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ANALYSIS OF VARIANCE(ANOVA)

- BCT2053
- CHAPTER 6

CONTENT

- 6.1 Analysis of Variance Purpose
- and Procedure
- 6.2 One-Way ANOVA
- 6.3 Two-Way ANOVA

OBJECTIVE

- After completing this chapter you should be able

to - Explain the purpose of ANOVA
- Identify the assumptions that underlie the ANOVA

technique - Describe the ANOVA hypothesis testing procedure
- Use the one-way ANOVA technique to determine if

there is a significance difference among three or

more means - Use the two-way ANOVA technique to determine if

there is an effect of interaction between two

factors experiment

6.1 Analysis of Variance Purpose and

Procedure

- ANOVA
- the approach that allows us to use sample data to

see if the values of three or more unknown

population means are likely to be different - Also known as factorial experiments
- this name is derived from the fact that in order

to test for statistical significance between

means, we are actually comparing (i.e.,

analyzing) variances. - Example of problems
- A manager want to evaluate the performance of

three (or more) employees to see if any

performance different from others - A marketing executive want to see if theres a

difference in sales productivity in the 5 company

region - A teacher wants to see if theres a difference in

students performance if he use 3 or more

approach to teach.

The Procedural Steps for an ANOVA Test

- State the Null and Alternative hypothesis
- Select the level of significance
- Determine the test distribution to use
- Define rejection or critical region
- State the decision rule
- Compute the test statistic
- Make the statistical decision

6.2 One-Way ANOVA

- Only one classification factor is considered

Response/ outcome/ dependent variable (samples)

(The level of the factor)

Replicates (1,,j) The object to a given

treatment

The resulting input grid of factorial experiment

where, i 1, 2, a is the number of levels

being tested. j 1, 2, is the number of

replicates at each level.

Assumptions

- To use the one-way ANOVA test, the following

assumptions - must be true
- The population under study have normal

distribution - The samples are drawn randomly, and each sample

is independent of the other samples. - All the populations from which the samples values

are obtained, have the same unknown population

variance, that is for k number of populations,

The Null and Alternative hypothesis

(All population means are equal)

If Ho is true we have k number of normal

populations with

(Not all population means are equal)

If H1 is true we may have 3 normal populations

with

The format of a general one-way ANOVA table

T kn

- Reject Ho if

Example 1

- The data shows the Maths test score for 4 group

of student with 3 different methods of study.

Test the hypothesis that theres no difference

between the Maths score at significance level

0.05.

Example 2

- An experiment was performed to determine whether

the annealing temperature of ductile iron affects

its tensile strength. Five specimens were

annealed at each of four temperatures. The

tensile strength (in ksi) was measured for each

temperature. The results are presented in the

following table. Can you conclude that there are

differences among the mean strengths?

6.3 Two-Way ANOVA

- Only two classification factor is considered

Assumptions

- The standard two-way ANOVA tests are valid under

the following conditions - The design must be complete
- Observations are taken on every possible

treatment - The design must be balanced
- The number of replicates is the same for each

treatment - The number of replicates per treatment, k must be

at least 2 - Within any treatment, the observations

- are a simple random sample from a normal

population - The sample observations are independent of each

other (the samples are not matched or paired in

any way) - The population variance is the same for all

treatments.

The format of a general two-way ANOVA table

Row effect

Column effect

Interaction effect

Procedure for Two-Way ANOVA

Ho No interaction between two factors

Yes (Reject Ho)

No (Accept Ho)

Ho No effects from the row factor (the row means

are equal)

Ho No effects from the column factor (the column

means are equal)

Example 1

- A chemical engineer is studying the effects of

various reagents and catalyst on the yield of a

certain process. Yield is expressed as a

percentage of a theoretical maximum. 4 runs of

the process were made for each combination of 3

reagents and 4 catalysts. Construct an ANOVA

table and test is there an interaction effect

between reagents and catalyst.

Example 2

- A study was done to determine the effects of two

factors on the lather ability of soap. The two

factors were type of water and glycerol. The

outcome measured was the amount of foam produced

in mL. The experiment was repeated 3 times for

each combination of factors. The result are

presented in the following table. Construct an

ANOVA table and test is there an interaction

effect between factors.

Summary

- The other name for ANOVA is experimental design.
- ANOVA help researchers to design an experiment

properly and analyzed the data it produces in

correctly way.

Thank You

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