Create an individual value plot of your data. Does there appear to be an optimal

Question

Create an individual value plot of your data. Does there appear to be an optimal shape and weight for distance of flight (use minitab) Many factors will influence the distance a paper airplane will fly. Weight and shape are two factors. Through this project we will attempt to determine the optimal shape and weight for distance. We will also compare regression and ANOVA to better understand key differences between the methods. In your data, you have four groups and the groups will probably have different means. We will use ANOVA and regression to test whether these differences are statistically significant. The ANOVA model for this project is: y,,-+ a, +6,, fr i= 1,2,3,4 and j=1,2,3,4,5 where E,-N(0,2) It's possible that the 4 means will lie on a line. If so, a test that the slope is zero (Ho : A = 0 ) will also test for a difference in the four means. A linear regression model may also be appropriate: + For the data you collected n = 20 because you did 5 trials with each of 4 different models. HB 4. HB 48 7.ST HB HB

Explanation / Answer

ANOVA

Back-up Theory

With the terminology already given in the question,

Null hypothesis: H0: 1 = 2 = 3 = 4 = 0 Vs Alternative: HA: H0 is false, i.e., at least one of the i’s is different from others.

Now, to work out the solution,

Terminology:

Row total = yi.= sum over j of yij

Grand total = G = sum over i of yi.

Correction Factor = C = G2/N, where N = total number of observations = sum of ni’s

Total Sum of Squares: SST = (sum over i,j of yij2) – C

Group Sum of Squares: SSR = {sum over i of (yi.2/ni)} – C

Error Sum of Squares: SSE = SST – SSR

Degrees of Freedom (DF): for SST – total number of observations – 1; for SSR – number of rows – 1; for SSE – (DF for SST - DF for SSR)

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

Fcrit: upper % point of F-Distribution with degrees of freedom n1 and n2, where n1 is the DF for SSR and n2 is the DF for SSE

Significance: Fobs is significant if Fobs > Fcrit

       ANOVA TABLE      

Source of

Variation

Degrees of Freedom (DF)

Sum of squares (SS)

Mean Sum

of squares

(MS = SS/DF)

Fobs

Fcrit*

Significance**

Group

r - 1

SSR

MSR/MSE

Error

N - rr

SSE

Total

N - 1

SST

     NOTE:

     * Fcrit: upper % 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: Fobs is significant if Fobs > Fcrit

Summary Excel Calculations

NOTE: Question says 4 groups of 5 values each. But, actual values given add to 22. The data is taken as given and analysis is based on 22 values

j

LP

HP

LD

HD

1

11.1

12.7

5.3

8.9

2

7.4

9.3

12.1

4.8

3

8.1

10.2

9.1

7.2

4

7.5

9.1

9.5

9.8

5

9.1

9.1

8.0

8.1

6

10.0

13.7

alpha

0.05

#treat

4

n1

6

n4

6

n2

5

n3

5

n

22

y1.

53.2

y4.

52.5

y2.

50.4

y3.

44

G = y..

200.1

C

1820

Sy1j^2

482.64

Sy4j^2

503.43

Sy2j^2

517.44

Sy3j^2

411.56

Syij^2

1915.07

Syi.^2/ni

1826.31

SST

95.0695

SSR

6.31321

SSE

88.7563

ANOVA TABLE

Source

DF

SS

MS

F

Fcrit

p-value

Activity

3

6.3132

2.104404

0.426778

3.159908

0.73621

Error

18

88.756

4.930907

Total

21

95.07

4.527121

Decision

Since Fcal < Fcrti, null hypothesis is accepted.

=> there is not enough evidence to suggest that the means of the different groups are different.

DONE

Source of

Variation

Degrees of Freedom (DF)

Sum of squares (SS)

Mean Sum

of squares

(MS = SS/DF)

Fobs

Fcrit*

Significance**

Group

r - 1

SSR

MSR/MSE

Error

N - rr

SSE

Total

N - 1

SST

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