Question 1

What is the term for an experimental plan that creates one treatment group for each treatment, and then randomly assigns each experimental unit to one of these groups?

Accepted Answer

The term for an experimental plan that creates one treatment group for each treatment, and then randomly assigns each experimental unit to one of these groups is a completely randomized design. Option B, randomized block design, involves creating blocks of experimental units that are similar in some way and then randomly assigning the treatments within each block. Option C, matched-pairs design, involves pairing similar experimental units and randomly assigning one from each pair to each treatment group.

Question 2

Which of the following is used as the test statistic of the single-factor ANOVA?

Accepted Answer

The mean square for treatments (MS_treatments) divided by the mean square for error (MS_error) is used as the test statistic for single-factor ANOVA. This is commonly referred to as the F-ratio or F-statistic.

Question 3

Which of the following correctly describes the treatment sum of squares in one-way ANOVA?

Accepted Answer

The treatment sum of squares in one-way ANOVA measures the variation between the means of different treatments. It is calculated as the sum of the squared deviations between each treatment sample mean and the grand mean, multiplied by the number of observations made for each treatment. This measures how much variability among the sample means can be attributed to the different treatments being tested.

Question 4

Which of the following values may be analyzed using one-way ANOVA?

Accepted Answer

One-way ANOVA is used to analyze the difference between more than two population means.

Question 5

Which of the following are underlying assumptions in order for an ANOVA completely randomized design to be used?

Accepted Answer

ANOVA assumes that the populations have a common variance (homogeneity of variance) and that samples are independent and randomly selected. It does not assume populations are F-distributed.

Question 6

In a one-way ANOVA, if the null hypothesis is  then what would the alternative hypothesis be?

Accepted Answer

The answer of In a one-way ANOVA, if the null...

Question 7

Which of the following is equal to the test statistic of the completely randomized ANOVA design?

Accepted Answer

The test statistic for a completely randomized ANOVA design is the F-ratio, which is calculated as the Mean Square for Treatments divided by the Mean Square for Error.

Question 8

In a one-way ANOVA, if  and a sample of size 25 is selected at random from each of the 5 populations, then which of the following is the correct critical value at

Accepted Answer

For a one-way ANOVA with 5 groups, the critical value is calculated using the F-distribution. The degrees of freedom for the numerator are dfn = k-1 = 4 and the degrees of freedom for the denominator are dfd = N-k = 115, where N is the total sample size (25*5 = 125) and k is the number of groups (k=5). Using a significance level of 0.05, the critical value for F is found to be 2.45. Therefore, choice C is the correct answer.

Question 9

How many degrees of freedom are there for a denominator in a one-way ANOVA test that includes three population means with ten observations sampled from each population?

Accepted Answer

The answer of How many degrees of freedom are there...

Question 10

In a completely randomized design for ANOVA, the number of degrees of freedom for the numerator and denominator for test statistic are 3 and 16, respectively. Which of the following is equal to the total number of observations?

Accepted Answer

The total number of observations is equal to the sum of the degrees of freedom for the numerator and denominator plus one: 3 + 16 + 1 = 20.

Question 11

In a one-way ANOVA test, the test statistic is F = 4.25. The rejection region is F > 3.06 for the 5% level of significance, F > 3.8 for the 2.5% level, and F > 4.89 for the 1% level. For this test, which of the following is a valid statement about the approximate p-value?

Accepted Answer

The test statistic F = 4.25 falls between the critical values for the 2.5% level (3.8) and the 1% level (4.89), indicating that the p-value is between 0.01 and 0.025.

Question 12

Which of the following correctly describes the F statistic for the completely randomized design?

Accepted Answer

The F statistic for the completely randomized design is equal to the ratio of the mean squares for treatments (MST) to the mean squares for error (MSE). This is because in the completely randomized design, we are interested in comparing the variability between groups (treatments) to the variability within groups (error). Therefore, the F statistic is calculated as MST/MSE.

Question 13

Which of the following is the distribution of the test statistic for analysis of variance?

Accepted Answer

The distribution of the test statistic for analysis of variance is the F distribution.

Question 14

In one-way ANOVA, suppose that there are four treatments with  ,  ,  , and  . What is the rejection region for this test at the 5% level of significance?
A)
B) 
C) 
D)

Accepted Answer

The answer of In one-way ANOVA, suppose that there are...

Question 15

In one-way ANOVA, which of the following statistics is used to measure the amount of total variation that is unexplained?

Accepted Answer

The sum of squares for error is used to measure the amount of total variation that is unexplained, as it represents the variation within each treatment group that cannot be accounted for by the treatment itself. The sum of squares for treatments measures the variation between the treatment groups, and the total sum of squares is the sum of both the sum of squares for treatments and the sum of squares for error. The degrees of freedom are used to calculate the sum of squares for each component.

Question 16

Which of the following is a characteristic of all members of the F distribution family?

Accepted Answer

All F distributions are positively skewed between values of 0 and infinity, with the peak of the distribution to the right of 1. The normal distribution is not a special case of the F-distribution family, as it has a different shape and only has one parameter (mean) whereas the F distribution has two parameters (degrees of freedom). Therefore, choices a, c, and d are incorrect.

Question 17

In one-way ANOVA, suppose that there are three treatments with  ,  , and  . What is the rejection region for this test at the 5% level of significance?

Accepted Answer

The rejection region for a one-way ANOVA test with three treatments at a 5% level of significance is F > F(2, 27) = 3.81. This critical value is obtained from a table or calculator, with numerator degrees of freedom = k - 1 = 3 - 1 = 2 and denominator degrees of freedom = N - k = 33 - 3 = 30.

Question 18

In the completely randomized design of ANOVA where there are k treatments and n observations, what are the degrees of freedom for the F statistic?

Accepted Answer

The degrees of freedom for the denominator (within groups) is n - k and the degrees of freedom for the numerator (between groups) is k - 1. Therefore, the correct answer is D, k - 1 and n - k.

Question 19

In one-way ANOVA, what is the sum of the squared deviations of each individual sample observation (regardless of the sample to which it belongs) from the mean of all observations?

Accepted Answer

The sum of the squared deviations of each individual sample observation from the mean of all observations is referred to as the total sum of squares (SS total) and is used in the calculation of the F statistic in one-way ANOVA.

Question 20

Which of the following is NOT a required condition for one-way ANOVA?

Accepted Answer

Equal sample sizes are desirable but not required for one-way ANOVA. The other three conditions are required: normality of populations, equality of population variances, and random and independent sampling.

Quiz 11: A: The Analysis of Variance