Statistics and Hypothesis Testing

When both the independent and dependent variables are interval/ratio level, which statistical test should be used?

Pearson's r.

Why is the term 2r(SEM1)(SEM2) not included in the independent-samples t-test?

Because r is assumed to be 0 in the independent-samples t-test.

The term −2r(SEM1)(SEM2) in the paired-samples t-test causes the t-ratio to be _____ than it would if the term were excluded.

Larger.

With respect to the paired-samples t-test, as the size of r increases, the t-ratio _____, making it _____ to reject the null hypothesis.

Increases, making rejection more likely.

If Pearson's r between variables X and Y is −.50, how much of the variance in Y can be predicted by X?

25%.

Which of the following, if any, is true with respect to Pearson's r? (Ordinal, ranges ±1 SD, nonlinear)

None of the above. Pearson's r measures linear relationship strength and direction.

Which of the following is used in determining the degrees of freedom between in a one-way ANOVA?

The number of levels of the independent variable.

In the F statistic, what influences the sum of squares between?

Differences between each group mean/grand mean and the number of groups.

In an experiment, the numerator in the F statistic is a measure of what?

The effect of the independent variable on the dependent variable.

In a one-way ANOVA, as the sum of squares within increases, what happens to the probability of accepting the null hypothesis?

It increases.

If the F statistic is significant in a one-way ANOVA, what is the next step?

Reject the null hypothesis and conduct a post-hoc test.

Why do we conduct a post-hoc test if we find a significant F in a one-way ANOVA?

The overall F test does not tell you which means differ from each other.

True or false: Less likely to reject null with a paired-samples t-test than an independent-samples t-test?

False.

True or false: In a one-way ANOVA, if SS between stays constant and means compared increase, significance is more likely?

False.

True or false: In a one-way ANOVA, IV levels and sample size influence degrees of freedom within?

True.

True or false: t-test has one df number because df between must always be 1?

True.

True or false: Independent and paired-samples t-tests with the same subjects have the same df?

False.

True or false: In a factorial ANOVA, if both main effects are significant, the interaction must be significant?

False.

True or false: In a factorial ANOVA, when neither main effect is significant, the interaction may still be significant?

True.

True or false: When ANOVA rejects the null hypothesis, between-group variability must exceed within-group variability?

True.

What is the effect size statistic for a one-way ANOVA?

Eta-squared (η²).

What is the effect size statistic for a paired-samples t-test?

Cohen's d.

The coefficient of determination is equal to _____ and tells us _____.

r², the proportion of variability in Y accounted for by X.

Compared with a placebo, a drug reduces hyperactivity in one age group but increases it in another. How is this described?

An interaction between the drug and age.

In the drug and age example, which F value would most likely be significant?

The F for the interaction.

In a paired-samples t-test, the value of t tells you how many _____ you are from _____.

Estimated standard errors of the difference from 0.

What is the relationship between t and F for equivalent tests of a single contrast?

F = t².

When both variables are interval-ratio level and the question concerns their linear relationship, which test should be used?

Pearson's r.

What does Pearson's r measure?

The strength and direction of a linear relationship between two variables.

What requirements for Pearson's r does the textbook emphasize?

Both variables are interval-ratio, the hypothesized relationship is linear, the data are homoscedastic, and the ranges should not be severely restricted.

What is the null hypothesis for Pearson's r?

The population correlation equals zero: H₀: ρ = 0. There is no linear relationship in the population.

What values can Pearson's r take?

Any value from −1 to +1, inclusive.

Which part of r indicates strength, and which part indicates direction?

The absolute value indicates strength. The positive or negative sign indicates direction.

Which relationship is stronger: r = −.70 or r = +.50?

r = −.70, because .70 is farther from zero than .50.

What does a positive correlation mean?

Higher X scores tend to accompany higher Y scores, and lower X scores tend to accompany lower Y scores.

What does a negative correlation mean?

Higher X scores tend to accompany lower Y scores, and lower X scores tend to accompany higher Y scores.

As points scatter farther from the best-fitting straight line, the absolute value of r _____.

Decreases toward zero. The linear relationship becomes weaker.

True or false: r = 0 rules out every possible relationship between two variables.

False. A nonlinear relationship, such as an inverted U, may still exist.

What is range restriction, and why is it a problem for correlation?

Range restriction means studying a narrow range of scores on X or Y. It can weaken or hide a relationship—for example, studying SAT scores only among nearly perfect scorers.

What does homoscedasticity mean for Pearson's r?

The spread of scores around the relationship is roughly consistent across values of X. The textbook describes this as relatively consistent relationship strength across the range.

What are the degrees of freedom for Pearson's r?

df = N − 2, where N is the number of pairs of scores.

With 25 people each measured on X and Y, what are the degrees of freedom for Pearson's r?

23, because 25 − 2 = 23.

How do you decide whether an obtained Pearson's r is significant in this course?

Use alpha and df = N − 2 to find the critical value in Table C. Reject H₀ when the absolute obtained r exceeds the critical r.

What is the coefficient of determination, and what does it tell us?

r². It tells us the proportion of variability in Y accounted for by its linear relationship with X. Multiply by 100 for a percentage.

If r = −.50, what percentage of variability in Y is accounted for by X?

25%, because (−.50)² = .25.

True or false: r = .80 means 80% of variability in Y is accounted for by X.

False. It means 64%, because .80² = .64.

True or false: Using Pearson's r automatically makes a study non-experimental.

False. The research design determines whether the study is experimental. An experiment can randomly assign quantitative drug dosages and analyze the results using Pearson's r.

What research question does Fisher's Z test answer in Chapter 6?

Whether two population correlations differ, using correlations from two independent samples.

What is the null hypothesis for Fisher's Z test comparing independent correlations?

The two population correlations are equal: H₀: ρ₁ = ρ₂.

How is Fisher's Z different from testing a single Pearson's r?

Pearson's r tests whether one correlation differs from zero. Fisher's Z tests whether two correlations differ from each other.

True or false: Two sample correlations that look different must differ significantly.

False. Their difference could reflect sampling error.

True or false: If two correlations are individually significant, their difference must also be significant.

False. Both can differ from zero without differing significantly from each other.

What must happen to the two r values before using the Fisher's Z formula?

Convert each r to a Fisher Zᵣ value using Table D. Subtract the transformed values, not the raw correlations.

What is the Fisher's Z formula for comparing two independent correlations?

Z = (Z_r1 - Z_r2) / sqrt(1/(N_1-3) + 1/(N_2-3)). Preserve the signs of the correlations when transforming them.

Holding the transformed correlation difference constant, increasing both sample sizes makes the absolute Fisher Z _____.

Larger, because the standard error decreases. Detecting a difference becomes easier.

What critical values are used for Fisher's Z in the textbook?

1.96 at α = .05 and 2.58 at α = .01 for the two-sided tests shown.

If Fisher's Z is not significant, what should you conclude, and does Chapter 6 give a separate effect-size formula for this comparison?

There is insufficient evidence that the population correlations differ. Chapter 6 gives no separate effect-size formula for Fisher's Z; r and r² describe the underlying correlations.

When should a paired-samples t test be used?

When comparing two related means—the same people measured twice or participants deliberately organized into matched pairs.

What other names does the textbook use for the paired-samples design?

Repeated-measures and within-subjects designs.

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

The population mean difference is zero: μbefore = μafter, or μdifference = 0.

True or false: The main question in a paired-samples t test is whether the two sets of scores are correlated.

False. It tests whether the population means differ. Correlation contributes to the standard-error calculation.

What are the numerator and denominator of the paired-samples t statistic?

Numerator: The mean difference, minus the null difference of zero. Denominator: The estimated standard error of that mean difference.

What does the value of a paired-samples t tell you?

How many estimated standard errors the observed mean difference is from zero.

Why are repeated measurements on the same people often positively correlated?

People often maintain similar relative positions across conditions. Someone with a high score before treatment may still score relatively high afterward, even if everyone improves.

Where does the term −2r(SEM₁)(SEM₂) appear in the paired t calculation?

Inside the square root in the standard error of the difference: SE_diff = sqrt(SEM₁² + SEM₂² - 2r(SEM₁)(SEM₂)).

Why is the correlation term omitted for independent samples?

In the textbook's formulation, r is assumed to equal zero, making the entire correlation term zero.

Holding means, SEMs, and sample size constant, increasing the positive correlation between paired scores makes SEdiff _____ and absolute t _____.

Smaller; larger. Rejecting H₀ becomes more likely.

Holding other quantities constant, decreasing the correlation between paired scores makes absolute t _____ and rejection of H₀ _____.

Smaller; less likely.

With the mean difference, SDs, and correlation held constant, increasing sample size makes absolute t _____.

Larger. A larger sample reduces the standard error.

With sample size, correlation, and mean difference held constant, increasing both SDs proportionally makes absolute t _____.

Smaller. More variability increases the standard error.

Holding the standard error constant, increasing the absolute mean difference makes absolute t _____.

Larger. Rejecting H₀ becomes more likely.

What are the degrees of freedom for a paired-samples t test with 20 pairs of scores?

19, because df = N − 1. Twenty people measured twice still provide 20 pairs.

What is the effect-size statistic for a paired-samples t test in this textbook?

Cohen's d, calculated as d = t_obtained / sqrt(N). Use the obtained t, not the critical t.

What are the textbook's small, moderate, and large benchmarks for Cohen's d?

Small: .20; moderate: .50; large: .80.

What is a major advantage and a major disadvantage of a within-subjects design?

Advantage: Positive pairing can reduce error and make differences easier to detect. Disadvantage: Repeated testing can create demand characteristics or reveal the study's purpose.

True or false: Different people in the two conditions always require an independent-samples t test.

False. Deliberately matched participants can justify a paired-samples t test.

According to the specific procedure on textbook page 43, what happens if the paired-score correlation is not significant?

The course says to omit the correlation adjustment, calculate t as for independent samples, and retain df = N − 1. This is a course-specific procedure; standard paired-t analysis generally retains the paired design regardless of correlation significance.

When is a one-way ANOVA appropriate, and what does one-way mean?

When comparing means across levels of one categorical IV with a quantitative DV, especially with more than two groups. One-way means one IV.

Why use ANOVA instead of repeatedly conducting ordinary t tests for every pair of groups?

Repeated unadjusted t tests increase the familywise risk of a Type I error.

What is the null hypothesis in a one-way ANOVA?

All population group means are equal: μ₁ = μ₂ = μ₃ = ...

What is the F ratio in a one-way ANOVA?

F = MS_between / MS_within. Each mean square equals its sum of squares divided by its degrees of freedom.

What does SSbetween reflect, and what can produce differences between group means?

It reflects how far group means are from the grand mean, weighted by group size. IV effects and sampling error can both produce between-group differences.

What does SSwithin reflect?

Differences between individual scores and their own group mean, squared and summed across groups. It represents within-group error variability.

Why can't variation within a treatment group be attributed to differences in the assigned IV level?

Everyone in the group receives the same IV level, so within-group differences must come from other sources.

Which quantities are used to calculate SSbetween and SSwithin?

SSbetween: Group means, the grand mean, and group sizes. SSwithin: Individual scores and their own group means.

What are the degrees of freedom in a one-way ANOVA?

df_between = k − 1 and df_within = N − k, where k is the number of groups and N is total sample size.

A one-way ANOVA has 4 groups and 40 total participants. What are its degrees of freedom?

df_between = 3; df_within = 36. Report F(3, 36).

Holding SSwithin and degrees of freedom constant, increasing SSbetween makes F _____.

Larger. Rejecting H₀ becomes more likely.

Holding SSbetween and degrees of freedom constant, increasing SSwithin makes F _____.

Smaller. Failing to reject H₀ becomes more likely.

True or false: Increasing SSwithin changes dfwithin by itself.

False. Degrees of freedom depend on sample size and number of groups, not the numerical value of SSwithin.

Holding SSbetween constant, adding levels to an IV increases dfbetween and therefore makes MSbetween _____.

Smaller, because MSbetween = SSbetween ÷ dfbetween. The overall F also depends on MSwithin.

How do you look up the critical F value in the textbook?

Use numerator df for the column, denominator df for the row, and the specified alpha. Table E uses ordinary type for .05 and bold for .01. If the exact df is absent, the course uses the next lower listed df.

True or false: A significant one-way F means every group mean differs significantly from every other mean.

False. It means at least one population mean differs, not that all pairs differ.

When does this course call for Tukey's HSD after a one-way ANOVA?

After a significant overall F when comparing more than two means.

What question does Tukey's HSD answer?

Which specific pairs of means differ significantly. HSD stands for Honestly Significant Difference.

How do you decide whether a pair of means differs using Tukey's HSD?

Compare their absolute mean difference with the HSD. In the course's procedure, differences exceeding HSD are significant.

Which inputs are used for Tukey's HSD in the equal-group-size problems in this course?

HSD = q * sqrt(MS_within / n). Find q using the number of means, df_within, and alpha. Here n is the number of scores contributing to each mean.

What is the effect-size statistic for a one-way ANOVA, and what is its formula?

Eta-squared: η² = SS_between / (SS_between + SS_within). It describes the proportion of total sample variability associated with group differences.

What are the textbook's small, moderate, and large benchmarks for eta-squared?

Small: .01; moderate: .06; large: .14. The course asks for effect sizes when the corresponding test is significant.

What is the relationship between F and t when they test the same two-mean comparison?

F = t² for the equivalent tests with numerator df = 1.