Statistics: Single, Paired & Independent-Samples t Tests—Formulas and Design Selection

What is a Single-Sample t Test used for?

To determine if the mean of one sample is significantly different from a known population mean.

When should you use a Single-Sample t Test?

When you have one sample and are comparing its mean to a known population mean.

Provide an example of a Single-Sample t Test situation.

A researcher compares the average stress score of students at a university to the national average of 50.

What is a Paired-Samples t Test used for?

To compare two scores from the same group of people measured at two different times or conditions.

When should you use a Paired-Samples t Test?

When the same participants provide two scores that are naturally paired.

Provide an example of a Paired-Samples t Test situation.

A researcher measures students' anxiety before and after an exam-preparation program.

What is an Independent-Samples t Test used for?

To determine if the means of two separate, independent groups differ.

When should you use an Independent-Samples t Test?

When you have two separate groups of people, each belonging to only one group.

Provide an example of an Independent-Samples t Test situation.

A researcher compares test scores from students in Classroom A with those in Classroom B.

What is the main question for a Single-Sample t Test?

Does this one sample differ from a known population mean?

What is the main question for a Paired-Samples t Test?

Did the same people change between Time 1 and Time 2 or between two conditions?

What is the main question for an Independent-Samples t Test?

Are these two separate groups different from each other?

What is the fastest way to identify which t Test to use?

Look at how many groups/sets of people you have and whether the same people are measured more than once.

What data structure is required for a Single-Sample t Test?

One sample of scores compared to one known population mean.

What data structure is required for a Paired-Samples t Test?

One group of people providing two measurements.

What data structure is required for an Independent-Samples t Test?

Two separate groups with one measurement for each person.

What makes paired data paired?

The same subjects provide both measurements, creating a natural pairing.

What makes independent data independent?

The two samples consist of separate groups of people.

What is attrition in research?

The loss of participants between different points in a study.

What must be accounted for in between-subjects data?

Differences between the samples, including means and standard deviations.

What is the formula for a Single-Sample t Test?

t = (M − μ) / σM

What is the standard error formula for a Single-Sample t Test?

σM = s / √N

What is the degrees of freedom formula for a Single-Sample t Test?

df = N − 1

What is the hypothesis example for a Single-Sample t Test?

H₀: μ = 80; H₁: μ ≠ 80.

What is the significance of a two-tailed test in a Single-Sample t Test?

It is used when testing if the sample mean is different from the population mean.

What does it mean to fail to reject H₀ in hypothesis testing?

The evidence is not strong enough to conclude that the sample mean is different from the population mean.

What is the formula for the paired-samples difference score?

D = X₂ − X₁

How is the paired-samples mean difference calculated?

M_D = ΣD / N

What is the formula for the paired-samples standard deviation?

s_D = √[Σ(D − M_D)² / (N − 1)]

How is the paired-samples standard error calculated?

σ_D = s_D / √N

What is the paired-samples t formula?

t = (M_D − μ_D) / σ_D

How do you calculate degrees of freedom for paired samples?

df = N − 1

What is the null hypothesis for paired-samples t test?

H₀: μ_D = 0

What is the alternative hypothesis for paired-samples t test?

H₁: μ_D ≠ 0

What is the main idea of the paired-samples t test?

It is a single-sample t test performed using difference scores instead of raw scores.

What is the critical t value for a two-tailed test with df = 3 at α = .05?

±3.182

What is the formula for independent-samples mean difference?

M₁ − M₂

How is the sum of squares calculated for independent samples?

SS = Σ(X − M)²; calculate SS separately for each sample.

What is the pooled variance formula for independent samples?

s²₀ = (SS₁ + SS₂) / (n₁ + n₂ − 2)

What is the formula for the standard error of independent samples?

σM₁−M₂ = √[s²₀(1/n₁ + 1/n₂)]

What is the degrees of freedom formula for independent samples?

df = n₁ + n₂ − 2

What is the null hypothesis for independent-samples t test?

H₀: μ₁ = μ₂

What is the alternative hypothesis for independent-samples t test?

H₁: μ₁ ≠ μ₂

What is the critical t value for independent samples with df = 6 at α = .05?

±2.447

What distinguishes a paired-samples t test from an independent-samples t test?

Paired samples use the same participants for both conditions, while independent samples use different participants.

What are key words associated with paired-samples t test?

Same people, before/after, pretest/posttest, difference scores.

What are key words associated with independent-samples t test?

Two groups, separate groups, different people.

What test is used when comparing one group to a known population mean?

Single-sample t test.

What test is appropriate for measuring the same participants before and after an intervention?

Paired-samples t test.

What test is used when comparing two different groups?

Independent-samples t test.

What is the decision rule if |t| < critical t value?

Fail to reject the null hypothesis.

What indicates a one-tailed test in hypothesis testing?

The wording 'greater than' or 'less than' in the research question.

What indicates a two-tailed test in hypothesis testing?

The wording 'different from' in the research question.

What should you consider if the same people are in both groups?

Use a paired-samples t test.

What should you consider if the groups are different?

Use an independent-samples t test.

What does it mean if a question says 'less than'?

It indicates a directional alternative hypothesis and suggests a one-tailed test.

What does it mean if a question says 'different from'?

It indicates a nondirectional alternative hypothesis and suggests a two-tailed test.

How can you identify a two-tailed test?

If H₁ uses ≠, the test is two-tailed.

How can you identify a right-tailed test?

If H₁ uses >, the test is right-tailed.

How can you identify a left-tailed test?

If H₁ uses <, the test is left-tailed.

What determines the critical value?

It is determined using alpha, degrees of freedom, and whether the test is one-tailed or two-tailed.

What are the steps to find the critical t value?

1. Identify α. 2. Determine one-tailed or two-tailed. 3. Calculate df. 4. Find the appropriate df row in the t table. 5. Use the appropriate alpha/tail column.

What is a two-tailed critical value?

There are two critical values: one positive and one negative.

What is a one-tailed critical value?

There is one critical cutoff in the direction of the alternative hypothesis.

What is the first step in hypothesis testing?

Determine which statistical test you are conducting.

What is the second step in hypothesis testing?

State the null and alternative hypotheses appropriate for the test.

What is the third step in hypothesis testing?

Calculate the test statistic.

What is the fourth step in hypothesis testing?

Find the critical value.

What is the fifth step in hypothesis testing?

Compare the test statistic to the critical value.

What is the sixth step in hypothesis testing?

State your decision regarding the null hypothesis.

What is the seventh step in hypothesis testing?

Write an interpretation of what your decision means.

When do you reject H₀?

Reject the null hypothesis when the calculated test statistic falls in the rejection region beyond the appropriate critical value.

When do you fail to reject H₀?

Fail to reject the null hypothesis when the calculated test statistic does not fall in the rejection region.

What is the two-tailed decision rule?

Reject H₀ if the absolute value of the calculated t is greater than the positive critical t value.

What does a negative t value indicate?

It indicates the direction of the difference based on the order of subtraction.

What does a positive t value indicate?

It indicates the direction of the difference based on the order of subtraction.

Why can t be negative?

If the first mean is smaller than the second mean, subtracting M₂ from M₁ produces a negative numerator.

Can a negative t value be significant?

Yes, for a two-tailed test, a negative t can still be significant; compare its absolute value with the positive critical value.

What is the formula for a single-sample mean?

M = ΣX/N

What is the formula for a single-sample t?

t = (M − μ) / σM

What is the formula for single-sample standard error?

σM = s/√N

What is the formula for paired t?

t = (M_D − μ_D)/σ_D

What is the formula for independent t?

t = (M₁ − M₂)/√[s²(1/n₁ + 1/n₂)]

What is the memory trick for test selection?

ONE vs POPULATION = SINGLE. SAME PEOPLE = PAIRED. SEPARATE PEOPLE = INDEPENDENT.

What is the final test-selection rule?

If you can answer 'ONE sample?', 'SAME people?', or 'SEPARATE groups?' you can usually identify the correct test.