Behavioral Stats Textbook Chapter 11 Notes PDF

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Chapter 11: The Independent-Samples t Test Introduction • We can “de-bias” many experiments by equalizing initial conditions through random assignment to independent groups. • Single-sample t test: Comparing one sample to a population for which we know the mean but not the standard deviation • Paired-samples t test: Before/after design in which the same participants are in both groups. • Independent-samples t test: A two-group study in which each participant is in ONLY one group. The scores for each group are independent of what happens in the other group. ◦ Aka between-groups t test Conducting an Independent-Samples t Test • The independent-samples t test is used to compare two means for a betweengroups design, a situation in which each participant is assigned to only one condition. • An independent-samples t test is used when we have two groups and a between-groups research design - that is, every participant is in only one of the two groups. • This test uses a distribution of differences between means. • It takes a bit more work when calculating a t test by hand. **You have to estimate the appropriate standard error.** A Distribution of Differences Between Means • Because we have different people in each condition of the study, we cannot create a difference score for each person. • A distribution of difference between means: Looking at the overall differences between two independent groups. ◦ There are two samples, so there are two sample means, but we’re building just one distribution of differences between those means. ◦ Notice that we don’t need to have the same number of participants in each sample, although it is best if the sample sizes are fairly similar. • Find the mean of first sample, find the mean of the second sample, then subtract second mean from the first mean. ◦ After doing this multiple times, eventually, we would have many differences between means—some positive, some negative, and some right at 0—and could plot them on a curve.

The Six Steps of the Independent-Samples t Test • STEP 1: Identify the populations, distribution, and assumptions. ◦ Two populations ◦ Comparison distribution = Distribution of differences between means ◦ As usual, the comparison distribution is based on the null hypothesis. As with the paired-samples t test, the null hypothesis for the independent-samples t test posits no mean difference. So the mean of the comparison distribution would be 0; this reflects a mean difference between means of 0. ◦ Same three assumptions as single-sample t test and paired-samples t test ‣ Assumption 1: The dependent variable is assessed using a scale measure. ‣ Assumption 2: The participants are randomly selected. ‣ Assumption 3: The distribution of the population of interest must be approximately normal. • STEP 2: State the null and research hypothesis. ◦ This step for an independent-samples t test is identical to that for the previous t tests. ◦ Null hypothesis: H0: μ1 = μ2 ◦ Research hypothesis: H1: μ1 ≠ μ2 • STEP 3: Determine the characteristics of the comparison distribution. ◦ Determine the appropriate mean and the appropriate standard error of the comparison distribution - the distribution based on the null hypothesis. ◦ According to the null hypothesis, no mean difference exists between the populations; that is, the difference between means is 0. So the mean of the comparison distribution is always 0, as long as the null hypothesis posits no mean difference ◦ Steps to calculating the approximate measure of spread of the distribution

‣ A) Calculate the corrected variance for each sample. • Steps to calculate corrected variance ◦ 1) Calculate the variance for X. Symbol = s^2 X ◦ 2) Repeat for Y.

‣ B) Pool the variances - Using the degrees of freedom, we calculat a sort of average variance. Pooled variance is a weighted average of the two estimates of variance—one from each sample—that are calculated when conducting an independent-samples t test. • Steps to calculate pooled variance: ◦ 1) Calculating the proportion of degrees of freedom represented by each sample. ◦ 2) We also calculate a total degrees of freedom that sums the degrees of freedom for the two samples.

◦ 3) Use the pooled variance formula.

◦ (Note: If we had exactly the same number of participants in each sample, this would be an unweighted average—that is, we could compute the average in the usual way by summing the two sample variances and dividing by 2.) ‣ C) Convert the pooled variance from squared standard deviation (variance) to squared standard error (another version of variance) by dividing the pooled variance by the sample size, first for one sample and then again for the second sample.

‣ D) Add the two variances (squared standard errors), one for each distribution of sample means, to calculate the estimated variance of the distribution of differences between means.

‣ E) Calculate the square root of this form of variance (squared standard error) to calculate the standard deviation of the distribution of differences between means.

• Summary of calculating spread of the distribution of difference between means

• STEP 4: Determine critical values, or cutoffs. ◦ Use the total degrees of freedom, df total

• STEP 5: Calculate the test statistic. ◦ Long version of formula:

◦ Because the population difference between means (according to the null hypothesis) is almost always 0, many statisticians choose to eliminate the latter part of the formula. So the formula for the test statistic is often abbreviated as:

• STEP 6: Make a decision. ◦ Fail to reject the null hypothesis ◦ Reject the null hypothesis ◦ If we reject the null hypothesis, we need to examine the means of the two conditions so that we know the direction of the effect.

Reporting the Statistics • To report the statistics as they would appear in a journal article, follow standard APA format. • Be sure to include the degrees of freedom, the value of the test statistic, and the p value associated with the test statistic. ◦ Example:

• In addition to the results of hypothesis testing, we would also include the means and standard deviations for the two samples. ◦ Example:

Reviewing the Concepts: • When we conduct an independent-samples t test, we cannot calculate individual difference scores. That is why we compare the mean of one sample with the mean of the other sample. • The comparison distribution is a distribution of differences between means. • We use the same six steps of hypothesis testing that we used with the z test and with the single-sample and paired-samples t tests. • Conceptually, the t test for independent samples makes the same comparisons as the other t tests. However, the calculations are different, and the critical values are based on degrees of freedom from two samples.

Beyond Hypothesis Testing • Two ways that researchers can evaluate the findings of a hypothesis test are by calculating a confidence interval and an effect size.

Calculating a Confidence Interval for an Independent-Samples t Test • Confidence intervals for an independent-samples t test are centered around the difference between means (rather than the means themselves). ◦ Population mean difference = (μX − μY) ◦ Sample mean difference = (MX − MY)sample ◦ Standard error for the difference between means = s difference • Formulas:

Steps to Calculating Confidence Interval for an Independent-Samples t Test • STEP 1: Draw a normal curve with the sample difference between means in the center. • STEP 2: Indicate the bounds of the confidence interval on either end, and write the percentages under each segment of the curve.

• STEP 3: Look up the t statistics for the lower and upper ends of the confidence interval in the t table. • STEP 4: Convert the t statistics to raw differences between means for the lower and upper ends of the confidence interval. ◦ Example:

◦ The confidence interval is [−2.96, −0.04].

• STEP 5: Check your answer. ◦ Each end of the confidence interval should be exactly the same distance from the sample mean. ◦ Example:

◦ The interval checks out. The bounds of the confidence interval are calculated as the difference between sample means, plus or minus 1.46. ◦ Also, the confidence interval does not include 0; thus, it is not plausible that there is no difference between means. ◦ When we conducted the independent-samples t test earlier, we rejected the null hypothesis and drew the same conclusion as we did with the confidence interval. But the confidence interval provides more information because it is an interval estimate rather than a point estimate. • ****FORMAT OF HOW TO WRITE CONFIDENCE INTERVAL**** ◦ The 95% confidence interval around the difference between means of _____ is [___,___]. ◦ The 95% confidence interval around the observed mean difference of _____ is [____, ____].

Calculating Effect Size for an Independent-Samples t Test • As with all hypothesis tests, it is recommended that the results be supplemented with an effect size. • For an independent-samples t test, as with other t tests, we can use Cohen’s d as the measure of effect size. • Pooled standard deviation = square root of the pooled variance

• Formula:

• EXAMPLE:

Cohen’s Conventions for Effect Sizes: d

The Bayesian Approach to Data Analysis • Bayes (1763) was a non-conformist minister whose statistical idea proposed that we should not just rely on the data at hand, but should also consider what we already know. • A modern-day proponent of Bayesian analysis, points out that in real life we examine new data and determine “how much we should change our beliefs relative to our prior beliefs.” • So Bayesian statistics help us to take both prior beliefs—“priors” for short—and probabilities into account. • The traditional approach to statistical inference assumes that we never learn from prior experience—the comparison distribution is always some version of chance. ◦ For example, with an independent-samples t test, we typically analyze the data against a null hypothesis of no difference, regardless of what previous research has found.

Reviewing the Concepts: • A confidence interval can be created with a t distribution around a difference between means. • We can calculate an effect size, Cohen’s d, for an independent-samples t test. • Bayesian statistics approach statistical analysis in a different way from traditional hypothesis testing. The main difference is that they take both prior beliefs—“priors” for short—and probabilities into account, whereas traditional hypothesis testing is based just on probabilities.

Conducting an Independent-Samples t Test • We use independent-samples t tests when we have two samples and different participants are in each sample. • Because the samples comprise different people, we cannot calculate difference scores, so the comparison distribution is a distribution of differences between means. • Because we are working with two separate samples of scores (rather than one set of difference scores) when we conduct an independent-samples t test, we need additional

steps to calculate an estimate of spread. • As part of these steps, we calculate estimates of variance from each sample, and then combine them to create a pooled variance.

Beyond Hypothesis Testing • A confidence interval can be created around a difference between means using a t distribution. • To understand the importance of a finding, we must also calculate an effect size. • Bayesian statistics approach statistical analysis in a different way from traditional hypothesis testing. The main difference is that they take both prior beliefs—“priors” for short—and probabilities into account, whereas traditional hypothesis testing is based just on probabilities.

Reporting Statistics in APA style • Fail to reject the null hypothesis (statistically not significant) ◦ t(df total) = ______ , p > 0.05 • Reject the null hypothesis (statistically significant) ◦ t(df total) = ______ , p < 0.05...