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Wednesday, December 31, 2014 1

Compute by hand and interpret

Dependent samples t test

Single sample t test

Use SPSS to compute the same tests

and interpret the output

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The Slides discuss

• Comparing two means to ascertain which mean is of greater statistical significance.

• The statistical test used for this type of analysis is the t test, and the statistic that is computed is called a t value.

• Three research situations in which the t test can be used to analyze the data and compare the means from

a) Two paired samples (e.g., pretest and posttest scores); and

b) a sample and a population (e.g., comparing a mean of a sample to the mean of the population).

• Examples with numerical data to illustrate each of the three types of t tests.

• After t value is computed, you have decided whether it is statistically significant and whether your research hypothesis was confirmed.

STEPS IN THE PROCESS OF HYPOTHESIS TESTING

1. Formulating a research question for the study.

2. Stating a research hypothesis (i.e., an alternative hypothesis). The hypothesis should represent the

researcher’s prediction about the outcome of the study and should be testable. Note that a null

hypothesis for the study is always implied, but it is not formally stated in most cases. The null

hypothesis predicts no difference between groups or means or no relationship between variables.

3. Designing a study to test the research hypothesis. The study’s methodology should include plans

for selecting one or more samples from the population that in interest to researcher; selecting or

designing instruments to gather the numerical data; carrying out the study’s procedure (and

intervention in experimental studies); and determining the statistical test(s) to be used to analyze

the data.

4. Conducting the study and collecting numerical data.

5. Analyzing the data and calculating the appropriate test statistics (e.g., Pearson correlation

coefficient, t test value or chi square value).

6. Determining the appropriate p value.

7. Deciding whether to retain or reject the null hypothesis. A p value of 0.05 is the most commonly

used benchmark to consider the results statistically significant. If the results are statistically

significant, the researcher may also wish to calculate the effect size (EF) to determine the

practical significance of the study’s results.

8. Making a decision whether to confirm the study’s alternative hypothesis (i.e., the research

hypothesis ) and how probable it is that the results were obtained purely by chance. This decision

is based on the decision made regarding the null hypothesis.

9. Summarizing the study’s conclusion, addressing the study’s research question.

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Assumptions of t-Test

• The groups are independent of each other.

• A person (or case) may appear in only one group.

• When the two group are approximately the same size, there is no need for the homogeneity of variance. This assumption is called the assumption of the homogeneity of variance.

• We compare the two variances to determine if there is a statistically significant difference between them.

• To test for the assumption of the homogeneity of the variances, we divide the larger variance by the smaller variance and obtain a ratio, called the F value.

• When the F value is statistically significant we cannot assume that the variance are equal.

Assumptions of t-Test

• A test for the equality of variance, such as the levene’s is used to test the significance of the F value. An F value that is not statistically significant (p>0.05) indicates that the assumption for the homogeneity of variance is not violated and, therefore, equal variances can be assumed , on the other hand, a significant F value (p< 0.05) indicates that the assumption for the homogeneity of variance was violated. The t test statistical results are adjusted for the unequal variance.

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Assumptions of t-Test

• The independent variables are interval or ratio.

• The population from which samples are drawn is normally distributed.

• Samples are randomly selected.

• The groups have equal variance (Homogeneity of variance).

• The t-statistic is robust (it is reasonably reliable even if assumptions are not fully met). Therefore, even if the assumption of the homogeneity of variance is not fully met, the researcher can probably still use the test to analyze the data.

A t-test is used to compare two means in three different situations

T Test

t-test for independent samples

- The two groups whose means are being compared are independent of each other

Ex: a comparison of experimental and control groups.

T-test for paired samples (t-test for dependent, matched or correlation

samples)

The two means represent two sets of scores that are paired.

Ex: a comparison of pretest and posttest scores obtained from one group of people

t-test for a single sample

The t-test is used when the mean of a sample is compared to the mean of a population

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Dependent Samples

t-tests

Dependent Samples t-test

• A t test for paired samples (dependent sample t-test) is used

when the two means being compared come from two sets of

scores that are related to each other.

• It is used, for example, in experimental research to measure

the effect of an intervention (a treatment) by comparing the

posttest to the pretest scores.

• The most important requirement for conducting this t test is

that the two sets of scores are paired: they belong to the

same individuals.

• Useful to control individual differences. Can result in more

powerful test than independent samples t-test.

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Computing the paired-samples t-test

• First find for each person the difference (D) between the

two scores (e.g., between pretest and posttest) and sum up

those difference (∑ D). Usually, the lower scores (e.g.,

pretest) are subtracted from the higher scores (e.g., posttest)

so D values are always positive.

• We also need to compute the sum of the squares of the

difference (∑ D2). Finally the t value is computed using the

formula:

The example that follows demonstrates the computation of a paired-

samples t test. To simplify the computations , we use scores of eight

students only. Of course, in conducting real-life research, it is

recommended that larger samples (thirty or more) be used.

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An example of a t Test

for dependent samples or paired samples Research has documented the potential effect of students’ positive self-concept on

their self-reflections, attitudes toward self, schoolwork, and general development. A

special program is developed by school psychologists and primary grade teachers to

enhance the self-concept of young-age children. The program is implemented in

Sunny Bay School with four groups of first-and second-grade students. The

intervention lasts six weeks and involves various activities in the class and at home.

The instrument used to assess the effectiveness of the program is comprised of forty

pictures, and scores can range from 0 to 40. The program coordinator conducts a

series of workshops to train five graduate students to administer the instrument. All of

the children in the program are tested before the start of the program and then again

during the first week after the end of the program. A t test for paired samples is used

to test the hypothesis that students’ self-concept would improve significantly on the

posttest, compared with their pretest scores. The research hypothesis is:

HA: Mean POSTTEST > Mean PRETEST

Ho: Mean POSTTEST = Mean PRETEST

Review 6 Steps for Significance

Testing

1. Set alpha (p level).

2. State the Research

question

3. Set up hypotheses,

Null and Alternative.

4. Calculate the test

statistic (sample

value).

4. Find the critical

value of the statistic.

5. State the decision

rule.

6. State the conclusion.

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Pretest and Posttest Scores of Eight Students

Pretest

X1

Posttest

X2

Posttes

t –

Pretest

D

(Posttest-Pretest)2

D2

30 31 +1 1

31 32 +1 1

34 35 +1 1

32 40 +8 64

32 32 0 0

30 31 +1 1

33 35 +2 4

34 37 +3 9

D= 17 D2 =81

The table shows the numerical data we

used to compute the t value for the eight

students selected at random from the

program participants.

The table shows the pretest and posttest

scores for each students, as well as the

means on the pretest and posttest.

The third column in the table shows the

difference between each pair of scores

(D) and is created by subtracting the

pretest from the posttest for each

participant.

The gain scores are then squared and

recorded in the fourth column (D2).

The scores in these last two columns are

added up to create ∑D and ∑D2

respectively. These values are used in the

computation of the t value:

Dependent Samples t Example

1. Set alpha = .05

2. Null hypothesis: H0: Posttest = Pretest. Alternative is HA: Mean Posttest > Pretest.

3. Calculate the test statistic:

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Dependent Samples t Example 4. Determine the critical value of t. Alpha =.05, one tailed

test df = N(pairs)-1 =8-1=7 ; Critical value is 1.895

5. Decision rule: The obtained t value of 2.37 exceeds the

critical values of 1.895 for one tailed test under p=0.05

6. Conclusion. We reject the null hypothesis that states that

there is no difference between the pretest and the posttest

scores. The chance that these results were obtained purely

by chance is less than 5 percent. We confirm the research

hypothesis that predicted that the posttest mean score

would be significantly higher than the pretest mean score.

Table A section from the Table of Critical

Value for t Tests

Level of significance (p value) for one-tailed test

p values 0.10 0.05 0.025 0.01 0.005

Level of significance (p value) for two-tailed test

p values 0.20 0.10 0.05 0.02 0.01

df=7 1.415 1.895 2.365 2.998 3.499 1.895

0.05

df=7

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Interpretation

According to this study, the self-concept

enhancement program was effective in

increasing the self-concept of the first-and

second-grade students.

Our data seem to indicate that the intervention to increase the

self-concept of the primary grade students was effective.

However, those conducting the research or those reading

about it should still decide for themselves whether the

intervention is worth while. The question to be asked is

whether an increase of 2.13 (34.13-32.00) points (out of

forty possible points on the scale) is worth the investment of

time, money, and effort.

Using SPSS for dependent t-test

• Open SPSS

• Open file “SPSS Examples” (same as before)

• Go to:

– “Analyze” then “Compare Means”

– Choose “Paired samples t-test”

– Choose the two IV conditions you are

comparing. Put in “paired variables box.”

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Dependent t- SPSS output

Paired Samples Statistics

Mean N Std.

Deviation Std. Error

Mean Pair 1 Pretest 32.00 8 1.604 .567

Postest 34.13 8 3.227 1.141

Paired Samples Correlations

N Correlation Sig. Pair 1 Pretest & Postest 8 .635 .091

Paired Samples Test

Paired Differences

95% Confidence Interval of the

Difference

Mean

Std. Deviation

Std. Error Mean Lower Upper

Pair 1

Pretest - Postest

-2.125 2.532 .895 -4.242 -.008

Paired Samples Test

t df

Sig. (2-tailed)

Pair 1

Pretest - Postest

-2.374 7 .049

Task of t-test for dependent samples

The data below shows the lengths ( in arbitrary units)

of syllables containing a particular vowel in two

different environments. Each of the ten subjects was

asked to read two sentences, each containing the

vowel in one environment, and a pair of lengths was

thus obtained for each subject. The researcher predicts

greater mean length in environment 2 than in

environment 1.

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Subject no. Environment 1 Environment 2

1 22 26

2 18 22

3 26 27

4 17 15

5 19 24

6 23 27

7 15 17

8 16 20

9 19 17

10 25 30

Table of Lengths of a vowel in two

environments

Do 6 Steps for Significance Testing

1. Set alpha (p level).

2. State the Research

question

3. Set up hypotheses,

Null and Alternative.

4. Calculate the test

statistic (sample

value).

4. Find the critical

value of the statistic.

5. State the decision

rule.

6. State the conclusion.

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t-Tests for a Single

Sample

t-Test for a Single Sample

• A t test for single sample is used when the mean of a sample

is compared to the mean of a population.

• Occasionally, a researcher is interested in comparing a

single group (a sample) to a larger group (a population).

• For example, a high school teacher of a freshmen-

accelerated English class may want to confirm that the

students in that class had obtained higher scores on an

English placement test compared with their peers.

• In order to carry out this kind of a study the researcher must

know prior to the start of the study the mean value of the

population. In this example, the mean score of the

population is the overall mean of the scores of all freshmen

on the English placement test.

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Computing t-Test for Single sample

• The t value is computed using the formula:

An example of a t Test

for Single sample A kindergarten teacher in a school commented to her colleague that the students in

her class this year seem to be less bright than those she had in the past. Her colleague

disagrees with her, and to test whether the first-graders this year are really different

from those in previous years, they conduct a test for a single sample. The scores used

are from the Wechsler Preschool and Primary Scale of Intelligence Third Edition

(WPPSI-III), which is given every year to all kindergarten students in the district. In

this example, we consider the district to be the population to which we compare the

mean of the current kindergarten class. Although the mean IQ score of the population

at large is 100 (µ=100), this district’s mean IQ score is 110 (µ= 110), and this mean is

used in the analysis. The research hypothesis is stated as a null hypothesis and

predicts that there is no statistically significant difference in the mean IQ score of this

year’s kindergarten students (the sample) and the mean IQ score of all kindergarten

students in the district (the population) that were gathered and recorded over the last

three years.

Ho: Mean Class = Mean District

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Review 6 Steps for Significance

Testing

1. Set alpha (p level).

2. State the Research

question

3. Set up hypotheses,

Null and Alternative.

4. Calculate the test

statistic (sample

value).

4. Find the critical

value of the statistic.

5. State the decision

rule.

6. State the conclusion.

Table IQ scores of Ten Students

Scores

115 118

135 113

105 98

107 120

112 99

∑X= 1,122

¯X=112.20

SD= 10.94

In order to test their hypothesis, the two teachers randomly select IQ scores of ten students from this year’s kindergarten class.

These IQ scores are listed in the table followed by the computation of the t value.

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Dependent Samples t Example 1. Is there any differences between this year’s

kindergarten class and the “typical kindergarten students in the district?

2. Null hypothesis: H0: Class = District : there

is no significant difference between this year’s

kindergarten class and the “typical” kindergarten

students in the district.

3. Alternative is HA: Class ≠ District. 4. Calculate the test statistic:

Single Samples t Example 4. Determine the critical value of t. Alpha =.05, two tailed

test df = N(pairs)-1 =10-1=9 ; Critical value is 1.895

5. Decision rule: The obtained t value of 0.64 does not

exceeds the critical values of 2.262 for two tailed test

under p=0.05 (because our hypothesis is non-directional)

6. Conclusion. We retain the null hypothesis that states that

there is no significant difference between this year’s

kindergarten class and the “typical” kindergarten students

in the district.

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Table A section from the Table of Critical

Value for df=9

Level of significance (p value) for one-tailed test

p values 0.10 0.05 0.025 0.01 0.005

Level of significance (p value) for two-tailed test

p values 0.20 0.10 0.05 0.02 0.01

df=9 1.383 1.833 2.262 2.821 3.250 2.262

0.05

df=9

Interpretation

The mean IQ score of this year’s

kindergarten students (mean= 112.20) is

actually slightly higher than the mean

score of the district (mean=110).

The research hypothesis that was

stated in a null form (i.e., predicting

no difference between two means) is

confirmed.

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Using SPSS for dependent t-test

• Open SPSS

• Open file “SPSS Examples” (same as before)

• Go to:

– “Analyze” then “Compare Means”

– Choose “Paired samples t-test”

– Choose the two IV conditions you are

comparing. Put in “paired variables box.”

Single Sample- SPSS output

One-Sample Statistics

N Mean Std. Deviation Std. Error Mean IQ Scores 10 112.20 10.942 3.460

One-Sample Test

Test Value = 0

95% Confidence Interval

of the Difference

t df Sig. (2-tailed)

Mean Difference Lower Upper

IQ Scores 32.425 9 .000 112.200 104.37 120.03

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Task of t-test for Single samples A report published by the Antaretic Academy of

Neurolinguistics indicates the left-handed people may be

better language learners than right-handed people. To test this

hypothesis, you did the following experiments:

In an ESL center, 250 students were randomly selected

and then assigned to two groups (left-handed vs. right-

handed). After equal amounts of instruction, you administered

a battery of language tests to all students. The information you

obtained was:

If the mean on the test for the population of ESL students is 50

a. Test whether left-handed people are better than the

population

b. Test whether right-handed people are better than the

population.

Review 6 Steps for Significance

Testing

1. Set alpha (p level).

2. State the Research

question

3. Set up hypotheses,

Null and Alternative.

4. Calculate the test

statistic (sample

value).

4. Find the critical

value of the statistic.

5. State the decision

rule.

6. State the conclusion.

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References Main Sources

Coolidge, F. L.2000. Statistics: A gentle introduction. London: Sage.

Kranzler, G & Moursund, J .1999. Statistics for the terrified. (2nd ed.). Upper Saddle

River, NJ: Prentice Hall.

Butler Christopher.1985. Statistics in Linguistics. Oxford: Basil Blackwell.

Hatch Evelyn & Hossein Farhady.1982. Research design and Statistics for Applied Linguistics.

Massachusetts: Newbury House Publishers, Inc.

Ravid Ruth.2011. Practical Statistics for Educators, fourth Ed. New York: Rowman &

Littlefield Publisher, Inc.

Quirk Thomas. 2012. Excel 2010 for Educational and Psychological Statistics: A Guide

to Solving Practical Problem. New York: Springer.

Other relevant sources

Agresi A, & B. Finlay.1986. Statistical methods for the social sciences. San Francisco,

CA: Dellen Publishing Company.

Bachman, L.F. 2004. Statistical Analysis for Language Assessment. New York: Cambridge University

Press.

Field, A. (2005). Discovering statistics using SPSS (2nd ed.). London: Sage.

Moore, D. S. (2000). The basic practice of statistics (2nd ed.). New York: W. H.

Freeman and Company.

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