In: Statistics and Probability
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 | 22 | 34 | 37 | 42 | 46 |
---|---|---|---|---|---|
Number of Bids | 4 | 5 | 6 | 8 | 9 |
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=46. Round your answer to three decimal places.
Step 4 of 6: Determine if the statement "Not all points predicted by the linear model fall on the same line" is true or false.
Step 5 of 6: Determine the value of the dependent variable y^ at x=0 (b0, b1, x, y)
Step 6 of 6: Find the value of the coefficient of determination. Round your answer to three decimal places.
Solution:
Correlation coefficient = r = [n∑xy - ∑x∑y]/sqrt[(n∑x^2 – (∑x)^2)*(n∑y^2 – (∑y)^2)]
The calculation table is given as below:
No. |
x |
y |
x^2 |
y^2 |
xy |
1 |
22 |
4 |
484 |
16 |
88 |
2 |
34 |
5 |
1156 |
25 |
170 |
3 |
37 |
6 |
1369 |
36 |
222 |
4 |
42 |
8 |
1764 |
64 |
336 |
5 |
46 |
9 |
2116 |
81 |
414 |
Total |
181 |
32 |
6889 |
222 |
1230 |
Mean |
36.2 |
6.4 |
From above table, we have
n = 5
∑x = 181
∑y = 32
∑x^2 = 6889
∑y^2 = 222
∑xy = 1230
Xbar = ∑x/n = 181/5 = 36.2
Ybar = ∑y/n = 32/5 = 6.4
r = [n∑xy - ∑x∑y]/sqrt[(n∑x^2 – (∑x)^2)*(n∑y^2 – (∑y)^2)]
r = [5*1230 – 181*32]/sqrt[(5*6889– (181)^2)*(5*222– (32)^2)]
r = 0.940725
Step 1 of 6: Find the estimated slope.
Slope = b = b1 = (1230 – 5*36.2*6.4)/(6889 – 5*36.2^2)
b = 0.212589
Step 2 of 6: Find the estimated y-intercept.
Y-intercept = b0 = Ybar – b*Xbar
b0 = 6.4 - 0.212589*36.2
b0 = -1.29572
Regression equation is given as below:
y = b0 + b1x
y = -1.29572 + 0.212589*x
Step 3 of 6: Find the estimated value of y when x=46.
y = -1.29572 + 0.212589*46
y = 8.483
Step 4 of 6: Determine if the statement "Not all points predicted by the linear model fall on the same line" is true or false.
Given statement is true, because we can use the regression equation only for the points that are lies within the given range of independent variable. The use of regression equation for the points fall on the same line but out of range of independent variable is said as extrapolation.
Step 5 of 6: Determine the value of the dependent variable y^ at x=0
y = -1.29572 + 0.212589*x
y = -1.29572 + 0.212589*0
y = -1.29572
y = b0
Step 6 of 6: Find the value of the coefficient of determination.
Coefficient of determination= R^2 = r^2 = 0.940725^2 = 0.88496
Coefficient of determination= 0.885
Coefficient of determination= 88.5%