PSY 2030 Paired Samples t Test: Formulas, Steps & Practice

What is a paired samples t test?

A statistical test used with within-subjects data to determine whether there is a significant difference between two sets of scores from the same subjects.

What is within-subjects data?

Data collected from the same people/subjects at two different points or conditions.

Give an example of within-subjects data.

Measuring the same people's scores before and after an experimental intervention.

Why are the two sets of scores not independent?

Because the scores come from the same people.

What are independent observations?

Observations that are not related to or dependent on one another.

What is a problem with within-subjects data?

The two measurements from the same person are not independent observations.

How do we make within-subjects data independent?

Convert the paired observations into difference scores.

What is a difference score?

A score created by subtracting the Time 1 score from the Time 2 score.

What is the difference score formula?

D = X₂ − X₁.

What does D represent?

The difference score for a participant.

What does X₁ represent?

The participant's score at Time 1.

What does X₂ represent?

The participant's score at Time 2.

What does a positive difference score indicate?

A difference score greater than 0, meaning the Time 2 score is higher than the Time 1 score.

What does a negative difference score indicate?

A difference score less than 0, meaning the Time 2 score is lower than the Time 1 score.

What does a zero difference score mean?

A difference score of 0, meaning the Time 1 and Time 2 scores are equal.

Why calculate difference scores?

To convert the paired observations into a set of difference scores that can be treated as independent from one another.

What is the sampling distribution of the mean difference?

The sampling distribution used to analyze the mean of the difference scores.

What is the mean difference?

The average of the difference scores in a sample.

What does M_D represent?

The sample mean of the difference scores.

What is the population mean difference?

The mean difference between the two populations.

What does μ_D represent?

The population mean difference.

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

H₀: μ_D = 0.

What does H₀: μ_D = 0 mean?

The null hypothesis states that there is no population difference between Time 1 and Time 2.

What is the equivalent null hypothesis?

H₀: μ₁ = μ₂.

Why is μ_D = 0 equivalent to μ₁ = μ₂?

If the two population means are equal, their difference must be 0.

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

H₁: μ_D ≠ 0.

What does H₁: μ_D ≠ 0 mean?

The alternative hypothesis states that the population mean difference is not 0; there is a difference between the two conditions/time points.

What is the standard error of the mean difference?

The estimated standard error of the sampling distribution of the mean difference.

What is the standard error formula for paired samples?

σ_D = s_D / √N.

What does s_D represent?

The sample standard deviation of the difference scores.

What does N represent?

The number of paired observations/participants.

What is the sample standard deviation of difference scores formula?

s_D = √[Σ(X_D − M_D)² / (N − 1)].

What does X_D represent?

An individual difference score.

What does Σ(X_D − M_D)² represent?

The sum of the squared deviations of each difference score from the mean difference.

What is the degrees of freedom for a paired samples t test?

df = N − 1.

What is the complete paired samples t formula?

t = (M_D − μ_D) / σ_D.

What is the paired samples t formula when H₀ is true?

t = (M_D − 0) / σ_D.

What is the paired samples t formula expanded?

t = (M_D − μ_D) / (s_D / √N).

What is different about a paired samples t test compared with a single sample t test?

The paired samples t test uses difference scores instead of raw scores.

What happens to the original two scores in a paired samples t test?

Each participant's Time 1 and Time 2 scores are converted into one difference score.

What is the basic paired samples t test process?

1. Calculate each difference score. 2. Calculate M_D. 3. Calculate s_D. 4. Calculate σ_D. 5. Calculate t. 6. Calculate df. 7. Find the critical t value. 8. Compare the calculated t with the critical t.

What do you conclude if you reject H₀ in a paired samples t test?

You conclude that there is a difference between the Time 1 and Time 2 scores; the population mean difference is probably not 0.

What do you conclude if you fail to reject H₀ in a paired samples t test?

You conclude that there is not sufficient evidence of a difference between Time 1 and Time 2 scores.

What is the important interpretation of rejecting H₀?

Rejecting H₀ means there IS a difference; failing to reject H₀ means there is NO difference.

What is the critical t value?

The cutoff value used to determine whether the calculated t statistic is sufficiently extreme to reject H₀.

How do you find the critical t value?

Determine df, whether the test is one-tailed or two-tailed, identify α, find the correct df row in the t table, and use the appropriate alpha column.

What is the degrees of freedom for N = 4?

df = 4 − 1 = 3.

What is the degrees of freedom for N = 10?

df = 10 − 1 = 9.

What is the degrees of freedom for N = 20?

df = 20 − 1 = 19.

What is a two-tailed paired samples test?

If the alternative hypothesis is H₁: μ_D ≠ 0, the test is two-tailed because it predicts a difference in either direction.

What is a one-tailed paired samples test?

A paired-samples test in which the alternative hypothesis predicts a specific direction of the difference.

What is the direction of D in these slides?

D = Time 2 − Time 1.

What are the difference scores for the practice example?

Before: 15, 35, 45, 20. After: 25, 40, 50, 40. Difference scores D = After − Before = 10, 5, 5, 20.

What is the mean difference for the practice example?

For D = 10, 5, 5, 20: M_D = (10 + 5 + 5 + 20)/4 = 40/4 = 10.

What are the deviations from M_D for the practice example?

For D = 10, 5, 5, 20 and M_D = 10: deviations are 0, −5, −5, and 10.

What are the deviations for D = 10, 5, 5, 20 with M_D = 10?

0, −5, −5, and 10.

What are the squared deviations for deviations 0, −5, −5, 10?

0, 25, 25, and 100, with a sum of 150.

How is s_D calculated for D = 10, 5, 5, 20?

s_D = √(150/3) = √50 ≈ 7.071.

What is the formula for standard error (σ_D)?

σ_D = s_D/√N.

What is the t statistic formula?

t = (M_D − μ_D)/σ_D.

How is degrees of freedom (df) calculated?

df = N − 1.

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

±3.182.

What is the conclusion for the reading intervention with t ≈ 2.828?

Fail to reject H₀ because |2.828| < 3.182.

What are the difference scores for the nature walk before and after ratings?

D = After − Before = 1, 0, 1, 1.

What is the mean difference (M_D) for the nature walk?

M_D = (1 + 0 + 1 + 1)/4 = 0.75.

What are the deviations for D = 1, 0, 1, 1 with M_D = 0.75?

0.25, −0.75, 0.25, and 0.25.

What are the squared deviations for the nature walk?

0.0625, 0.5625, 0.0625, and 0.0625, with a sum of 0.75.

What is s_D for the nature walk?

s_D = √(0.75/3) = 0.50.

What is the standard error for the nature walk?

σ_D = 0.50/√4 = 0.25.

What is the t statistic for the nature walk?

t = (0.75 − 0)/0.25 = 3.00.

What is the conclusion for the nature walk with t = 3.00?

Fail to reject H₀ because |3.00| < 3.182.

What is the formula for the difference score?

D = X₂ − X₁.

What is the formula for mean difference (M_D)?

M_D = ΣD/N.

What is the formula for standard deviation of the difference scores (s_D)?

s_D = √[Σ(D − M_D)²/(N − 1)].

What is the formula for t statistic?

t = (M_D − μ_D)/σ_D.

What is the formula for degrees of freedom (df)?

df = N − 1.

What is the most important concept of the paired samples t test?

It analyzes the difference between two scores from the same subjects.

What are the most important hypotheses for the paired samples t test?

H₀: μ_D = 0; H₁: μ_D ≠ 0.

What is the common mistake regarding the two sets of raw scores?

Using the raw scores directly instead of calculating difference scores first.

What is the correct direction for calculating difference scores?

Time 2 − Time 1.

What is the significance of the null hypothesis being 0?

If there is no difference, the mean difference must equal 0.

What does rejecting the null hypothesis indicate?

It indicates a difference between the two conditions/time points.

What does failing to reject the null hypothesis mean?

Insufficient evidence to conclude a difference exists.

What is the main takeaway from the lecture on paired samples?

Same subjects provide data at two points; analyze using difference scores.

What is the sequence of steps for the paired-samples t test?

Calculate differences, find mean difference, find s_D, find standard error, calculate t, df = N − 1, find critical t, compare.