Question 1

In employing the randomized block design, the primary interest lies in reducing sum of squares for blocks (SSB).

Accepted Answer

The primary interest in employing the randomized block design is to reduce the sum of squares due to error (SSE), not the sum of squares for blocks (SSB), by accounting for variability due to the blocks.

Question 2

In a two-way ANOVA, there are 4 levels for factor A, 3 levels for factor B, and 3 observations within each of the 12 factor combinations. The number of treatments in this experiment will be 36.

Accepted Answer

The number of treatments in a two-way ANOVA is equal to the product of the number of levels for each factor. Therefore, the number of treatments in this experiment is 4 x 3 = 12.

Question 3

A randomized block design ANOVA has two treatments. The test to be performed in this procedure is equivalent to dependent samples t-test.

Accepted Answer

A randomized block design ANOVA is used to compare the means of two treatments while controlling for the effects of a blocking variable. This test is mathematically equivalent to a dependent samples t-test, which is used to compare the means of two related samples. Therefore, the answer is "True".

Question 4

In a two-way ANOVA, where a is the number of factor A levels and b is the number of factor B levels, the number of the degrees of freedom for the "error term" is:

Accepted Answer

The degrees of freedom for the error term in a two-way ANOVA is calculated as $nab - a - b + 1$, where $n$ is the number of observations per group, $a$ is the number of levels of factor A, and $b$ is the number of levels of factor B. None of the provided options exactly matches this formula, but option B, $nab$, is the closest in form. However, it's important to note that the correct formula for the degrees of freedom for the error term actually includes subtracting the degrees of freedom for the main effects and the interaction (if applicable), which is not perfectly represented by any of the options given. Therefore, my initial selection based on the options provided might not accurately reflect the precise calculation for degrees of freedom for the error term in a two-way ANOVA. The correct formula for the degrees of freedom for the error term is indeed not directly listed among the options.

Question 5

If we first arrange test units into similar groups before assigning treatments to them, the test design we should use is the randomized block design.

Accepted Answer

The randomized block design is a test design where test units are first grouped based on their similarities and then treatments are assigned to these groups. This helps to decrease the variability among the test units and increase the precision of the experiment. Therefore, if we first arrange test units into similar groups, we should use the randomized block design.

Question 6

When the problem objective is to compare more than two populations, the experimental design that is the counterpart of the matched pairs experiment is called the randomized block design.

Accepted Answer

The randomized block design is used when the objective is to compare more than two populations, and it is the counterpart of the matched pairs experiment in this context.

Question 7

Three tennis players, a beginner, an experienced, and a professional, have been randomly selected from the membership of a large city tennis club. Using the same ball, each person hits four serves with each of five racquet models, with the five racquet models selected randomly. Each serve is clocked with a radar gun and the result recorded. Among ANOVA models, this setup is most like the randomized block design.

Accepted Answer

The situation described involves randomly assigning five racquet models to each of the three tennis players, meaning each player serves with the same racquet model four times in a row (a block). The order in which the racquet models are assigned is randomized, making it similar to a randomized block design. The ANOVA model is appropriate for analyzing the differences in serve speeds across the different racquet models and between the different players.

Question 8

Blocking is a procedure that lets extraneous factors operate during an experiment but assures - by virtue of the random selection of experimental units and their subsequent random assignment to experimental and control groups - that each treatment has an equal chance to be enhanced or handicapped by these factors.

Accepted Answer

Blocking is a technique used to control for the effects of extraneous factors by grouping similar experimental units together, not by random assignment, to ensure that these factors do not confound the results.

Question 9

In a two-way ANOVA, there are 4 levels for factor A, 3 levels for factor B, and two observations within each of the 12 factor combinations. The number of treatments in this experiment will be 12.

Accepted Answer

In a two-way ANOVA, the number of treatments is determined by multiplying the levels of factor A by the levels of factor B. Here, with 4 levels for factor A and 3 levels for factor B, there are indeed 4*3 = 12 treatments.

Question 10

The purpose of designing a randomized block experiment is to reduce the between-treatments variation (SST) to more easily detect differences between the treatment means.

Accepted Answer

The purpose of designing a randomized block experiment is to reduce the within-treatment variation by accounting for variability due to other factors (blocks), thereby making it easier to detect differences between treatment means. It does not directly aim to reduce the between-treatments variation (SST), but rather to control for extraneous variation.

Question 11

The randomized block design is also called the two-way analysis of variance.

Accepted Answer

The randomized block design is a type of experimental design that involves dividing subjects into blocks based on some characteristic, and then randomly assigning treatments within each block. This design is often used when there is a potential confounding variable that may affect the response variable, and the goal is to control for this variable. The analysis of this design involves using a two-way ANOVA with one factor being the treatment and the other factor being the blocking variable. Therefore, the randomized block design is also referred to as the two-way ANOVA.

Question 12

The randomized block design is a two-way classification design.

Accepted Answer

The randomized block design is a two-way classification design since it involves the classification of treatments and blocks. The treatments are the main factor of interest, while the blocks represent any other factor that can influence the response variable. Therefore, it is important to control the effect of the blocks by randomly assigning the treatments to the blocks in order to reduce the noise and improve the precision of the estimates.

Question 13

The randomized block design with two treatments is equivalent to a non-directional dependent samples z-test.

Accepted Answer

The randomized block design is an experimental design used to control for the effects of known potential confounding variables, and it typically involves more than two treatments or compares different treatments, not just two. A non-directional dependent samples z-test is used to compare the means of two related groups, assuming the difference scores are normally distributed. These are different statistical methods used for different types of data and research questions.

Question 14

A researcher at Michigan State University (MSU) wanted to determine whether different building signs (building maps versus wall signage) affect the total amount of time visitors require to reach their destination and whether that time depends on whether the starting location is inside or outside the building. Three subjects were assigned to each of the combinations of signs and starting locations, and travel time in seconds from beginning to destination was recorded. A partial computer output of the appropriate analysis is given below:  The degrees of freedom for the different building signs are ______________.
The degrees of freedom for the different starting location are ______________.
The degrees of freedom for the interaction between the levels of signs and starting location are ______________.
The error degrees of freedom are ______________.
The mean squares value for starting location is ______________.
The F test statistic for testing the main effect of types of signs is ______________.
The F test statistic for testing the interaction effect between the types of signs and the starting location is ______________.
In order to determine the critical value of the F ratio against which to test for differences between the levels of factor A, we should use numerator df = ______________, and denominator df = ______________.
In order to determine the critical value of the F ratio against which to test for differences between the levels of factor B, we should use numerator df = ______________, and denominator df = ______________.
In order to determine the critical value of the F ratio against which to test for interaction between levels of Factor A and levels of Factor B, we should use numerator df = ______________, and denominator of F = ______________.

Accepted Answer

The answer of A researcher at Michigan State University (MSU)...

Question 15

The F-test of the randomized block design of the analysis of variance has the same requirements as the independent samples design; that is, the random variable must be normally distributed and the population variances must be equal.

Accepted Answer

The F-test in both randomized block design and independent samples design of analysis of variance requires that the data be normally distributed and that the variances across groups (or blocks) are homogenous (equal). These assumptions ensure the validity of the F-test results.

Question 16

The completely randomized design is an experimental plan that divides all available experimental units into blocks of fairly homogeneous units - each block containing as many units as there are treatments or some multiple of that number - and then randomly matches each treatment with one or more units within each block.

Accepted Answer

The description given is for a randomized block design, not a completely randomized design. In a completely randomized design, all experimental units are assigned randomly to treatments without forming blocks of homogeneous units.

Question 17

A randomized block experiment having five treatments and four blocks produced the following values: Total SS = 1500, SST = 275, SSE = 153. The value of MSB must be 268.

Accepted Answer

Given Total SS = 1500, SST = 275, SSE = 153, we can calculate SSB (Sum of Squares for Blocks) and then MSB (Mean Square for Blocks). First, find SSB using the formula SSB = Total SS - SST - SSE. SSB = 1500 - 275 - 153 = 1072. MSB is calculated by dividing SSB by the degrees of freedom for blocks, which is the number of blocks minus one (4-1=3). Therefore, MSB = 1072 / 3 = 357.33, not 268.

Question 18

A randomized block design ANOVA has five treatments and four blocks. The computed test statistic (value of F) is 6.25. With a 0.05 significance level, the conclusion will be to accept the null hypothesis.

Accepted Answer

To draw a conclusion, one needs to compare the computed test statistic with the critical value of F. The critical value of F for a 0.05 significance level with four and 20 degrees of freedom (four blocks and five treatments) is 2.866. Since 6.25 is greater than 2.866, the null hypothesis is rejected in favor of the alternative hypothesis. Therefore, the answer is False, not True.

Question 19

Two samples of ten each from the male and female workers of a large company have been taken. The data involved the wage rate of each worker. To test whether there is any difference in the average wage rate between male and female workers a pooled-variances t-test will be considered. Another test option to consider is ANOVA. The most likely ANOVA to fit this test situation is the randomized block design.

Accepted Answer

The most appropriate ANOVA for this situation is the one-way ANOVA, not the randomized block design, because there is only one factor (gender) being compared across two groups (male and female workers).

Question 20

In the analysis of variance, the sum of the squared deviations between each block sample mean and the grand mean, multiplied by the number of observations made for each block, multiplied by the number of observations per cell (which may well equal 1) equals the blocks mean square.

Accepted Answer

In the analysis of variance, the description given mixes elements of calculating the sum of squares for blocks with the calculation of mean squares. The mean square for blocks is actually calculated by dividing the sum of squares for blocks by the degrees of freedom for blocks, not by multiplying deviations by the number of observations per cell.

Quiz 11: The Analysis of Variance