Two-Way ANOVAs
How many effects do you test for in a two-way between-subjects ANOVA?
3: 2 main effects (one for each factor) and 1 interaction effect (between the two factors)
What information does the correlation coefficient give you?
The strength and direction of the linear relationship between two variables
In an analysis of regression for a single predictor, what are the two sources of variation?
Regression variation and residual variation
Which scale of measurement is your data on when running a chi-square analysis?
nominal or ordinal
The physical fitness score for a population of firefighters is 72. A sample of 49 new firefighters records a mean fitness score of 74 with a standard deviation of 8. Which statistical test should you use to see whether the new recruits scored higher than the average firefighter? Explain.
one-sample t-test
You have one group and you're trying to compare statistic (on interval/ratio scale) to parameter; however, population SD is not known so you have to estimate the SE using the sample SD -- thus a one sample t-test instead of a one-sample z-test
How many cells do you have in a 3 x 5 two-way between subjects ANOVA?
15 (you multiply the number of levels in each factor)
What does it mean to have a positive vs. negative relationship?
Positive: as one variable increases, the other variable increases (same direction)
Negative: as one variable increases, the other variable decreases (opposite direction)
How do you interpret the slope and intercept in a regression equation?
Intercept: When the value of X is 0, Y will be the value of the intercept.
distribution-free tests
A researcher wanted to examine whether pet owners vs. non pet owners have lower levels of stress. The researcher sampled 100 people and recorded whether they were pet owners (yes vs. no), and their stress levels on a 1-9 scale. Which statistical test should the researcher use to analyze their data? Explain.
independent samples t-test OR one-way between subjects ANOVA
It's a between subjects design because participants experience only one level of the IV (nominal variable). The DV is on an interval scale.
How would you describe an interaction effect?
The effect of factor A on the DV varies across the levels of factor B
What are some considerations/limitations when interpreting a correlation coefficient?
1) correlations don't demonstrate cause
2) outliers can change the direction and strength of a relationship
3) you cannot generalize the direction and strength of a correlation beyond the range of data measured
What is the standard error of estimate?
It's an estimate of the standard deviation of distance that a set of data points fall from the regression line.
How do you calculate the degrees of freedom for a chi-square test? (State both for goodness-of-fit and test-for-independence)
goodness-of-fit: k -1
test-for-independence: (k1-1)(k2-1)
where k is the number of levels in each factor
Researchers for a clinical trial wanted to examine the effectiveness of a new anxiety drug. The researchers recruited 20 patients to partake in a 3 week study. Participants experienced 3 different drugs: one existing drug, one placebo, and the new anxiety drug. One drug was given each week; after each drug, participants reported their anxiety levels (scale of 1-7) at the end of the week. Which statistical test should the researchers use to examine whether the new drug is more effective than the existing drug and the placebo? Explain.
There is one factor (drug type) with 3 levels with a interval scale DV. Because there are more than 2 groups, we have to do an ANOVA instead of a t-test. We do a within subjects ANOVA because the participants experienced all 3 levels.
A researcher conducts a 2 x 4 between subjects ANOVA in which 12 participants were observed in each group. If SS for factor B is 18, and SS for error is 264. What decision would you make regarding factor B?
Total # of participants: 96
df for factor B: 3
df for error: 88
MS for factor B: 18/3 = 6
MS for error: 264/88 = 3
F for factor B: 6/3 = 2
F critical for (3, 88) = between 2.29 and 2.37
Because obtained F is less than F critical, we would retain the null hypothesis for Factor B
What is the coefficient of determination?
Your data yielded the following: SSX = 356, SSY = 96.40, SSXY = -48.
What is the proportion of variance in Y that can be explained by X?
r = -48/sqrt(356 * 96.40)
= 0.2591
r2 = .07
Approximately 7% of the variance in Y can be explained by X
You wanted to see if there were gender differences in pre-school enrollment. A total of 61 students enrolled at your local preschool. Twenty-one students were male, and 40 were female. Conduct a chi-square goodness-of-fit test. What do you conclude?
observed frequency: 21, 40
expected frequency: 30.5, 30.5
residual: -9.5, 9.5
residual2: 90.25, 90.25
residual2/expected frequency: 2.959, 2.959
sum of residual2/expected frequency (chi-square value): 5.92
chi square critical value: 3.84
Because the obtained chi-square value is greater than our chi-square critical value, we can reject the null hypothesis -- there is a significant difference in pre-school enrollment, with more females enrolled than males.
A researcher wanted to examine intersectionality of race and gender in academia. The researcher randomly assigned students to view a recorded lecture from a professor that was either White, Black, Latinx, or Asian, and Male or Female. The recorded lecture used the same script and Powerpoint. The students then completed an evaluation that yielded an overall competence score (1-5 scale). Which statistical test should the researcher use to test her hypotheses about the intersectionality of race and gender? Explain.
Two-way between subjects ANOVA
There are two factors (race and gender) that are nominal, with a single DV that is on an interval scale of measurement. This is a between subjects design because participants experience only one level of each factor.
A researcher computes a 2 x 3 between subjects ANOVA in which 11 participants were observed in each group. The researcher has calculated the SS for factor A = 10, SS for the interaction between factor A and B = 40, the SS error = 240, the SS total = 340. Which effects are significant?
df for factor A: 1
df for factor B: 2
df for interaction: 2
df for error: 60
SS for factor B: 340 - 240 - 40 - 10 = 50
MS for factor A: 10/1 = 10
MS for factor B: 50/2 = 25
MS for interaction: 40/2 = 20
MS for error: 240/60 = 4
F for factor A: 10/4 = 2.5
F for factor B: 25/4 = 6.25
F for interaction: 20/4 = 5.00
F critical (1, 60) = 4.00; F critical (2, 60) = 3.15
Factor B and the interaction effect between A and B are significant; Factor A is not significant
Your data yields the following: SSX = 320, SSY = 410, SSXY = 240
Calculate the Pearson correlation coefficient
240/sqrt(320 * 410)
= 0.66
You want to examine if math ability can predict verbal ability. You score 62 participants on their math and verbal ability. You know that the total SS is 1440 and the MSregression is 80. Should you retain or reject the null hypothesis? Explain.
df for regression = 1
df for residual = 60
thus, SS for regression = 80
SS for residual = 1440 - 80 = 1360
MS for residual = 1360/60 = 22.67
F = 80/22.67 = 3.53
F critical is 4.00. Because critical value is greater than obtained value, we retain the null hypothesis -- math ability does not predict verbal ability.
You want to examine whether there is a relationship between stress levels (low vs. high) and morning sickness (yes vs. no) during pregnancy. You find that 4 women feel low levels of stress and have morning sickness, 8 women have high levels of stress and have morning sickness, 8 women have low levels of stress and no morning sickness, and 5 women have high levels of stress and no morning sickness. Conduct a chi-square test-for independence. What is your conclusion?
Observed frequency: 4, 8, 8, 5
Expected frequency: 5.76, 6.24, 6.24, 6.76
Residual: -1.76, 1.76, 1.76, -1.76
Residual2: 3.0976, 3.0976, 3.0976, 3.0976
Residual2/Expected: .5378, .4964, .4964, .4582
Sum of Residual2/Expected (chi-square): 1.99
The critical chi-square value is 3.84. Since the obtained chi-square is less than the critical chi-square, we must retain the null-hypothesis. There is no relationship between stress levels and experience of morning sickness.
You were curious whether people of a certain race are more likely to be religious. You sample 100 people, record their race, and whether they practice a religion (yes vs. no). Which statistical test would you use to examine this relationship? Explain.
Chi-square test-for-independence.
You are examining the relationship between two factors that are both on a nominal scale.