Chapter 13 #1
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 |
26 |
27 |
34 |
41 |
45 |
Number of Bids |
1 |
3 |
4 |
5 |
8 |
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: Determine if the statement "Not all points predicted by the linear model fall on the same line" is true or false.
Step 4 of 6: According to the estimated linear model, if the value of the independent variable is increased by one unit, then the change in the dependent variable yˆ is given by?
(bo/b1/x/y)
Step 5 of 6: Find the estimated value of y when x=41. Round your answer to three decimal places.
Step 6 of 6: Find the value of the coefficient of determination. Round your answer to three decimal places.
Here to solve all these questions we required the following table
Price(x) | Bids(y) | x^2 | xy | y^2 | |
26 | 1 | 676 | 26 | 1 | |
27 | 3 | 729 | 81 | 9 | |
34 | 4 | 1156 | 136 | 16 | |
41 | 5 | 1681 | 205 | 25 | |
45 | 8 | 2025 | 360 | 64 | |
Sum | 173 | 21 | 6267 | 808 | 115 |
Step 1
Slope = [ n(Σxy) - (Σx)((Σy)]/ [ n (Σx^{2} ) - (Σx)^{2} ]
= [5 * 808 - 173 * 21]/[5 * 6267 - 173^{2}]
slope = 0.2895
Step 2
Here Intercept
Intercept = b_{0} = [(Σy) (Σx^{2} ) - (Σx) (Σxy)]/ [ n (Σx^{2} ) - (Σx)^{2} ]
= [21 * 6267 - 173 * 808]/ [5 * 6267 - 173^{2}]
Intercept = b_{1} = -5.8158
Step 3 of 6 : Here as we get the regression line
y^ = -5.8158 + 0.2895 x
so here we take first point (26, 1)
y^ = -5.8158 + 0.2895 * 26 = 1.7112
so here we can see that "Not all points predicted by the linear model fall on the same line"
Step 4 of 6
here According to the estimated linear model, if the value of the independent variable is increased by one unit, then the change in the dependent variable yˆ is given by b_{1} which is 0.2895.
Step 5 of 6
Here x = 41
y^ = -5.8158 + 0.2895 * 41 = 6.0526
Step 6 of 6
Here coefficient of determination
r^{2} = [nΣxy - ΣxΣy]^{2} / [n(Σx^{2} - (Σx)^{2}] [n(Σy^{2} - (Σy)^{2}]
r^{2} = [5 * 808 - 173 * 21]^{2}/ [{5 * 6267 - 173^{2}) * (5 * 115 - 21^{2})}]
r^{2} = 0.879
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