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Question 1:

ANOVA:

The data is divided into two parts, one part contains 8 observations and the other part contains 7 observations, the table below summarizes the data:

group 1

group 2

1

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30

0

2

30

0

3

20

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20

4

20

20

5

0

30

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6

0

30

7

0

30

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8

0

Total

100

130

230

N

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8

7

15

mean

12.5

18.57143

The mean for group one data is 12.5 and for group 2 the mean is 18.57, in order to construct the ANOVA table the square of the observations in each group is determined and totals calculated, the table below summarizes the results:

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group 1

x2

group 2

x2

1

30

900

0

0

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2

30

900

0

0

3

20

400

20

400

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4

20

400

20

400

5

0

0

30

900

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6

0

0

30

900

7

0

0

30

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900

8

0

0

0

Total

100

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130

230

N

8

7

15

mean

12.5

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18.57143

sum X2

2600

3500

6100

Sum x2 /n

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1250

2414.286

3664.286

From the above grand total of X = 230

Total n = 15

The following values are determined using the data above:

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value

A

(grand  total of X)2 / total n

= (230*230)/15 3526.667

B

total sum  of X2 for both groups

= 2600 +  3500

6100

C

sum of x2  group 1/n + sum of x2 for group 2/n

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= 1250 +  2414.286

3664.285714

Using the above values the ANOVA table is determined as follows:

variation source

SS

df

between

C-A

1

within

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B-C

13

total

B-A

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The ANOVA table is as follows:

variation source

SS

df

ms

f

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between

137.6190476

1

137.619

0.734506

within

2435.714286

13

187.3626

total

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2573.333333

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Given that the critical value from the F table when alpha = 0.05 = 4.6672, and that the F value calculated is 0.734506, the F critical value > F statistics value calculated, this means that there is no significant difference between the two halves.

Question 2:

Regression analysis:

The table below summarizes calculations undertaken to estimate the model of the form:

Y = a + BX

We use the formula:

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B = ∑xy/ ∑x2

And a = Y’ – BX’ where Y’ and X’ are the mean values,

independent

dependent

X

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Y

small y

small x

xy

x2

1

30

14.66667

-7

-102.667

49

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2

30

14.66667

-6

-88

36

3

20

4.666667

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-5

-23.3333

25

4

20

4.666667

-4

-18.6667

16

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5

0

-15.3333

-3

46

9

6

0

-15.3333

-2

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30.66667

4

7

0

-15.3333

-1

15.33333

1

8

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0

-15.3333

0

0

0

9

0

-15.3333

1

-15.3333

1

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10

0

-15.3333

2

-30.6667

4

11

20

4.666667

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3

14

9

12

20

4.666667

4

18.66667

16

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13

30

14.66667

5

73.33333

25

14

30

14.66667

6

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88

36

15

30

14.66667

7

102.6667

49

TOTALS

120

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230

110

280

MEAN

8

15.3333333

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The above table shows total sums of x2 and xy where x = X-X’ and y=X-X’ where X and Y’ are the mean values.

∑xy=110 and ∑x2=280

B = 110/ 280

B =0.392857

Given that a = Y’ – BX’,

Then a =15.3333333-(0.392857*8)

a = 12.1904762

The estimated model is therefore:

y = 12.19 + 0.393X

This model means that as we increase X by one unit then Y increases by 0.393 units and that if the value of x was zero than Y value will be 12.19 given that we hold all other factors constant.

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REFERENCE:

Mendenhall, W. (2003) Introduction to statistics, Prentice Hall press, New  Jersey

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