The number of initial public offerings of stock issued in a 10-year period and the total proceeds of these offerings (in millions) are shown in the table. The equation of the regression line is ModifyingAbove y with caret equals 47.302 x plus 18 comma 672.15. Complete parts a and b. Issues, x 402 461 679 495 486 389 55 50 176 174 Proceeds, y 19 comma 678 28 comma 329 42 comma 933 30 comma 445 66 comma 807 66 comma 245 20 comma 654 12 comma 175 31 comma 030 27 comma 693 Question content area bottom Part 1 (a) Find the coefficient of determination and interpret the result. enter your response here (Round to three decimal places as needed.)

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The coefficient of determination, denoted as

2

R

2

, measures the proportion of the variation in the dependent variable (proceeds) that is predictable from the independent variable (number of issues) in a regression model. It ranges from 0 to 1, where 1 indicates a perfect fit and 0 indicates no linear relationship between the variables.

To find the coefficient of determination, we can use the equation provided:

2

=

explained variation

total variation

R

2

=

total variation

explained variation

In this case, the regression equation is given as:

^

=

47.302

+

18

,

672.15

y

^

=47.302x+18,672.15

Now, let's calculate the coefficient of determination using the provided data and the regression equation.

First, we need to calculate the sum of squares total (SST), which represents the total variation in the dependent variable (proceeds).

=

∑

(

−

ˉ

)

2

SST=∑(y

i

−

y

ˉ

)

2

Where:

y

i

represents each individual value of proceeds.

ˉ

y

ˉ

represents the mean of all proceeds.

Then, we'll calculate the sum of squares error (SSE), which represents the unexplained variation in the dependent variable.

=

∑

(

−

^

)

2

SSE=∑(y

i

−

y

^

i

)

2

Where:

^

y

^

i

represents the predicted value of proceeds based on the regression equation.

Finally, we'll use these values to calculate the coefficient of determination:

2

=

1

−

R

2

=1−

SST

SSE

Let's proceed with the calculations.

First, we need to calculate the mean of the proceeds (

ˉ

y

ˉ

):

ˉ

=

1

∑

=

1

y

ˉ

=

n

1

∑

i=1

n

y

i

Where

n is the number of data points.

ˉ

=

19

,

678

+

28

,

329

+

42

,

933

+

30

,

445

+

66

,

807

+

66

,

245

+

20

,

654

+

12

,

175

+

31

,

030

+

27

,

693

10

y

ˉ

=

10

19,678+28,329+42,933+30,445+66,807+66,245+20,654+12,175+31,030+27,693

ˉ

=

335

,

019

10

y

ˉ

=

10

335,019

ˉ

=

33

,

501.9

y

ˉ

=33,501.9

Now, let's calculate the sum of squares total (SST):

=

∑

(

−

ˉ

)

2

SST=∑(y

i

−

y

ˉ

)

2

=

(

19

,

678

−

33

,

501.9

)

2

+

(

28

,

329

−

33

,

501.9

)

2

+

⋯

+

(

27

,

693

−

33

,

501.9

)

2

SST=(19,678−33,501.9)

2

+(28,329−33,501.9)

2

+⋯+(27,693−33,501.9)

2

After calculating SST, we'll use the provided regression equation to find the predicted values of proceeds (

^

y

^

i

).

Then, we'll calculate the sum of squares error (SSE):

=

∑

(

−

^

)

2

SSE=∑(y

i

−

y

^

i

)

2

Once we have SST and SSE, we can use the formula to find

2

R

2

:

2

=

1

−

R

2

=1−

SST

SSE

Let's proceed with these calculations.