Question

A study is to be conducted to determine customer preference for short-travel push buttons on an automobile radio. There are two types: tact and rubber. The tact push buttons have a hard, firm feel, whereas the rubber push buttons have a soft, smooth feel. 10 people will be random selected and divided into two groups. Each group will be asked to rate one type of button on a scale from 1 to 10, with 10 denoting the highest level of satisfaction with performance of the push buttons.

A. Prepare a data layout.

B. Write a mathematical model for this experiment.

C. Prepare a partial ANOVA summary. Include the expected mean squares. Determine the appropriate tests that can be run and comment.

Answer 1

Solution

Let xij represent the jth observation in the ith row, j = 1,2,…,n; i = 1,2,……,r

Then the ANOVA model is: xij = µ + αi + εij, where µ = common effect, αi = effect of ith row, and εij is the error component which is assumed to be Normally Distributed with mean 0 and variance σ².

Null hypothesis: H₀: α₁ = α₂ = α₃ = 0 Vs Alternative: H_A: H₀ is false [at least one αi is different from others]

Part (A)

Button Type	Ratings
Tact	x11, x12,…….., x1n1
Rubber	X21, x22,…….., x2n2

xij represents the rating of the jth person for the ith type of button, j = 1, 2, ….., ni, i = 1, 2; n1 = number of persons in the group rating Tact and n2 = number of persons in the group rating Rubber. ANSWER

Part (b)

The mathematical model is: xij = µ + αi + εij, where µ = common effect, αi = effect of ith type of button, and εij is the error component which is assumed to be Normally Distributed with mean 0 and variance σ².

Null hypothesis: H₀: α₁ = α₂ Vs Alternative: H_A: H₀ is false [α1 is different from α2]

ANSWER

Part (c)

ANOVA TABLE      

Source of Variation	Degrees of Freedom (DF)	Sum of squares (SS)	Mean Sum of squares (MS = SS/DF)	Fobs	Fcrit*	Significance**
Row	r - 1	SSR		MSR/MSE
Error	rn - r	SSE
Total	rn - 1	SST

NOTE:

     * F_crit: upper α% point of F-Distribution with degrees of freedom n1 and n2, where n1

     is the DF for Row and n2 is the DF for Error

     ** Significance: F_obs is significant if F_obs > F_crit

Now, to work out the solution,

Terminology:

Row total = xi.= sum over j of xij

Grand total = G = sum over i of xi.

Correction Factor = C = G²/N, where N = total number of observations = r x n =

Total Sum of Squares: SST = (sum over i,j of xij²) – C

Row Sum of Squares: SSR = {(sum over i of xi.²)/(n)} – C

Error Sum of Squares: SSE = SST – SSR

Mean Sum of Squares = Sum of squares/Degrees of Freedom

DONE