Hypothesis Testing Using the Two-Sample t-Test Chapter 9 Hypothesis Testing Using the Two-Sample t-Test

Going Forward Your goals in this chapter are to learn: The logic of a two-sample experiment The difference between independent samples and related samples When and how to perform the independent-samples t-tests When and how to perform the related-samples t-test What effect size is and how it is measured using Cohen’s d or

Understanding the Two-Sample Experiment

Two-Sample Experiment Participants’ scores are measured under two conditions of the independent variable Condition 1 produces sample mean representing Condition 2 produces sample mean representing

Two-Sample t-Test The parametric statistical procedure for determining whether the results of a two-sample experiment are significant is the two-sample t-test The two versions of the two-sample t-test are The independent-samples t-test The related-samples t-test

Relationship in the Population in a Two-sample Experiment

The Independent Samples t-Test

Independent Samples t-Test The parametric procedure used for testing two sample means from independent samples Independent samples result when we randomly select participants for a condition without regard to who else has been selected for either condition

Assumptions of the Independent Samples t-Test The dependent scores are normally distributed interval or ratio scores. The populations have homogeneous variance. Homogeneity of variance means the variance of the populations being represented are equal.

Statistical Hypotheses For a two-tailed test, the statistical hypotheses are H0 implies both samples represent the same population of scores Ha implies the means from our conditions each represent a different population of scores

Sampling Distribution The sampling distribution of differences between the means is the distribution of all possible differences between two means when both samples are drawn from the one raw score population that H0 says we are representing.

Performing the Independent Samples t-Test Compute the mean and estimated population variance for each condition Remember: The formula for the estimated variance in each condition is

Performing the Independent Samples t-Test Compute the pooled variance using the formula

Performing the Independent Samples t-Test Compute the standard error of the difference. This is the standard deviation of the sampling distribution of differences between means. The formula is

Performing the Independent Samples t-Test Compute tobt for two independent samples using the formula

One-Tailed Tests The statistical hypotheses for a one-tailed test of independent samples are OR If m1 is expected to If m2 is expected to be larger than m2 be larger than m1

One-Tailed Tests Conduct one-tailed tests only when you can confidently predict the direction the dependent scores will change.

One-Tailed Tests Decide which and corresponding m is expected to be larger Arbitrarily decide which condition to subtract from the other Decide whether the difference will be positive or negative Create Ha and H0 to match this prediction Locate the region of rejection Complete the t-test as described previously

Critical Values Critical values for the independent samples t-test (tcrit) are determined based on degrees of freedom df = (n1 – 1) + (n2 – 1), the selected a, and whether a one-tailed or two-tailed test is used

Interpreting the Independent-Samples t-Test In a two-tailed t-test of independent samples, reject H0 if tobt is greater than (beyond) +tcrit or if tobt is less than (beyond) –tcrit Otherwise, fail to reject H0

The Related Samples t-Test

Related-Samples t-Test The related-samples t-test is used when we have two sample means from two related samples Related samples occur when we pair each score in one sample with a particular score in the other sample Two types of research designs producing related samples are the matched-samples design and the repeated-measures design

Matched-Samples Design The researcher matches each participant in one condition with a particular participant in the other condition We do this so we have more comparable samples

Repeated-Measures Design Each participant is tested under both conditions of the independent variable That is, each participant is measured under condition 1 and again under condition 2

Transforming the Raw Scores In a related samples t-test, the raw scores are transformed by finding each difference score The difference score is the difference between the two raw scores in a pair The symbol for a difference score is D

Statistical Hypotheses The statistical hypotheses for a two-tailed related-samples t-test are

Sampling Distribution The sampling distribution of mean differences shows all possible values of the population mean of the difference scores ( ) that occur when samples are drawn from the population of difference scores that H0 says we are representing.

Performing the Related-Samples t-Test Compute the estimated variance of the difference scores ( ) using the formula where N equals the number of difference scores

Performing the Related-Samples t-Test Compute the standard error of the mean difference ( ) using the formula

Performing the Related-Samples t-Test Find tobt using the formula

One-Tailed Tests The statistical hypotheses for a one-tailed t-test of related samples are If we expect the If we expect the difference to be difference to be larger than 0 less than 0

Critical Values The critical value (tcrit) is determined based on degrees of freedom df = N – 1 where N is the number of difference scores the selected a, and whether a one-tailed or two-tailed test is used

Interpreting the Related-Samples t-Test In a two-tailed test of related samples, reject H0 if tobt is greater than (beyond) +tcrit or if tobt is less than (beyond) –tcrit Otherwise, fail to reject H0

Describing Effect Size

Effect Size Effect size indicates the amount of influence changing the conditions of the independent variable had on dependent scores The larger the effect size, the more scientifically important the independent variable is

Computing Effect Size Cohen’s d is used to compute effect size Independent Samples t-Test Related Samples t-test

Interpreting Effect Size We interpret the Cohen’s d using a small, medium, or large effect size classification d = 0.2 is a small effect d = 0.5 is a medium effect d = 0.8 is a large effect

Proportion of Variance Accounted For The proportion of variance accounted for is the proportion of the differences in scores that can be attributed to changing the conditions in the independent variable We use the formula for the squared point-biserial correlation coefficient

Example 1 Using the following data set, conduct an independent-samples t-test. Use a = 0.05 and a two-tailed test. Sample 1 Sample 2 14 13 15 11 10 12 17

Example 1

Example 1 The standard error of the difference is

Example 1

Example 1 tcrit for df = (9 – 1) + (9 – 1) = 16 with a = .05 and a two-tailed test is 2.120. Reject H0 if tobt is greater than +2.120 or if tobt is less than –2.120. Because tobt of – 0.285 is not beyond the –tcrit of –2.120, it does not lie within the rejection region. We fail to reject H0.

Example 2 Using the following data set, conduct a related-samples t-test. Use a = 0.05 and a two-tailed test. Sample 1 Sample 2 14 13 15 16 10 12 17 18 19

Example 2 First, we find the differences between the matched scores Sample 1 Sample 2 Differences 14 13 15 16 -1 -2 10 12 -3 -4 17 18 19

Example 2

Example 2 Using a = 0.05 and df = 8, tcrit = 2.306. Reject H0 if tobt is greater than +2.306 or if tobt is less than –2.306. Because tobt of –6.260 is beyond the –tcrit value of –2.306, it lies within the rejection region. We reject H0.

Example 2 Effect size

Example 2 Proportion of variance accounted for