standard Deviation of the Distribution of Differences of the Distribution of Differences Between MeansBetween Means• Figure the estimated population variances based on each sample.

• S2 = [∑(X – M)2] / (N – 1)• Figure the pooled estimate of the population variance.

S2Pooled = df1(S2

1) + df2(S22)

df Total df Total

df1 = N1 – 1 and df2 = N2 – 1; dfTotal = df1 + df2

• Figure the variance of each distribution of means.S2

M1 = S2

Pooled / N1

S2M2

= S2 Pooled / N2

• Figure the variance of the distribution of differences between means.• S2

Difference = S2M1

+ S2M2

• Figure the standard deviation of the distribution of differences between means.• SDifference =√ S2

Difference


The Shape of the Distribution The Shape of the Distribution of Differences Between of Differences Between MeansMeansSince the distribution of

differences between means is based on estimated population variances:◦ The distribution of differences between means

is a t distribution.◦ The variance of this distribution is figured

based on population variance estimates from two samples. The degrees of freedom of this t distribution are

the sum of the degrees of freedom of the two samples. dfTotal = df1 + df2


The The tt score for the Difference score for the Difference Between the Two Actual MeansBetween the Two Actual Means

Figure the difference between your two samples’ means.

Figure out where this difference is on the distribution of differences between means.◦ t = M1 – M2 / SDifference


Hypothesis Testing with a Hypothesis Testing with a tt Test for Independent MeansTest for Independent Means

The comparison distribution is a distribution of differences between means.

The degrees of freedom for finding the cutoff on the t table is based on two samples.

Your samples’ score on the comparison distribution is based on the difference between your two means.



Example from text page 301◦You have a sample of 10 students who

were recruited to take part in the study. The researcher wants to see if the group who received the experimental procedure would perform any differently than those who did not receive any experimental procedure.

◦5 students were randomly assigned to the experimental group and 5 were randomly assigned to the control group. All of the students rated their overall level of

emotional adjustment on a scale from 0 (very poor) to 10 (very positive).

Example of The t Test for Independent Means: Background

Step One: Restate Question Step One: Restate Question into Research and Null into Research and Null HypothesesHypotheses◦ Population 1: students in the experimental group

◦ Population 2: students not in the experimental group (control group)

◦Research hypothesis: Population 1 students would rate their adjustment differently from Population 2 students (two-tailed tests).

◦Null hypothesis: Population 1 students would rate their adjustment the same as Population 2 students.


Step Two: Determine Characteristics Step Two: Determine Characteristics of Comparison Distributionof Comparison Distribution

The comparison distribution is a distributions of differences between means. Its mean = 0.

Figure the estimate population variances based on each sample.• S2

1 = 1.5 and S22 = 2.5

Figure the pooled estimate of the population variance.• S2

Pooled = 2.0

Figure the variance of each distribution of means.• S2

Pooled / N = S2M

• S2M1 = .40

• S2M2 = .40

Figure the variance of the distribution of differences between means.• Adding up the variances of the two distributions of means would come out to

S2Difference = .80

Figure the standard deviation of the distribution of difference between means.• S Difference= √S2

Difference = √.80 = .89

The shape of the comparison distribution will be a t distribution with a total of 8 degrees of freedom.


Step Three: Determine cutoff scores Step Three: Determine cutoff scores from from tt table table

Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.◦You will use a two-tailed test.◦If you also chose a significance level of

.05, the cutoff scores from the t table would be 2.306 and -2.306.


Determine your sample’s score on the comparison distribution.◦t = (M1 – M2) / SDifference

◦(7 – 4) / .89 ◦3 / .89 = 3.37

Step Four: Determine your sample’s score (calculate t)


Step Five: Decide whether to reject Step Five: Decide whether to reject the nullthe null

◦Compare your samples’ score on the comparison distribution to the cutoff t score.

◦Your samples’ score is 3.37, which is larger than the cutoff score of 2.306.

◦You can reject the null hypothesis.

Review of the Review of the tt Test for a Single Sample, Test for a Single Sample, tt Test Test for Dependent Means, and the for Dependent Means, and the tt Test for Test for Independent MeansIndependent Means• t Test for a Single Sample

Population Variance is not known. Population mean is known. There is 1 score for each participant. The comparison distribution is a t distribution. df = N – 1 Formula t = (M – Population M) / Population SM

t Test for Dependent Means Population variance is not known. Population mean is not known. There are 2 scores for each participant. The comparison distribution is a t distribution. t test is carried out on a difference score. df = N – 1 Formula t = (M – Population M) / Population SM

t Test for Independent Means Population variance is not known. Population mean is not known. There is 1 score for each participant. The comparison distribution is a t distribution. df total = df1 + df2 (df1 = N1 – 1; df2 = N2 – 1) Formula t = (M1 – M2) / SDifference

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Aron chpt 9 ed f2011

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