A researcher compares mean test scores for students taught with four different methods. Which procedure tests whether the four population means are equal?
One-way ANOVA
✓
Chi-square test of independence
Linear regression
Test of a single variance
Correct Answer
One-way ANOVA
One-way ANOVA tests whether several population means are equal, using one categorical factor (teaching method) and a numerical response (test score). Chi-square test of independence works with counts of two categorical variables, not means. Linear regression models a numerical y from a numerical x. Test of a single variance concerns the spread of one population.
Category: ANOVA Basics
Question 2
For an ANOVA comparing k groups, which statement is the alternative hypothesis?
All k population means are equal
At least two of the population means differ
✓
Every population mean differs from every other
The k sample means are all different
Correct Answer
At least two of the population means differ
The alternative hypothesis says the means are not all equal, which only requires at least two of them to differ. All k population means are equal is the null hypothesis. Every population mean differs from every other is far stronger than the alternative claims. The k sample means are all different is about sample statistics, which almost always differ by chance.
Category: ANOVA Basics
Question 3
Three groups each have 5 observations. The variance of the three sample means is 4, and the average of the three sample variances is 10. For equal group sizes, F = n(variance of the sample means) / (average sample variance), with n the group size. What is F?
0.4
0.5
2
✓
20
Correct Answer
2
With equal group sizes, F = 5(4) / 10 = 20 / 10 = 2. The value 0.4 divides 4 by 10 without multiplying by the group size of 5. The value 0.5 flips the ratio, 10 / 20. The value 20 is the numerator alone, before dividing by the average sample variance.
Category: F-Ratio
Question 4
An ANOVA compares 5 groups with 6 observations in each group, 30 observations in all. How many degrees of freedom does the denominator (within groups) have?
29
4
30
25
✓
Correct Answer
25
The within-groups degrees of freedom are n - k = 30 - 5 = 25. The value 29 is n - 1, the total degrees of freedom. The value 4 is k - 1, the numerator degrees of freedom. The value 30 is just the number of observations.
Category: ANOVA Table
Question 5
An ANOVA table shows MS between = 18 and MS within = 6. What is the F statistic? Round to two decimal places if needed.
3
✓
0.33
12
108
Correct Answer
3
The F statistic is MS between divided by MS within, so F = 18 / 6 = 3. The value 0.33 divides in the wrong order, 6 / 18. The value 12 subtracts the mean squares instead of dividing. The value 108 multiplies them.
Category: F-Ratio
Question 6
An ANOVA compares k = 3 groups and has SS between = 40. What is MS between? Round to two decimal places if needed.
40
20
✓
13.33
80
Correct Answer
20
MS between is SS between divided by its degrees of freedom, k - 1 = 2, so MS between = 40 / 2 = 20. The value 40 is SS between before dividing. The value 13.33 divides by k = 3 instead of k - 1. The value 80 multiplies by 2 instead of dividing.
Category: ANOVA Table
Question 7
An ANOVA comparing several groups gives F = 1.02. What does this value suggest?
The group means clearly differ
The within-group variance is zero
Between-group variation is about what chance alone produces
✓
The sample sizes are unequal
Correct Answer
Between-group variation is about what chance alone produces
When H0 is true, MS between and MS within estimate the same variance, so F near 1 is what equal population means would produce. The group means clearly differ would need F well above 1. The within-group variance is zero would make F undefined, not close to 1. The sample sizes are unequal cannot be read from the F value.
Category: F-Ratio
Question 8
Which F values lead to rejecting H0 in an ANOVA?
Values close to 0
Values close to 1
Negative values
Large values in the right tail
✓
Correct Answer
Large values in the right tail
Differences among group means make MS between large compared with MS within, so the ANOVA test is right-tailed and rejects H0 for large F. Values close to 0 mean the group means are even closer than chance would suggest. Values close to 1 are typical when H0 is true. Negative values cannot occur, since F is a ratio of nonnegative quantities.
Category: F Distribution
Question 9
An ANOVA comparing weight loss on three diets gives p = 0.02 with alpha = 0.05. Which conclusion is correct?
At least two diet means appear to differ
✓
All three diet means are equal
Every diet mean differs from the others
The diets have equal variances
Correct Answer
At least two diet means appear to differ
Because 0.02 is less than 0.05, H0 is rejected and the data suggest that the population means are not all equal. All three diet means are equal is the null hypothesis that was just rejected. Every diet mean differs from the others claims more than ANOVA shows, since it does not identify which means differ. The diets have equal variances is an assumption of the test, not its conclusion.
Category: ANOVA Basics
Question 10
Which of these is an assumption of an ANOVA comparing several group means?
The sample sizes are all equal
The populations have equal standard deviations
✓
The response variable is categorical
The samples are paired
Correct Answer
The populations have equal standard deviations
ANOVA assumes the populations have equal standard deviations, along with normal populations and independent random samples. The sample sizes are all equal is not required, since ANOVA handles unequal group sizes. The response variable is categorical is backward, because the response must be numerical. The samples are paired contradicts the requirement that samples be independent.
Category: ANOVA Assumptions
Question 11
In an ANOVA, what type of variable is the factor that defines the groups?
Numerical
Continuous
Categorical
✓
Ratio-level
Correct Answer
Categorical
The factor sorts individuals into groups, such as teaching method or diet, so it is categorical. Numerical describes the response variable, which is measured on each individual. Continuous also describes a kind of numerical variable, not a grouping variable. Ratio-level is a level of measurement for numerical data.
Category: ANOVA Assumptions
Question 12
An ANOVA table shows SS total = 100 and SS between = 30. What is SS within? Round to two decimal places if needed.
130
30
3.33
70
✓
Correct Answer
70
Total variation splits into between and within parts, so SS within = SS total - SS between = 100 - 30 = 70. The value 130 adds the two sums instead of subtracting. The value 30 repeats SS between. The value 3.33 divides SS total by SS between.
Category: ANOVA Table
Question 13
An ANOVA has k = 3 groups, n = 23 total observations and SS within = 60. What is MS within? Round to two decimal places if needed.
3
✓
2.73
20
30
Correct Answer
3
MS within is SS within divided by n - k = 23 - 3 = 20, so MS within = 60 / 20 = 3. The value 2.73 divides by n - 1 = 22. The value 20 is the degrees of freedom, not the mean square. The value 30 divides by k - 1 = 2, the between-groups degrees of freedom.
Category: ANOVA Table
Question 14
What does MS between measure in an ANOVA?
Variation of values inside each group
Variation of group means around the overall mean
✓
The gap between the largest and smallest values
The average size of the samples
Correct Answer
Variation of group means around the overall mean
MS between measures how spread out the group means are around the grand mean, so it grows when the population means differ. Variation of values inside each group is what MS within measures. The gap between the largest and smallest values is the range. The average size of the samples is not part of either mean square.
Category: F-Ratio
Question 15
What is the shape of an F distribution?
Symmetric and bell-shaped
Skewed to the left
Skewed to the right
✓
Uniform
Correct Answer
Skewed to the right
F curves start at zero and have a long tail toward large values, so they are skewed to the right. Symmetric and bell-shaped describes the normal curve, which F approaches only when both degrees of freedom are large. Skewed to the left would put the long tail toward small values, which cannot happen for a variable that starts at zero. Uniform would give a flat curve.
Category: F Distribution
Question 16
An ANOVA compares 3 groups using 18 total observations. Which distribution does its F statistic follow when H0 is true?
F(3, 18)
F(15, 2)
F(2, 17)
F(2, 15)
✓
Correct Answer
F(2, 15)
The numerator df is k - 1 = 2 and the denominator df is n - k = 18 - 3 = 15, so F follows F(2, 15). F(3, 18) uses the raw counts of groups and observations. F(15, 2) reverses the numerator and denominator. F(2, 17) uses n - 1 for the denominator instead of n - k.
Category: F Distribution
Question 17
Two independent samples from normal populations have sample variances s1^2 = 12 and s2^2 = 4. With s1^2 in the numerator, what is the F statistic for testing equal population variances? Round to two decimal places if needed.
3
✓
0.33
8
16
Correct Answer
3
When H0 says the population variances are equal, the F statistic is the ratio of sample variances, 12 / 4 = 3. The value 0.33 puts s2^2 in the numerator, 4 / 12. The value 8 subtracts the variances. The value 16 adds them.
Category: Two Variances
Question 18
Sample 1 has 11 values and sample 2 has 16 values. With sample 1's variance in the numerator, which distribution does the F statistic for comparing the two variances follow?
F(11, 16)
F(10, 15)
✓
F(15, 10)
F(9, 14)
Correct Answer
F(10, 15)
Each sample contributes its size minus 1, so the numerator df is 11 - 1 = 10 and the denominator df is 16 - 1 = 15, giving F(10, 15). F(11, 16) forgets to subtract 1. F(15, 10) reverses the numerator and denominator. F(9, 14) subtracts 2 from each sample size.
Category: Two Variances
Question 19
Why do many statisticians use the F test that compares two population variances with caution?
It can only be right-tailed
It requires equal sample sizes
It is very sensitive to departures from normality
✓
It works only for categorical data
Correct Answer
It is very sensitive to departures from normality
When the populations are not normal, the p-values of this test can be too high or too low in unpredictable ways. It can only be right-tailed is wrong because the test can be left-tailed, right-tailed or two-tailed. It requires equal sample sizes is wrong because each sample has its own degrees of freedom. It works only for categorical data is wrong because variances require numerical data.
Category: Two Variances
Question 20
A coffee chain wants to know whether the variability of customer wait times differs between two of its stores. Which test fits this question?
One-way ANOVA
Test of independence
Two-sample t-test
F test for two variances
✓
Correct Answer
F test for two variances
A question about whether two populations differ in spread is answered by the F test for two variances, which compares the ratio of the sample variances with 1. One-way ANOVA compares means of several groups. Test of independence relates two categorical variables. Two-sample t-test compares two means, not two spreads.
Category: Two Variances