Question

The table below gives the list price and the number of bids received for five randomly selected items sold through online auctions. Using this data, consider the equation of the regression line, yˆ=b0+b1x , for predicting the number of bids an item will receive based on the list price. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant. Price in Dollars 25 27 28 30 45 Number of Bids 1 4 6 8 9 Table Step 1 of 6: Find the estimated slope. Round your answer to three decimal places. Step 2 of 6: Find the estimated y-intercept. Round your answer to three decimal places. Step 3 of 6: Find the estimated value of y when x=30 . Round your answer to three decimal places Step 4 of 6: Determine the value of the dependent variable yˆ at x=0 Step 5 of 6: Substitute the values you found in steps 1 and 2 into the equation for the regression line to find the estimated linear model. According to this model, if the value of the independent variable is increased by one unit, then find the change in the dependent variable yˆ. Step 6 of 6: Find the value of the coefficient of determination. Round your answer to three decimal places.

Answer 1

X	Y	XY	X²	Y²
25	1	25	625	1
27	4	108	729	16
28	6	168	784	36
30	8	240	900	64
45	9	405	2025	81

Ʃx =	155
Ʃy =	28
Ʃxy =	946
Ʃx² =	5063
Ʃy² =	198
Sample size, n =	5
x̅ = Ʃx/n = 155/5 =	31
y̅ = Ʃy/n = 28/5 =	5.6
SSxx = Ʃx² - (Ʃx)²/n = 5063 - (155)²/5 =	258
SSyy = Ʃy² - (Ʃy)²/n = 198 - (28)²/5 =	41.2
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 946 - (155)(28)/5 =	78

Null and alternative hypothesis:

Ho: ρ = 0 ; Ha: ρ ≠ 0

Correlation coefficient, r = SSxy/√(SSxx*SSyy) = 78/√(258*41.2) = 0.7565

Test statistic :  

t = r*√(n-2)/√(1-r²) = 0.7565 *√(5 - 2)/√(1 - 0.7565²) = 2.0038

df = n-2 = 3

p-value = T.DIST.2T(ABS(2.0038), 3) = 0.1388

Conclusion:

p-value > α , Fail to reject the null hypothesis. There is no correlation between x and y.

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1.

Slope, b = SSxy/SSxx = 78/258 = 0.3023256 = 0.302

2.

y-intercept, a = y̅ -b* x̅ = 5.6 - (0.30233)*31 = -3.772093 = -3.772

Regression equation :

ŷ = -3.772 + (0.302) x

3.

Predicted value of y at x = 30

ŷ = -3.7721 + (0.3023) * 30 = 5.2977

4.

Predicted value of y at x = 0

ŷ = -3.7721 + (0.3023) * 0 = -3.7721

5.

Change on dependent variable = 0.302

6.

Coefficient of determination, r² = (SSxy)²/(SSxx*SSyy) = (78)²/(258*41.2) = 0.572