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# T-Tests

## Parametric Statistics

To addresses parametric tests, which allow researchers to make assumptions regarding estimates of traits or characteristics manifested by specific populations. These assumptions usually involve the mean or standard deviation of the population when the distribution is normal and the research employs interval or ratio data. Conversely, nonparametric tests do not involve the utilization of any population parameters, so the mean and the standard deviation are not necessary, and they most often employ nominal or ordinal level data (Jackson, 2011).

In a two-group research design, there are two samples that represent two populations, which are compared by having the control group not receive anything (no treatment), while the other experimental group receives some level of the designated independent variable. In a two-group design, it is also possible to incorporate two experimental groups without a control group, where members of each group receive different levels of the independent variable. For example, in a drug trial, one experimental group may receive dosages of 5 mg while the other experimental group may receive dosages of 10 mg (Jackson, 2011).

There are two types of t-tests covered here: the t-test for independent groups and the t-test for correlated groups. The t-test for independent groups compares the sample means of two independent groups of scores, while the t-test for correlated groups compares the means of two related groups or samples (within-participants or matched-participants). The independent-groups t-test indicates whether the performance of the two samples is so similar that the researchers conclude that they are likely to have originated from the same population, or the performance of the two samples is so different that the researchers conclude that they represent two different populations. The correlated-groups t-test also compares the performance of the subjects in two groups; however, the same subjects are utilized in each group in a within-participants design, or different subjects are utilized in each group in a matched-participants design. The t-test indicates whether a difference in the sample means exists and whether that difference is based on chance. In a correlated-groups t-test, there are two scores for each person or for a matched pair if it is a matched-participants research design instead of the usual one score per participant (Jackson, 2011).

Each type of t-test is based upon its own set of assumptions where the assumptions of the t-test for independent groups are:

2 PSY 223 Module Four

Interval-ratio data

Bell-shaped distributions

Independent observations

Homogeneous variance

In contrast, the assumptions of the t-test for correlated groups are:

Interval-ratio data

Bell-shaped distributions

Dependent observations

Homogeneous variance

Note that the only difference between the two types of t-tests is the observations (Jackson, 2011).

## Reference

Jackson, S. L. (2011). Research methods: A modular approach. Belmont, CA: Wadsworth Cengage Learning.

**Under what circumstances would it be appropriate for researchers to conduct a t-test for independent groups? When would it be appropriate to conduct a t-test for correlated groups? **

A *t* test is a” hypothesis-testing procedure in which the population variance is unknown; it compares t scores from a sample to a comparison distribution called a *t* distribution” (Aron, Coups, & Aron, 2013, p. 227). A researcher could decide to conduct a t-test for independent groups under circumstances where the researcher needs to compare one independent variable with two different characteristics. For instance t-tests are often used to compare scores and grades between different genders. One example of this would be a researcher deciding to use a t-test to discover which gender has a higher G.P.A. in a certain high school. Using a t-test for correlated groups is when a sample group is tested numerous times under more than one condition. For example this test could be used to determine the model and brand of car that has the quickest braking ability. To do this multiple cars’ braking speed would be timed under different conditions such as: clear weather, rain, snow, ice, and hail.

## 5 Steps of Hypothesis Testing

**Step 1: Restate the question as a research hypothesis and a null hypothesis about the populations. ** A research hypothesis is a statement about the predicted relationship between the two populations. The null hypothesis is opposite of the research hypothesis. If one of the hypotheses is true, the other cannot be. For example, if we were to research the health of children who eat fresh vegetables versus children who do not eat fresh vegetables, the research hypothesis would be as follows. “Children who eat fresh vegetables are healthier than those who don’t eat vegetables.” The null hypothesis would be the following. "Children who eat fresh vegetables are *not* healthier than those who don't eat fresh vegetables."

The two populations that we would be comparing would be children who eat fresh vegetables (population 1) and children who don't eat fresh vegetables (population 2). ** Step 2: Determine the characteristics of the comparison distribution.** The comparison distribution represents the population situation if the null hypothesis is true. It is the distribution to which one can compare the score based on the sample's results.** Step 3: Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.** The cutoff sample score is the point if reached or exceeded by the sample score that you reject the null hypothesis. In this step, you set the z score at a score that would be unlikely if the null hypothesis is true. For example, the researchers testing the vegetable hypothesis may decide that if a result were less than 5% then they would reject the null hypothesis. ** Step 4: Determine your sample’s score on the comparison distribution.** This is the point where the study is carried out and the actual results for the sample are obtained. ** Step 5: Decide whether to reject the null hypothesis.** Compare the actual sample’s z score to the cut off z score. If the actual score is higher than the cut off score, the null hypothesis would be rejected.

## References

Aron, A., Aron. E., Coups. E. (2014). *Statistics for Psychology* Pearson Education Inc.

2014.

## Chapter 4: Introduction to Hypothesis Testing

Core Logic of Hypothesis Testing

Considers the probability that the result of a study could have come about if the experimental procedure had no effect

If this probability is low, scenario of no effect is rejected and the theory behind the experimental procedure is supported

The Hypothesis Testing Process

Restate the question as a research hypothesis and a null hypothesis about the populations.

Population 1

Population 2

Research hypothesis

Null hypothesis

The Hypothesis Testing Process

Determine the characteristics of the comparison distribution.

Comparison distribution

The Hypothesis Testing Process

Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.

Cutoff sample score

Conventional levels of significance:

p < .05, p < .01

The Hypothesis Testing Process

Determine your sample’s score on the comparison distribution.

Decide whether to reject the null hypothesis.

One-Tailed and Two-Tailed Hypothesis Tests

Directional hypotheses

One-tailed test

Nondirectional hypotheses

Two-tailed test

Determining Cutoff Points With Two-Tailed Tests

Divide up the significance between the two tails

Controversies and Limitations

Criticisms of basic logic of significance tests

Misuse of significance tests

“... rigorous research requires continued use of significance testing in the appropriate context…and adherence to …recommendations that promote its rational use…” (Balluerka et al, 2005)

Hypothesis Tests in

Research Articles

Reported with regard to specific statistical procedures

“Near significant trend”

Not significant, ns

Hypothesis Tests in Research Articles

Shown as asterisks in a table of results

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