Question

w... Solved: Use the following information to 1.5 pts Question 11 A random sample of ten professional athletes produced the following data. The first rowis the number of endorsements the athlete has andthe second row is the amount of money made in millions of dollars) by the athlete. What is they-intercept of the line of best fit? What does it represent in the context of the problem? 0 3 2 1 5 5 4 3 0 4 Number of Endorsements 10 Profit (in Millions) 2 8 7 3 13 12 9 9 3 (0.2.2): The point represents the increase in profit of the athlete once the athlete starts signing endorsements. (0,2.2): The point represents the profit of the athlete if the athlete has no endorsements. 1.988: 2.2: The increase in profit per endorsement for an athlete. 1.988; The point represents the profit of the athlete if the athlete has no endorsements. 2.2; The increase in profit per endorsement for an athlete. 2.2: The profit of the athlete. (0,1.988): The point represents the profit of the athlete if the athlete has no endorsements.

for this question, could you include written work of the

1) independant and dependant variables?

2) is it reasonable to use linear regression and why?

3) what is the slope of the line best fit?

4) what is the y-intercept of ^ and what does it represent?

5) what is the linear coefficent?

6) is the linear coefficent significant and why?

7) if a professional athlete did six endorsements, what could they expect a profit according to the regression equation?

8) if a professional athlete did to endorsements, what is the residual for this data point?

Answer 1

1)

Independent variable: Profit

Dependent variable: Number of Endorsements

2)

To see whether the linear relationship exists between the independent and the dependent variable, the scatterplot is obtained in excel. The screenshot is shown below,

Profit 2 3 4 # of Endorsements 5 6

From the scatterplot we can observe the two variables are linearly associated hence the linear regression can be used here.

3)

Answer:

Slope = 1.9875

Explanation:

The regression equation is defined as,

Y = bo + b1X

where b0 = intercept and b1 = slope of the best fit line.

The least-square estimate of the slope is obtained using the following formula,

$b_1=\frac{n(\sum XY) -\sum X \sum Y}{n(\sum X^2)-(\sum X)^2}$

From the data values,

X	Y	X^2	Y^2	X*Y
0	2	0	4	0
3	8	9	64	24
2	7	4	49	14
1	3	1	9	3
5	13	25	169	65
5	12	25	144	60
4	9	16	81	36
3	9	9	81	27
0	3	0	9	0
4	10	16	100	40
Sum = 27	76	105	710	269

$b_1=\frac{10\times 269-27\times 76}{10\times 105-(27)^2}$

bi = 1.9875

4)

Answer:

Slope = 2.2336

The expected profit is approximately $ 2.2336 million when the number of endorsements is zero.

Explanation:

The least-square estimate of the slope is obtained using the following formula,

ΣΥΣΧ2 –ΣΧΣ ΧΥ ηΣ Χ2) - (ΣΧ)2

From the data values,

$b_0=\frac{76\times 105-27\times 269}{10\times 76-(27)^2}$

$b_0=2.2336$

The intercept value gives the expected value of the dependent variable the independent variable = 0 hence the expected profit is approximately $ 2.2336 million when the number of endorsements is zero.

5)

Answer:

linear correlation coefficient = 0.9786

Explanation:

The linear correlation coefficient is obtained using the following formula,

From the data values,

$r=\frac{10\times 269-27\times 76}{\sqrt{[10\times 105-(27)^2][10\times 710-(76)^2]}}$

$r=0.9786$

6)

The hypothesis test for significance of the correlation coefficient is performed in the following steps,

Hypotheses

Let the significance level,

Test statistic

The t statistic is obtained using the formula,

$t=\frac{0.9786\sqrt{10-2}}{\sqrt{1-(0.9786)^2}}=13.4652$

P-value

The p-value for the t statistic = 13.4652 is obtained from the t distribution table for the degree of freedom = n-2=8

P-value = 0.0000

Conclusion

Since the p-value = 0.0000 is less than 0.05 at a 5% significance level, the null hypothesis is rejected hence there is sufficient evidence to conclude that the Profit and the number of endorsements are linearly correlated.

7)

Answer:

Expected profit = $ 14.1589 million

Explanation:

The regression equation is defined as,

$\widehat{Y}=2.2336+1.9875X$

For X = 6

$\widehat{Y}=2.2336+1.9875\times 6=14.1589$

8)

Answer:

Residual = $ 0.7913 million

Explanation:

The regression equation is defined as,

$\widehat{Y}=2.2336+1.9875X$

For X = 2

$\widehat{Y}=2.2336+1.9875\times 2=6.2087$

From the data values, the actual value of profit for the number of endorsement, X = 2 is 7

$\text{Residual}=Y_i-\widehat{Y}=7-6.2087=0.7913$