QBM 3200 - Quiz 3

Analysis of variance (ANOVA)

1. can be used to analyze the data obtained from experimental or observational studies. 2. can be used to test for the equality of three or more population means. 3. can be used to analyze the data obtained from studies on 3 or more populations. Ho: Mu1 = Mu2 = Mu3 .... Ha: Not all population means are equal Rejection Rule: If H0 is rejected, we cannot conclude that all population means are different. Rejecting H0 means that at least two population means have different values. 4. can be viewed as the process of partitioning the total sum of squares and the degrees of freedom into their corresponding sources: treatments and error See (Assumptions of ANOVA)

Between-Treatments is Within-Treatments is

Estimate of Population Variance. Sigma Squared

12. An experimental design that permits statistical conclusions about one factor is a a. factorial design b. randomized block design c. completely randomized design d. either completely randomized design or randomized block design

completely randomized design

The ANOVA procedure is a statistical approach for testing a. equality of means of two or more populations b. equality of variances of two or more populations c. equality of standard deviations of two or more populations d. none of the other three options is correct

equality of means of two or more populations

A treatment

is a level of a factor. (example: an exact temp. variable of 350 degrees or time of 20 min)

A factor

is a variable that the experimenter has selected for investigation. (example: temp or time in cooking)

A completely randomized design

is an experimental design in which the treatments are randomly assigned to the experimental units.

In an observational study

no attempt is made to control the factors.

In a single factor experimental design where the experimental units are randomly assigned to the treatments when heterogeneity of experimental units is not suspected is known as a. randomized block design b. random factor design c. factor block design d. none of the other three alternatives is correct

none of the other three alternatives is correct. completely randomized design

In an experimental study

one or more factors are controlled so that data can be obtained about how the factors influence the variables of interest

SST divided by its degrees of freedom nT - 1 is the

overall sample variance that would be obtained if we treated the entire set of observations as one data set.

The repetition of experimental condition is known as a. replication b. partition c. experimental condition d. factor

replication

In the analysis of variance procedure (ANOVA), variable refers to a. the independent variable b. the dependent variable c. level or a treatment d. the critical value of F

the independent variable

In the ANOVA, factor levels are a. treatments b. experimental units c. the dependent variable d. the response

treatments

An ANOVA procedure is used for Completely Randomized Design data obtained from two samples, each comprised of 10 observations. The numerator and denominator (respectively) degrees of freedom for the critical value of F are a. 2 and 19 b. 1 and 19 c. 1 and 18 d. 2 and 18

1 and 18 If the null hypothesis is true and the ANOVA assumptions are valid, the sampling distribution of MSTR/MSE is an F distribution with MSTR d.f. equal to k-1 and MSE d.f. equal to nT-k. If the means of the k populations are not equal, the value of MSTR/MSE will be inflated because MSTR overestimates σ2

Assumptions for ANOVA

1. For each population, the response (dependent) variable is normally distributed. 2. The variance of the response variable, denoted sigma squared, is the same for all of the populations. Meaning the spread of the distribution is the same for all populations 3. The observations must be independent

In an analysis of variance problem involving 3 treatments and 5 observations per treatment, SSE = 120.0. The MSE for this situation is a. 10.0 b. 12.0 c. 8.0 d. 60.0

10.0 MSE= SSE/Nt-K 120/(15-3)=10

In an analysis of variance problem involving Completely Randomized Design, if SST = 20 and SSTR = 8, then SSE is a. 12 b. 28 c. 20 d. 8

12 SST-SSTR=SSE 20-8=12

An ANOVA procedure is used for Completely Randomized Design data that was obtained from five sample groups each comprised of six observations. The degrees of freedom for the critical value of F are a. 4 and 29 b. 5 and 29 c. 4 and 25 d. 5 and 30

4 and 25 / Treatments-SSTR-DF (K-1) / and Error-SSE-DF(Nt-k)

Testing for the Equality of k Population Means: A Completely Randomized Design

Hypotheses H0: Mu1 = Mu2 = Mu3 Ha: Not all population means are equal Test Statistic: F=MSTR/MSE Rejection Rule P-value Approach Reject H0 if p-value < a Critical Value Approach Reject H0 if F > Fa where the value of F-alpha is based on an F distribution with k - 1 numerator d.f. and nT - k denominator d.f.

Randomized Block Design

If the experimental units are heterogeneous, blocking can be used to form homogeneous groups, resulting in a randomized block design. A completely randomized design is an experimental design in which the treatments are randomly assigned to the experimental units. For a randomized block design the sum of squares total (SST) is partitioned into three groups: sum of squares due to treatments, sum of squares due to blocks, and sum of squares due to error. SST = SSTR + SSBL + SSE The total degrees of freedom, nT - 1, are partitioned such that k - 1 degrees of freedom go to treatments, b - 1 go to blocks, and (k - 1)(b - 1) go to the error term.

Factorial Experiment

In some experiments we want to draw conclusions about more than one variable or factor. Factorial experiments and their corresponding ANOVA computations are valuable designs when simultaneous conclusions about two or more factors are required. The term factorial is used because the experimental conditions include all possible combinations of the factors. For example, for a levels of factor A and b levels of factor B, the experiment will involve collecting data on ab treatment combinations.

mean square due to treatments

Numerator is called the sum of squares due to treatments (SSTR). Denominator is the degrees of freedom associated with SSTR.

Which of the following is not a required assumption for the analysis of variance? a. Populations have equal means b. The variance associated with the response variable must be the same for all populations c. The observations must be independent d. The random variable of interest for each population has a normal probability distribution.

Populations have equal means

Sampling Distribution of Given H0 is True

Sample means are close together because there is only one sampling distribution when H0 is true.

Sampling Distribution of Given H0 is False

Sample means come from different sampling distributions and are not as close together when H0 is false.

Two-Factor Factorial Experiment

The ANOVA procedure for the two-factor factorial experiment is similar to the completely randomized experiment and the randomized block experiment. We again partition the sum of squares total (SST) into its sources. SST = SSA + SSB + SSAB + SSE The total degrees of freedom, nT - 1, are partitioned such that (a - 1) d.f. go to Factor A, (b - 1) d.f. go to Factor B, (a - 1)(b - 1) d.f. go to Interaction, and ab(r - 1) go to Error.

Comparing the Variance Estimates:

The F Test

In an analysis of variance, one estimate of σ2 is based upon the variation of the sample observation

The estimate ofσ2 based on the variation of the sample observations within each sample is called the mean square error and is denoted by MSE. Numerator is called the sum of squares due to error (SSE) Denominator is the degrees of freedom associated with SSE

The independent variable of interest in an ANOVA procedure is called a. a factor b. a treatment c. either a partition or a treatment d. a partition

a factor

Experimental units

are the objects of interest in the experiment. (example: what you are collecting data on. The subject or test object)

Statistical studies can be classified as being either

experimental or observational

Cause-and-effect relationships are easier to establish in

experimental studies than in observational studies

5. In factorial designs, the effects of the response produced at different levels of a factor is known as a. interaction b. replication c. main effect d. none of the other three alternatives is correct

main effect

The estimate of sigma squared based on the variation of the sample means is called the

mean square due to treatments and is denoted by MSTR.

The required condition for using an ANOVA procedure on data from several populations is that the a. sampled populations have equal means b. sampled populations are all uniform c. the selected samples are dependent on each other. d. none of the other three options is correct

none of the other three options is correct

15. In an analysis of variance, one estimate of σ2 is based upon the sum of squared deviations within the sample data and the a. overall mean b. sample means c. sum of observations d. populations have equal means

sample means. The estimate of σ2 based on the variation of the sample means is called the mean square due to treatments and is denoted by MSTR. Numerator is called the sum of squares due to treatments (SSTR) Denominator is the degrees of freedom associated with SSTR K number of treatments - 1 = DF

If the null hypothesis is true and the ANOVA assumptions are valid

the sampling distribution of MSTR/MSE is an F distribution with MSTR d.f. equal to k - 1 and MSE d.f. equal to nT - k

If the means of the k populations are not equal

the value of MSTR/MSE will be inflated because MSTR overestimates sigma^2

Dividing the sum of squares by the appropriate degrees of freedom provides

the variance estimates and the F value used to test the hypothesis of equal population means