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Inferential Statistics I: The t-test

Experimental Methods and Statistics

Department of Cognitive Science

Michael J. Kalsher

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OutlineDefinitions

Descriptive vs. Inferential Statistics

The t-test

- One-group t-test

- Dependent-groups t-test

- Independent-groups t-test

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The t-test: Basic Concepts

• Types of t-tests- Independent Groups vs. Dependent

Groups• Rationale for the tests

- Assumptions

• Interpretation• Reporting results• Calculating an Effect Size• t-tests as GLM

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William Sealy Gosset (1876–1937)

Famous as a statistician, best known by his pen name Student and for his work on Student's t-distribution.

Beer and Statistics: A Winning Combination!

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The One Group t testThe One-group t test is used to compare a sample mean to a specific value (e.g., a population parameter; a neutral point on a

Likert-type scale).

Examples:1. A study investigating whether stock brokers differ from the general population on

some rating scale where the mean for the general population is known.

2. An observational study to investigate whether scores differ from some neutral point on a Likert-type scale.

Calculation of ty : ty = Mean Difference

Standard Error (of the mean difference)Note: The symbol ty indicates

this is a t test for a single group mean.

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Assumptions

The one-group t test requires the following statistical assumptions:

1. Random and Independent sampling.

2. Data are from normally distributed populations. Note: The one-group t test is generally considered robust against violation of

this assumption once N > 30.

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Computing the one-group t test by hand

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Critical Values: One-Group t testNote: Degrees of Freedom = N - 1

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Computing the one-group t test using SPSS

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Move DV to box labeled “Test variable(s):

Type in “3” as a proxy for the population mean.

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SPSS Output

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Reporting the Results: One Group t test

The results showed that the students’ rated level of agreement with the statement “I feel good about myself” (M=3.4) was not significantly different from the scale’s neutral point (M=3.0), t(4)=.784. However, it is important to note several important limitations with this result, including the use of self-report measures and the small sample size (five participants). Additional research is needed to confirm, or refute, this initial finding.

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11. Select both Time 1 and Time 2, then move to the box labeled “Paired Variables.”

12. Next, “click”, “Paste”.

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The Independent Groups t test: Between-subjects designs

Assumption: Participants contributing to the two means come from different groups; therefore, each person contributes only one score to the data.

Calculation of t: t = Mean Difference

Standard Error (of the mean difference)

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Standard Error: How well does my sample represent the population?

When someone takes a sample from a population, they are taking one of many possible samples--each of which has its own mean (and s.d.).

We can plot the sample means as a frequency distribution or sampling distribution.

Sample Mean

Fre

quen

cy

5

4

3

2

1

0

6

10

Sampling Distribution

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Standard Error: How well does my sample represent the population?

The Standard Error, or Standard Error of the Mean, is an estimate of the standard deviation of the sampling distribution of means, based on the data from one or more random samples.

• Large values tell us that sample means can be quite different, and therefore, a given sample may not be representative of the population.

• Small values tell us that the sample is likely to be a reasonably accurate reflection of the population.

• An approximation of the standard error can be calculated by dividing the sample standard deviation by the square root of the sample size

SE = N

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Standard Error: Applied to Differences

We can extend the concept of standard error to situations in which we’re examining differences between means.

The standard error of the differences estimates the extent to which we’d expect sample means to differ by chance alone--it is a measure of the unsystematic variance, or variance not caused by the experiment.

An estimate of the standard error can be calculated by dividing the sample standard deviation by the square root of the sample size.

SE = N

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Computing the independent-groups t test by hand

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Sample ProblemA college administrator reads an article in USA Today suggesting that liberal arts professors tend to be more anxious than faculty members from other disciplines within the humanities and social sciences. To test whether this is true at her university, she carries out a study to determine whether professors teaching liberal arts courses are more anxious than professors teaching behavioral science courses.

Sample data are gathered on two variables: type of professor and level of anxiety.

Anxiety Scores

Liberal Arts

Behavioral Science

45 58

63 59

62 63

51 68

54 74

63 68

52 52

54 66

64 69

49 57

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Cri

tica

l Val

ues

:

Ind

epen

den

t G

rou

ps

t te

st

Note: Degrees of Freedom = N1 + N2 - 2

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On average, the mean level of anxiety among a sample of liberal arts professors (M = 55.7) was significantly lower than the mean level of anxiety among a sample of behavioral science professors (M = 63.4), t(18) = -2.54, p < .05, r2 = .26. The effect size estimate indicates that the difference in anxiety level between the two groups of professors represents a large effect.

Reporting the Results: Independent Groups t test

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Computing the independent-groups t test using SPSS

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Sample Problem

A researcher is interested in comparing the appetite suppression effects of two drugs, fenfluramine and amphetamine, in rat pups. Five-day-old rat pups are randomly assigned to be injected with one of the two drugs. After injection, pups are allowed to eat for two hours. Percent weight gain is then measured.

Compute the independent groups t-test using the data at right.

Is this a true experiment, quasi-experiment, or observational study?

Percent Weight Gain

Fenfluramine Amphetamine

2 8

3 10

3 4

4 7

4 9

5 3

6 7

6 12

6 6

7 8

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SPSS Output: Independent-Groups t test

Group Statistics

10 4.60 1.647 .521

10 7.40 2.675 .846

DrugType1

2

WeightgainN Mean Std. Deviation

Std. ErrorMean

Independent Samples Test

1.110 .306 -2.819 18 .011 -2.800 .993 -4.887 -.713

-2.819 14.964 .013 -2.800 .993 -4.918 -.682

Equal variancesassumed

Equal variancesnot assumed

WeightgainF Sig.

Levene's Test forEquality of Variances

t df Sig. (2-tailed)Mean

DifferenceStd. ErrorDifference Lower Upper

95% ConfidenceInterval of the

Difference

t-test for Equality of Means

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Calculating Effect Size: Independent Samples t test

r = t2

t2 + df

(-2.819)2

(-2.819)2 + 18

7.95

7.95 + 18

r =.5534

r2 = .306

Note: Degrees of freedom calculated by adding the two sample sizes and then subtracting the number of samples:

df = 10 + 10 – 2 = 18

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On average, the percent weight gain of five-day-old rat pups receiving amphetamine (M = 7.4, SE = .85) was significantly higher than the percent weight gain of rat pups receiving fenfluramine (M = 4.6, SE = .52), t(18) = -2.82, p < .05, r2 = .31. The effect size estimate indicates that the difference in weight gain caused by the type of drug given represents a large, and therefore substantive, effect.

Reporting the Results: Independent Groups t test

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