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Week 9:Independent t -test

t test for Two Independent Samples

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Independent Samples t - test

The reason for hypothesis testing is to gain knowledge about an unknown population.

Independent samples t-test is applied when we have two independent samples and want to make a comparison between two groups of individuals. The parameters are unknown.

How is this different than a Z-test and One Sample t-test?

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Independent t - test We are interested in the difference

between two independent groups. As such, we are comparing two populations by evaluating the mean difference.

In order to evaluate the mean difference between two populations, we sample from each population and compare the sample means on a given variable.

Must have two independent groups (i.e.samples) and one dependent variable that is continuous to compare them on.

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Examples:

Do males and females significantly differ on their level of math anxiety?

IV: Gender (2 groups: males and females)DV: Level of math anxiety Do older people exercise significantly

less frequently than younger people?IV: Age (2 groups: older people and

younger people)DV: Frequency of getting exercise

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Examples: Do 8th graders have significantly more

unexcused absences than 7th graders in Toledo junior highs?

IV: Grade (2 groups: 8th grade and 7th grade)

DV: Unexcused absences Note that Independent t-test can be

applied to answer each research question when the independent variable is dichotomous with only two groups and the dependent variable is continuous.

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Generate examples of research questions requiring an Independent Samples t-test:

What are some examples that you can come up with? Remember- you need two independent samples and one dependent variable that is continuous.

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Assumptions The two groups are independent of one another.

The dependent variable is normally distributed. Examine skewness and kurtosis (peak) of distribution

Leptokurtosis vs. platykurtosis vs. mesokurtosis

The two groups have approximately equal variance on the dependent variable. (When [equal sample sizes] ,the violation of this assumption has been shown to be unimportant.)

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Steps in Independent Samples t-test

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Step 1: State the hypotheses

Ho: The null hypothesis states that the two samples come from the same population. In other words, There is no statistically significant difference between the two groups on the dependent variable.

Symbols:

Non-directional: Ho: μ1 = μ2

Directional: or

• If the null hypothesis is tenable, the two group means differ only by sampling fluctuation – how much the statistic’s value varies from sample to sample or chance.

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:0 H21

:0 H

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Ha: The alternative hypothesis states that the two samples come from different populations. In other words, There is a statistically significant difference between the two groups on the dependent variable.

Symbols:Non-directional:

Directional:

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:1 H

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:1 H

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:1 H

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Step 2: Set a Criterion for Rejecting Ho

Compute degrees of freedom Set alpha level Identify critical value(s)

Table C. 3 (page 638 of text)

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Computing Degrees of Freedom

Calculate degrees of freedom (df) to determine rejection region.

df = sample size for sample1+ sample size for sample2 - 2• df describe the number of scores in a sample

that are free to vary. • We subtract 2 because in this case we have 2

samples.

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More on Degrees of Freedom

• In an Independent samples t-test, each sample mean places a restriction on the value of one score in the sample, hence the sample lost one degree of freedom and there are n-1 degrees of freedom for the sample.

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Set alpha level

Set at .001, .01 , .05, or .10, etc.

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Identify critical value(s) Directional or non-directional? Look at page 638 Table C.3. To determine your CV(s) you need

to know: df – if df are not in the table, use the

next lowest number to be conservative

directionality of the test alpha level

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Step 3: Collect data and Calculate t statistic

nnnnnsnsxxt

2121

2

2

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2

1

21

112

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Whereby: n: Sample size s2 = variance :Sample mean subscript1 = sample 1 or group 1

subscript2 = sample 2 or group 2

x df

variance

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Step 4: Compare test statistic to criterion

df = 18 α = .05 , two-tailed test in this example• critical values are ± 2.101 in this example

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Step 5: Make Decision

Fail to reject the null hypothesis and conclude that there is no statistically significant difference between the two groups on the dependent variable, t = , p > α.

OR

Reject the null hypothesis and conclude that there is a statistically significant difference between the two groups on the dependent variable, t = , p < α.

• If directional, indicate which group is higher or lower (greater, or less than, etc.).

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Interpreting Output Table:

Retrieved on July 12, 2007 from SPSSShortManual.html

t-value

Degrees of freedom

p - value

Levene’s tests the assumption of equal variances – if p < .05, then variances are not equal and use a different test to modify this:

Here, we have met the assumption so use first row.

Observed difference between the groups

Mean APGAR SCORE

CI

Sample size

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Interpreting APA table:

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Variable Math anxiety t

Gender

Male 3.66

Female 3.98 3.35***

Age

Under 40 years

3.32

Over 41 years 3.64 2.67**Note. **p < .01. ***p < .001.

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Examples and Practice See attached document. Create the following index cards from

this lecture: When to conduct a t-test (purpose,

conditions, and assumptions) t-test statistic formula for computation

t-test statistic formula df formula