t-Tests

Overview of t-Tests How a t-Test Works How a t-Test Works Single-Sample t Single-Sample t Independent Samples t Independent Samples t Paired t Paired t Effect Size Effect Size

How a t-Test Works The t-test is used to compare means. The t-test is used to compare means. The difference between means is divided by a standard error The difference between means is divided by a standard error The t statistic is conceptually similar to a z-score. The t statistic is conceptually similar to a z-score.

How a t-Test Works

The t-Test as Regression b o is the mean of one group b o is the mean of one group b 1 is the difference between means b 1 is the difference between means If b 1 is significant, then there is a significant difference between means If b 1 is significant, then there is a significant difference between means

Single Sample t-test Compare a sample mean to a hypothesized population mean (test value based on previous research or norms) Compare a sample mean to a hypothesized population mean (test value based on previous research or norms)

Assumptions for Single-Sample t 1. Independent observations. 1. Independent observations. 2. Population distribution is symmetrical. 2. Population distribution is symmetrical. 3. Interval or ratio level data. 3. Interval or ratio level data.

Sampling Distribution of the Mean The t distribution is symmetrical but flatter than a normal distribution. The t distribution is symmetrical but flatter than a normal distribution. The exact shape depends on degrees of freedom The exact shape depends on degrees of freedom

normal distribution t distribution

Degrees of Freedom Amount of information in the sample Amount of information in the sample Changes depending on the design and statistic Changes depending on the design and statistic For a one-group design, df = N-1 For a one-group design, df = N-1 The last score is not “free to vary” The last score is not “free to vary”

Independent Samples t-test Also called: Unpaired t-test Also called: Unpaired t-test Use with between-subjects, unmatched designs Use with between-subjects, unmatched designs

Sampling Distribution of the Difference Between Means We are collecting two sample means and finding out how big the difference is between them. We are collecting two sample means and finding out how big the difference is between them. The mean of this sampling distribution is the Ho difference between population means, which is zero. The mean of this sampling distribution is the Ho difference between population means, which is zero.

 1 -  2 x 1 -x 2 sampling distribution of the difference between means

Independent Samples t -test Assumptions Interval/ratio data Interval/ratio data Normal distribution or N at least 30 Normal distribution or N at least 30 Independent observations Independent observations Homogeneity of variance - equal variances in the population Homogeneity of variance - equal variances in the population

Levene’s Test Test for homogeneity of variance Test for homogeneity of variance If the test is significant, the variances of the two populations should not be assumed to be equal If the test is significant, the variances of the two populations should not be assumed to be equal

Independent Samples t-test Interpretation Sign of t depends on the order of entry of the two groups Sign of t depends on the order of entry of the two groups df = N 1 + N df = N 1 + N Use Bonferroni correction for multiple tests Use Bonferroni correction for multiple tests Divide alpha level by the number of tests Divide alpha level by the number of tests

Paired t-Test Also called: Dependent Samples or Related Samples t-test Also called: Dependent Samples or Related Samples t-test Compares two conditions with paired scores: Compares two conditions with paired scores: Within subjects design Within subjects design Matched groups design Matched groups design

Paired Samples t-Test Assumptions Interval/ratio data Interval/ratio data Normal distribution or N at least 30 Normal distribution or N at least 30 Independent observations Independent observations

Paired Samples t-test - Interpretation The sign of the t depends on the order in which the variables are entered The sign of the t depends on the order in which the variables are entered df = N-1 df = N-1 Use Bonferroni correction for multiple tests Use Bonferroni correction for multiple tests

Effect Size Statistical significance is about the Null Hypothesis, not about the size of the difference Statistical significance is about the Null Hypothesis, not about the size of the difference A small difference may be significant with sufficient power A small difference may be significant with sufficient power A significant but small difference may not be important in practice A significant but small difference may not be important in practice

Effect Size with r 2 Compute the correlation between the independent and dependent variables Compute the correlation between the independent and dependent variables This will be a point-biserial correlation This will be a point-biserial correlation Square the r to get the proportion of variance explained Square the r to get the proportion of variance explained

Computing r 2 from t

Example APA Format Sentence A paired samples t-test indicated a significant difference between the number of incorrect items (M = 2.64, SD = 2.54) and the number of lures recalled (M = 3.30, SD = 1.83), t(97) = 2.54, p =.013, r 2 =.06. A paired samples t-test indicated a significant difference between the number of incorrect items (M = 2.64, SD = 2.54) and the number of lures recalled (M = 3.30, SD = 1.83), t(97) = 2.54, p =.013, r 2 =.06.

Take-Home Points Every t-test compares a systematic difference to a measure of error Every t-test compares a systematic difference to a measure of error Effect size should be reported along with whether a difference is significant Effect size should be reported along with whether a difference is significant