Answers to Your Turn exercises

Answers to 5.3 Power

In the following table, determine where each of the pieces should go. Note that we have six things to populate but only four cells: each cell must contain at least one of the six things. Think critically here before testing your answers!

H0 is true H1 is true
p < .05 (statistically significant) A B
p > .05 (statistically non-significant) C D

Which cell should each of the following items go?

  1. alpha (answer = A)
  2. power (answer = B)
  3. type I error (answer = A)
  4. Type II error (answer = D)
  5. Correct inference (answer = B & C)

Let’s see what happens when we both increase power AND decrease alpha. Fill out the table on your own. When we assume the null and alternative hypotheses are 50% likely each, and we set our alpha to 1% and our power to 95%, how much more likely is it that the alternative hypothesis is true than the null hypothesis is true? (answer = 95 times more likely!)

Answers to 7.1 one-sample t-test

  1. Do the students in our dataset have a higher Writing score than than the passing score? (M = 70)?

    • Should you use a one-tailed or two-tailed hypothesis? (answer = one-tailed)

    • Which statistic should you use based on your assumptions? (answer = Wilcoxon rank one sample t-test)

    • Do the students in our dataset have a higher Writing score than than the passing score? (answer = yes)

  2. Do the students in our dataset have the same national average height of college students (M = 68 inches)?

    • Should you use a one-tailed or two-tailed hypothesis? (answer = two-tailed)

    • Which statistic should you use based on your assumptions? (answer = student one sample t-test)

    • Do the students in our dataset have the same national average height of college students)? (answer = no)

Answers to 7.2 independent samples t-test

  1. Does height differ by gender (Gender: male = 0, female = 1)?

    • Should you use a one-tailed or two-tailed hypothesis? (answer = two-tailed)

    • Which statistic should you use based on your assumptions? (answer = Mann Whitney U)

    • Does height differ by gender? (answer = yes)

  2. Do athletes (Athlete: athletes = 1, non-athlete = 0) have faster sprint times than non-athletes?

    • Should you use a one-tailed or two-tailed hypothesis? (answer = one-tailed)

    • Which statistic should you use based on your assumptions? (answer = Mann Whitney U)

    • Do athletes have faster sprint times than non-athletes? (answer = yes)

  3. Do students who live on campus (LiveOnCampus: on campus = 1, off campus = 0) have higher English scores than students who live off campus?

    • Should you use a one-tailed or two-tailed hypothesis? (answer = one-tailed)

    • Which statistic should you use based on your assumptions? (answer = Welch t-test)

    • Does students who live on campus have higher English scores? (answer = no)

  4. Does athletic status relate to math scores?

    • Should you use a one-tailed or two-tailed hypothesis? (answer = two-tailed)

    • Which statistic should you use based on your assumptions? (answer = independent t-test)

    • Does athletic status relate to math scores? (answer = yes)

Answers to 7.3 dependent t-test

Note: Technically, none of our data is suitable for a dependent t-test in this dataset. We will pretend that the four test score variables (English, Reading, Math, and Writing) are really four measurements of the same underlying test. In reality, we would analyze this data using correlation.

  1. Do students perform better on the English test than they do the Writing test?

    • Should you use a one-tailed or two-tailed hypothesis? (answer = one-tailed)

    • Which statistic should you use based on your assumptions? (answer = dependent t-test)

    • Do students perform better on the English test than they do the Writing test? (answer = yes)

  2. Does students’ English scores relate to their Reading scores?

    • Should you use a one-tailed or two-tailed hypothesis? (answer = two-tailed)

    • Which statistic should you use based on your assumptions? (answer = dependent t-test)

    • Does students’ English scores relate to their Reading scores? (answer = no)

Answers to 8.1 goodness of fit test

  1. Are there equal numbers of athletes and non-athletes? (Athlete variable)

    • Do you meet the assumptions? (answer = yes)

    • Are the observed frequencies similar to the expected frequencies? (answer = no)

    • What is your chi-square value, rounded to two decimal places? (answer = 10.32)

  2. I happen to know the school this data comes from has 40% athletes and 60% non-athletes. Does our data match the school population?

    • Change your Expected Proportions ratio to .6 for non-athletes and .4 for athletes.

    • Are the observed frequencies similar to the expected frequencies? (answer = yes)

    • What is your chi-square value, rounded to two decimal places? (answer = .96)

  3. Are there equal numbers of freshmen, sophomores, juniors, and seniors? (Rank variable)

    • Do you meet the assumptions? (answer = yes)

    • Are the observed frequencies similar to the expected frequencies? (answer = no)

    • What is your chi-square value, rounded to two decimal places? (answer = 33.94)

Answers to 8.2 chi-square test of independence

  1. Is Athlete related to Gender?

    • Do you meet the assumptions? (answer = yes)

    • Which test should you perform? (answer = chi-square)

    • Are the observed frequencies similar to the expected frequencies? (answer = no)

    • What is your chi-square value, rounded to two decimal places? (answer = 8.45)

  2. Is Gender related to Rank?

    • Do you meet the assumptions? (answer = yes)

    • Which test should you perform? (answer = chi-square)

    • Are the observed frequencies similar to the expected frequencies? (answer = yes)

    • What is your chi-square value, rounded to two decimal places? (answer = .61)

Answers to 9.1 one-way ANOVA

  1. Does students differ on English scores by rank (i.e., freshmen, sophomore, junior, senior)?

    • Do you satisfy the assumption of normality? (answer = yes)

    • Do you satisfy the assumption of homogeneity of variance? (answer = yes)

    • Which statistic should you use? (answer = one-way ANOVA)

    • Do students differ on English scores by rank? (answer = no)

  2. Does smoking status (Smoking: Nonsmoker = 0, Past smoker = 1, Current smoker = 2) relate to sprint time?

    • Do you satisfy the assumption of normality? (answer = no)

    • Do you satisfy the assumption of homogeneity of variance? (answer = yes)

    • Which statistic should you use? (answer = Kruskal-Wallis test)

    • Does smoking status relate to sprint time? (answer = yes)

Answers to 9.2 finding group differences

  1. Does students differ on English scores by rank (i.e., freshmen, sophomore, junior, senior)?

    • Perform Tukey’s post hoc tests. What are the results of the post hoc comparison? (answer = trick question! you wouldn’t perform them because the F-test is not significant)

  2. Does smoking status (Smoking: Nonsmoker = 0, Past smoker = 1, Current smoker = 2) relate to sprint time?

    • Perform Tukey’s post hoc tests. What are the results of the post hoc comparison? (answer = Nonsmokers had significantly faster sprint times than current smokers)

Answers to 9.3 repeated measures ANOVA

  1. Does students differ on their test scores (English, Reading, Math, Writing)?

    • Based on your understanding of the nature of the test scores, which statistic should you use? (answer = repeated measures ANOVA)

    • Should you apply a sphericity correction? If so, which one? (answer = yes, Huynh-Feldt)

    • Do students differ on their test scores? (answer = yes)

    • Should you perform a planned contrast or post hoc comparison? (answer = yes)

    • What are the results of the post hoc comparison?

Answers to 10.1 correlation

  1. Are there significant correlations among the four tests (English, reading, math, writing)?

    • Do you meet the assumption of normality for all four tests? (answer = yes for all but maybe not writing)

    • Do you meet the assumption of linearity for all four tests? (answer = yes)

    • Are the four tests significantly correlated among each other? (answer = yes)

    • Round your answers to two decimal places:

      • What is the correlation between reading and math? (answer = .52)

      • What is the correlation between writing and reading? (answer = .11)

      • What is the correlation between writing and English? (answer = .37)

Answers to 10.2 regression

  1. Perform a multiple regression examining how English, Reading and Writing, as well as Gender relate to the dependent variable Math.

    • Do you have any significant outliers? (answer = no)

    • Are your residuals normally distributed? (answer = yes)

    • Do you satisfy the assumption of linearity and homoscedasticity of your residuals (just check the Fitted residual plot)? (answer = yes)

    • Do you meet the assumption of independent residuals? (answer = yes)

    • Do you meet the assumption of no multicollinearity? (answer = yes)

    • Can you perform a regression with this data? (answer = yes)

    • What is your adjusted R-squared, rounded to two decimal places? (answer = .31)

    • Is the overall model statistically significant? (answer = yes)

    • Is English statistically significant? (answer = no)

    • Is Reading statistically significant? (answer = yes)

    • Is Writing statistically significant? (answer = yes)

    • Is Gender statistically significant? (answer = yes)

    • For Gender, do male (Gender = 0) or female (Gender = 1) students have higher math scores? (answer = male)