Independent Samples T-Test

Outline of Today’s Discussion 1.About T-Tests 2.The One-Sample T-Test 3.Independent Samples T-Tests 4.Two Tails or One? 5.Independent Samples T-Test: Excel 6.Independent Samples T-Test: SPSS Please refrain from typing, surfing or printing during our conversation!

Part 1 About T-Tests

Computing the ‘t’ Statistic Recall One of Our Themes: Correlational Research vs Differential Research

Computing the ‘t’ Statistic When we evaluate the difference between ANY two means We can’t just look at the mean-difference alone. We need to consider the mean-difference in Context!!! For t-tests…the context is the variability.

Computing the ‘t’ Statistic Which graph makes a more convincing case for Drug X, and why? For t-tests…the context is the variability.

Computing the ‘t’ Statistic Mantra: T-Tests compare means.

Computing the ‘t’ Statistic Mantra: T-Tests compare means (in the context of variability).

Part 2 The One Sample T-Test

1.In ancient times –before the number 2 was invented- caveman used a one sample t-test! ;) 2.A one sample t-test is an evaluation of a single mean, rather than two means. 3.The sample mean is compared to a ‘test value’ that is of interest to the research. 4.Example 1: Consider the proportion of m&m’s colors in the population. There would be a “test value” for each color…

Test Values In the Population: 24% blue 14% brown 16% green 20% orange 13% red 14% yellow

The One Sample T-Test 1.Example 2: Assume that you are a therapist who has received a ‘sample’ of 20 patients diagnosed with clinical depression. 2.After 5 weeks of your treatment, you might ask whether the mean of your sample is statistically indistinguishable from non-depressed patients, who have a mean of, say, 500 on a standard mood assessment.

The One Sample T-Test 1.The null hypothesis would be as follows: Ho: In the population, the mean mood score of patients who have completed 5 weeks of (my) therapy is equal to that of non- depressed people. 2.In this example, the test value would be equal to whatever the mean of the non-depressed population is…let’s say ‘500’. So, the test value = 500 here.

The One Sample T-Test 1.Example 3: Ho: In the population, the mean age when first married is equal to So, the test value = 20 here. 3.The steps in SPSS are simple, and are virtually identical to the two-mean case…

The One Sample T-Test SPSS Steps: Analyze  Compare Means  One Sample T-Test  Test Value Box  Enter the value that is to be compared to the sample mean The sample mean of 22.79, is compared to our test value, Say, … 20. The ‘ sig ’ value indicates that we should reject Ho, i.e., we should REJECT that the sample mean is equal to the test value.

The One Sample T-Test SPSS Steps: Analyze  Compare Means  One Sample T-Test  Test Value Box  Enter the value that is to be compared to the sample mean The sample mean of 22.79, is compared to our test value, Say, … The ‘ sig ’ value indicates that we should retain Ho, i.e., we should accept that the sample mean is equal to the test value.

The One Sample T-Test SPSS Steps: Analyze  Compare Means  One Sample T-Test  Test Value Box  Enter the value that is to be compared to the sample mean The sample mean of 22.79, is compared to our test value, Say, … The mean difference of is NOT significantly different from zero (Ho). Note: When Ho is true, than the 95% confidence interval contains a zero.

The One Sample T-Test 1.Typically, we prefer to run two samples…one experimental group and one control group. 2.However, that may not always be possible. 3.The one sample t-test allows for a statistical comparison to some abstract standard (the ‘test value’), rather than to a control group.

Part 3 Independent Samples T-Tests

Computing the ‘t’ Statistic Formula for the independent samples “t” statistic What does “x bar” represent, again?

Computing the ‘t’ Statistic Denominator term in the independent samples “t” statistic What does this “s” represent, again?

Computing the ‘t’ Statistic Denominator term in the independent samples “t” statistic Simplified!

Computing the ‘t’ Statistic Degrees of Freedom for the independent samples “t” statistic Where N equals the sum of the two sample sizes (n1 + n2). df = N - 2

Computing the ‘t’ Statistic Which Graph would be associated with the larger ‘t’ statistic, and why?

Computing the ‘t’ Statistic 1.The independent samples “t” statistic is based on 3 assumptions. 2.The first assumption is the distribution of scores should be bell-shaped. 3.The second assumption is that the two populations from which the samples are selected must have (at least approximately) equal variances. 4.The third assumption is independence (the value of any datum does not depend on the any other datum).

Computing the ‘t’ Statistic 1.How might we quantitatively assess the first assumption, i.e., the normalcy assumption? Hint: We’ve done it before (kind of). 2.There are ways to test the equal variance assumption quantitatively. SPSS will help us with that later. 3.The independence assumption requires that we investigate how the data were obtained. (No stats here.) The independence assumption does NOT pertain to the within-subject t-test.

Part 4 Two Tails or One?

1.Here’s are some hypotheses for t-tests H 0 : In the population, the means for the control and experimental groups will be equal. H 1 : In the population, the means for the control and experimental groups will NOT be equal. 2.The alternate hypothesis is said to be “two tailed” because it is non-directional…it does NOT state which of the two means will be larger. Two-Tailed Case

Two Tails or One? Two-Tailed Case.025 is lower than this.025 is higher than this Non-directional hypotheses are evaluated with a two-tailed test! We could reject the null hypothesis whether the observed t-statistic is in either critical region … but the observed “ t ” value must be at least 47.5 percentiles away from the mean! mean = 50 th percentile: = 2.5 percentile (left) : = 97.5 percentile (right)

Two Tails or One? 1.Here’s are some hypotheses for t-tests H 0 : In the population, the means for the control and experimental groups will be equal. H 1 : In the population, the mean for the control group will be greater than the mean for the experimental group. 2.The alternate hypothesis is said to be “one tailed” because it is directional…it states which mean will be larger. One-Tailed Case

Two Tails or One? One-Tailed Case Directional hypotheses are evaluated with a one-tailed test! We can only reject the null hypothesis when the observed t-statistic is in the predicted critical region … but the observed “ t ” value only needs to be at least 45 percentiles away from the mean! mean = 50 th percentile: = 95 percentile (right)

Two Tails or One? 1.When do we use two tails versus one tail? 2.It depends! 3.If you want to make a directional prediction, use one-tail, otherwise use two-tails. 4.Two-tails are often preferred because they are more conservative (47.5 percentiles from mean for two-tailed significance, versus a mere 45 percentiles from mean for one tailed significance).

Part 5 Independent Sample T-Tests in Excel

Part 6 Independent Sample T-Tests in SPSS

Independent Samples T-Tests in SPSS 1.SPSS Question – When conducting t-tests in SPSS, what is a synonym for “Test Variable”? What is a synonym for “Grouping Variable”? 2.SPSS Question– Consider the four types of measurement scales that we discussed earlier this semester. For which scale or scales would a t-test be appropriate? Explain your reasoning.

Independent Samples T-Tests in SPSS 1.SPSS Question – In your own words, explain why it is necessary to consider Levene’s test? 2.SPSS Question – In your own words, explain how to interpret Levene’s test in the SPSS output.