Question

the table below gives the list price and the number of birds received for five randomly selected items sold through online auctions. using this data,

price in dollars 23 26 40 42 47

number of birds 1 2 3 5 7

step 3 of 6 substitute the values you found in step 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 y.

step 4 of 6: determine if the statement "all points predicted by the linear model fall on the same line" is true o false.

step 6 of 6 find the value of the coefficient of determination.

Answer 1

Regressing Price (Y - Dependent variable) on No. of birds (X - Independent variable), we would obtain a fitted regression equaltion of the form:

where,

Substituting the values,

Price (Y)	No. of Birds (X)	(X-X bar)(Y-Y bar)	(X-X bar)2
23	1	32.76	6.76
26	2	15.36	2.56
40	3	-2.64	0.36
42	5	8.96	1.96
47	7	38.76	11.56
Average	Sum
35.6	3.6	93.2	23.2

And

Hence, the fitted regression is obtained as:

3. Here, Slope can be interpreted as the mean change in the response variable for a unit change in the predictor, other predictors in the model being constant.Here, the mean price is estimated to increase by 4.017 units for every increase in No. of birds purchased. (Here, change observed is an 'increase', since, the slope for No. of Birds is positive) .

Hence, according to this model, if the value of the independent variable is increased by one unit, then the change in the dependent variable y is 4.017 units.

4. Obtaining the predicted prices:

Price (Y)	No. of Birds (X)	Predicted Price
23	1	21.138+4.017(1) = 113.529
26	2	21.138+4.017(2) = 125.58
40	3	21.138+4.017(3) = 181.818
42	5	21.138+4.017(5) =189.852
47	7	21.138+4.017(7) = 209.937

All points obtained above are predicted using the fitted regression equation 21.138 + 4.017x, by substituting for each value of x. Hence, the predicted points are nothing but the co-ordinates of the line . Hence, the statement that "all points predicted by the linear model fall on the same line" is True.

6. The Coefficient of Determination is obtained using the formula:

Price (Y)	No. of Birds (X)	(X-X bar)(Y-Y bar)	(X-X bar)	(Y -Y bar)
23	1	32.76	6.76	158.76
26	2	15.36	2.56	92.16
40	3	-2.64	0.36	19.36
42	5	8.96	1.96	40.96
47	7	38.76	11.56	129.96
Average	Sum
35.6	3.6	93.2	23.2	441.2

This implies that the variation in No. of Birds explains about 84.9% of the variation in Price.