Conceptual: Quantitative Methods in Psychology Exam 2 (kilianski)

What type of statistic is used when both the IV and DV are interval-ratio level variables?

The Pearson's r correlation coefficient is used.

What type of variables are used with Pearson's r?

Both the independent variable (IV) and dependent variable (DV) must be interval-ratio level variables.

What does Pearson's r measure?

It measures the strength and direction of a linear relationship between two-interval ratio variables.

What does a "linear relationship" mean?

A relationship that can be described by a straight line, not curved or nonlinear.

What is the null hypothesis (H0) for Pearson's r?

H₀: r (population) = 0,

meaning there is NO relationship between the two variables in the population.

Again: How is the null hypothesis written for Pearson's r in statistical form?

H₀: r (population) = 0

What is the range of possible values for Pearson's r?

r can range from -1.00 to +1.00

What does the sign of r indicate?

The direction of the relationship (positive or negative)

What does the absolute value of r indicate?

The strength of the relationship (how strongly X and Y are related).

What does an r of +1.00 or -1.00 mean?

A perfect linear relationship.

Knowing X lets you predict Y with 100% accuracy.

What happens as the scatter of points increases in a scatterplot?

The strength of the relationship decreases and r moves closer to 0.

Give an example of a positive correlation.

As study time (X) increases, test scores (Y) increases.

Given an example of a negative correlation.

As stress level (X) increases, sleep hours (Y) decrease.

What is the formula for calculating Pearson's r?

A: r = Σ(ZxZy) / N

where:

Σ = sum

ZxZy = product of Z-scores on X and Y

N = number of pairs

How are degrees of freedom (df) for Pearson's r calculated?

df = N-2

How do we decide whether to reject H₀?

If the obtained r (absolute value) is greater than the critical value from the r table (Table C), reject H₀.

What does it mean if the obtained r is greater than the critical r value (in absolute value)?

It means the relationship is statistically significant, and we reject the null hypothesis

-> there is likely a relationship between the two variables.

What is the effect size in Pearson's r?

Pearson's r itself is an effect size;

it shows the strength of the relationship.

OR in the professor's terms: it tells you what percentage of variability in Y can be accounted for or predicted by X.

What does it mean if the obtained r is less than the critical r value?

The relationship is not statistically significant, so we fail to reject H₀, meaning there is probably no relationship in the population.

What does r^2 (r-squared) represent?

The Coefficient of Determination - the percentage of variability in Y that can be accounted for or predicted by X.

How is r^2 (r-squared) interpreted as a percentage?

Multiply r^2 by 100 to get the percentage of variance in Y accounted for by X.

(Example: r = .80 -> r^2 = .64 -> 64% of Y explained by X)

What are the requirements for using Pearson's r?

1. Both variables are interval/ratio level.

2. The relationship between X and Y MUST be linear.

3. The data are homoscedastic (strength of relationship b/w X and Y are relatively constant across the entire range of values).

What effects accuracy in Pearson's r?

Range of values is not restricted (b/c restricted range reduces variability -> lowers r).

Outliers may exert undue influence (as in outliers can inflate or deflate r or even change its direction)

What does "range restriction" mean?

When one variable has a narrow range of values. making it harder to detect a correlation.

Example of range restriction affecting r?

At Ivy League schools, SAT scores and GPA have little variation, so correlation appears weak even though it's strong in a general population.

What is homoscedasticity?

When the strength of the relationship between X and Y is consistent across all values of X.

What happens if data are not homoscedastic?

The relationship changes at different values of X, violating Pearson's r assumptions.

Why can't Pearson's r be used for curvilinear relationships?

Because r only measures linear relationships.

A curvilinear will make r appear near zero even if the relationship is strong.

What's the difference between a correlational method and a correlation statistic?

- Correlational method: A research design without random assignment or IV manipulation.

- Correlation Statistic (r): A numerical measure of relationship strength/direction that can be used in any design, even experiments.

Can you infer causation using Pearson's r?

Yes, but only if the research design is experimental (e.g., manipulating dosage levels).

If the method is non-experimental, you cannot infer causation.

What is the Fisher's Z-test used for?

To test if two correlation coefficients (r values) are significantly different from each other.

What is the formula for Fisher's Z-test?

Z = (Zr₁ - Zr₂) / √[(1 / (N₁ - 3)) + (1 / (N₂ - 3))]

(Zr values are taken from Table D for each r)

What are the critical Z values for Fisher's Z-test?

A: ±1.96 at α = .05 and ±2.58 at α = .01.

If the obtained Z is greater than the critical value, what do you conclude?

The two correlations are significantly different from each other.

What is an example of when to use Fisher's Z-test?

Comparing whether the correlation between blood pressure and longevity differs for men vs. women.

What are other types of correlation coefficients (besides Pearson's r)?

Spearman's rho (ρ): Used when one or both variables are ordinal.

Point-biserial: Used when one variable is interval-ratio and the other is dichotomous.

Phi coefficient (φ): Used when both variables are dichotomous.

note for computational exam: What should you round to when performing calculations?

Two decimals places after EACH step (products, division, etc).

What does a correlation of 0 mean?

There is no linear relationship between the two variables.

What are the main steps in using Pearson's r?

1. Compute r using Z scores.

2. Find df = N - 2.

3. Locate the critical r value.

4. Compare obtained r (absolute value) to the critical r.

5. Decide whether to reject or fail to reject H₀.

6. Interpret the strength and direction (as in, give the statement).

What does a negative correlation tell you about prediction?

As one variable increases, the other decreases, allowing you to predict the opposite outcomes.

Why is Pearson's r useful in research?

It allows researchers to quantify relationships and predict one variable from another in both experimental and non-experimental contexts.

What does "strength" refer to in Pearson's r?

"Strength" refers to how closely the data points cluster around a straight line, indicating how well one variable predicts the other.

What does "direction" refer to in Pearson's r?

"Direction" refers to whether the relationship is positive (both increase together) or negative (one increases while the other decreases).

What does it mean that Pearson's r deals with a "hypothetical linear relationship"?

It means Pearson's r assumes that the relationship between X and Y can be described by a straight line.

It tests how well that assumption fits the data.

When is Pearson's r NOT appropriate to use?

When either interval is not interval-ratio level or when the relationship between them in nonlinear (curved).

What kind of statistic is Pearson's r? descriptive or inferential?

Pearson's r is an inferential statistic, meaning it is used to make conclusions about a population based on sample data.

What is the main goal when using Pearson's r as an inferential test?

The goal is to determine whether there is a significant relationship between two interval-ratio variables in the population, not just in the sample.

What are the main steps when testing significance in Pearson's r?

1. Calculate the obtained value of r from sample data.

2. Find the critical value or r using the degrees of freedom (f) and alpha level (α).

3. Compare the obtained r with the critical value to decide whether to reject or fail to reject H₀.

When is an independent samples t-test used

When the two samples are composed of different individuals. This is referred to as a between-subjects design.

What is a between-subjects (Ss) design?

A design where each sample contains different participants, and there is no overlap or pairing between the groups.

Give an example of a between-subjects design.

A drug study where half of the participants get a placebo and half get the drug.

What is the correlation (r) between samples in a between-subjects design?

r = 0, because the samples are independent.

When is a paired-samples (or repeated measures / within-subjects) t-test used?

When the same participants are measured more than once, such as before and after an intervention.

What is a within-subjects (within-Ss) design?

A design where the same participants are tested under different conditions or at different times.

Give an example of a within-subjects design

A drug study where all participants are tested after taking a placebo and again after taking the drug.

In a within-subjects design, are the scores likely to be correlated?

Yes, because each person's scores are compared to themselves across conditions.

Why are within-subjects scores usually correlated?

Because people who score higher or lower on one measurement tend to score similarly on another measurement.

In the weight-loss drug example, what does it mean that participants' relative positions didn't change?

It means that even though everyone lost weight, their ranking (heaviest to lightest) stayed the same.

What would the Pearson's r be in a perfect within-subjects correlation example?

r = 1.0

Do correlations of r = 1.0 occur in real research?

No, that's hypothetical; in reality, correlations of 1.0 almost never happen.

What does the paired-samples t-test assume about before and after scores?

That they are significantly correlated, meaning a Pearson's r should be calculated and tested for significance.

How does the formula for the denominator (estimated SEDiff) differ in the paired-samples t-test compared to the independent samples t-test?

It includes the term 2(rAB)(est. SEMA)(est. SEMB) to account for the correlation between the two sets of scores.

What is the formula for the estimated SEDiff in a paired-samples t-test?

√[(est. SEMA)² + (est. SEMB)² - 2(rAB)(est. SEMA)(est. SEMB)]

What remains the same in the t formula for the paired-samples test compared to the independent samples test?

The numerator, which is the difference between the two means (before and after).

Why was the correlation term (-2rAB...) excluded in the independent samples t-test?

Because in independent samples, r = 0, so the term has no effect.

What is the formula for degrees of freedom (df) in the paired-samples t-test?

df = N - 1, where N is the number of pairs of scores.

How is effect size (Cohen's d) calculated for the paired-samples t-test?

d = t / √N, using the obtained t value, not the critical value.

Why is the estimated SEDiff smaller in a paired-samples t-test when before and after scores are correlated?

Because the correlation term subtracts from the denominator, making it smaller.

What does a smaller SEDiff mean for the t value and rejecting H₀?

A smaller SEDiff makes it larger, which increases the chance of rejecting the null hypothesis.

How does Pearson's r affect the SEDiff and t value?

The higher the r, the smaller the SEDiff, and the larger the t value.

Why do researchers often prefer within-subjects (paired) designs?

Because they make it easier to find significant results, which are more likely to be published or accepted at conferences.

What is a major disadvantage of the paired-samples (within-subjects) design?

It can create methodological problems because the same participants are tested repeatedly.

In the drug study example, what methodological problem can occur?

If participants notice side effects (or lack of them), they may guess whether they received the placebo or the real drug, which affects results.

What happens if a participant realizes they received a placebo?

The placebo effect may disappear, invalidating the study results.

In the salary discrimination study example, what problem arises?

Participants may realize the study's purpose (demand characteristics) and change their responses to appear fair or unbiased.

What are "demand characteristics"?

Clues in a study that make participants aware of the research purpose, influencing their behavior or responses.

What is the paired-samples t-test actually testing?

Whether the mean difference between before and after scores is due to sampling error.

What is the null hypothesis (H₀) for the paired-samples t-test?

H₀: µbefore = µafter

Is the paired-samples t-test a test of correlation between before and after scores?

No, the correlation is just part of the calculation process, not the main research question.

What should you do if the correlation (r) between before and after scores is not significant (meaning you fail to reject the null)?

Compute the t as for an independent samples t-test, but use df = N - 1.

How is a matched pairs design different from a repeated measures design?

Different people are in each condition, but they are matched based on a shared characteristic or score.

Why does a matched pairs design justify using a paired-samples t-test?

Because the paired subjects' scores are correlated, similar to a within-subjects design.

What is an example of a variable participants might be matched on in a matched pairs design?

IQ, baseline performance, or any other relevant variable related to the dependent variable.

How does a matched pairs design "look" and "work"?

It looks like an independent samples design (different people), but works like a within-subjects design (paired, correlated data).

From this point on, this is about 1-way ANOVA.

What statistical problem arises when researchers need to compare more than two group means? (For 1-way ANOVA).

The risk of inflated Type I error when running multiple t-tests.

Which statistical tests are limited to comparing only two means?

The z-test, 1-sample t-test, independent samples t-test, and paired-samples t-test.

Why is it incorrect to run a series of t-tests when comparing several groups?

Because each t-test adds to the chance of making a Type I error, increasing the overall (familywise) error rate.

How many pairwise comparisons exist when comparing 4 groups?

6 possible comparisons.

If α = .05 for each of 6 comparisons, what happens to the overall Type I error rate?

It rises to about 26% instead of 5%.

What is the formula for calculating familywise error probability?

p = 1 − (1 − α)ᵈ, where d is the number of comparisons.

What does ANOVA stand for?

Analysis of Variance.

What does ANOVA compare in order to test for group differences?

The between-group variance (differences among group means) and the within-group variance (differences within each group).

In ANOVA, what does the "signal" represent?

The effect of the independent variable on group means.

In ANOVA, what does the "noise" represent?

Variability caused by factors other than the independent variable (random or individual differences).

Why is ANOVA considered a "signal-to-noise" ratio?

Because it measures how large the between-group differences (signal) are relative to within-group differences (noise).

Why does using ANOVA prevent the familywise error problem?

It tests all means simultaneously in one overall (omnibus) test rather than multiple t-tests.

What is an example of a study where a 1-way ANOVA would be appropriate?

A drug study with four conditions: control, placebo, low dose, and high dose.

What conclusion can be drawn when a 1-way ANOVA is significant?

At least one group mean is significantly different from one or more others.

What is the statistic for the ANOVA?

The F-ratio.