Single and Dependent Samples - Statistics for the Behavioral Sciences - Lecture Slides, Slides of Behavioural Science

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Statistics for the Behavioral

Sciences

Single Sample t-Test

Dependent Sample t-Test

Student’s t-Test

 William Sealy Gossett published

under the name “Student” but was a chemist and executive at Guiness Brewery until 1935.

Comparison to Normal Distribution

 Both are symmetrical, unimodal, and bell-shaped.

 When df are infinite, the t distribution is the normal distribution.

 When df are greater than 30, the t distribution closely approximates it.

 When df are less than 30, higher frequencies occur in the tails for t.

The Shape Varies with the df (k)

Smaller df produce larger tails

Finding Critical Values of t

 Use the t-table NOT the z-table.

 Calculate the degrees of freedom.

 Select the significance level (.05, .01).

 Look in the column corresponding to the df and the significance level.

 If t is greater than the critical value, then the result is significant (reject the null hypothesis).

Link to t-Tables

http://www.statsoft.com/textboo k/sttable.html

Confidence Intervals for t

 Use the same formula as for z but:

 Substitute the t value (from the t- table) in place of z.  Substitute the estimated standard error of the mean in place of the calculated standard error of the mean.

 Mean ± (tconf)(s (^) x )

 Get t (^) conf from the t-table by

selecting the df and confidence level

Assumptions

 Use t whenever the standard

deviation is unknown.

 The t test assumes the underlying

population is normal.

 The t test will produce valid results

with non-normal underlying populations when sample size > 10.

What are Degrees of Freedom?

 Degrees of freedom (df) are the number of values free to vary given some mathematical restriction.

 Example – if a set of numbers must add up to a specific toal, df are the number of values that can vary and still produce that total.

 In calculating s (std dev), one df is used up calculating the mean.

Example

 What number must X be to make

the total 20?

5 100 10 200 7 300 X X 20 20

Free to vary

Limited by the constraint that the sum of all the numbers must be 20

So there are 3 degrees of freedom in this example.

Within Subjects Designs

 Two t-tests, depending on design:

 t-test for independent groups is for Between Subjects designs.  t-test for paired samples is for Within Subjects designs.

 Dependent samples are also called:

 Paired samples  Repeated measures  Matched samples

Examples of Paired Samples

 Within subject designs

 Pre-test/post-test

 Matched-pairs

Dependent Samples

 Each observation in one sample is paired one-to-one with a single observation in the other sample.

 Difference score (D) – the difference between each pair of scores in the two paired samples.

 Hypotheses:

 H 0 : μD = 0 μD ≤ 0  H 1 : μD ≠ 0 μD > 0

Repeated Measures

 A special kind of matching where the same subject is measured more than once.

 This kind of matching reduces variability due to individual differences.