Question

In: Statistics and Probability

DATAfile: WinePrices For a particular red wine, the following data show the auction price for a...

DATAfile: WinePrices

For a particular red wine, the following data show the auction price for a 750 milliliter bottle and the age of the wine in June of 2016.†

Age
(years)
Price
($)
36 256
20 142
29 212
33 255
41 331
27 173
30 209
45 297
34 237
22 182

(a)Develop a scatter diagram for these data with age as the independent variable.

a) A scatter diagram has a horizontal axis labeled "Age (years)" with values from 0 to 50 and a vertical axis labeled "Price ($)" with values from 0 to 350. The scatter diagram has 10 points. A pattern goes up and right from (20, 142) to (45, 331). The points are scattered moderately from the pattern.

b) A scatter diagram has a horizontal axis labeled "Price ($)" with values from 0 to 350 and a vertical axis labeled "Age (years)" with values from 0 to 50. The scatter diagram has 10 points. A pattern goes down and right from (119, 45) to (308, 20). The points are scattered moderately from the pattern.

c) A scatter diagram has a horizontal axis labeled "Age (years)" with values from 0 to 50 and a vertical axis labeled "Price ($)" with values from 0 to 350. The scatter diagram has 10 points. A pattern goes down and right from (20, 308) to (45, 119). The points are scattered moderately from the pattern.

d) A scatter diagram has a horizontal axis labeled "Price ($)" with values from 0 to 350 and a vertical axis labeled "Age (years)" with values from 0 to 50. The scatter diagram has 10 points. A pattern goes up and right from (142, 20) to (331, 45). The points are scattered moderately from the pattern.

(b)What does the scatter diagram developed in part (a) indicate about the relationship between age and price?

a) The scatter diagram indicates a nonlinear relationship between age and price.

b) The scatter diagram indicates a positive linear relationship between age and price.  

c) The scatter diagram indicates a negative linear relationship between age and price.

d)The scatter diagram indicates no apparent relationship between age and price.

(c) Develop the least squares estimated regression equation. (Let x = age (in years), and let y = price (in $). Round your numerical values to two decimal places.)

ŷ =

(d) Provide an interpretation for the slope of the estimated equation.

a) For every additional year of age, the price of the wine decreases by the amount of the slope.

b) For every additional dollar of price, the age of the wine decreases by the amount of the slope.    

c) For every additional dollar of price, the age of the wine increases by the amount of the slope.

d) For every additional year of age, the price of the wine increases by the amount of the slope.

e) The slope is the ratio of the average price of the wine in dollars to the average age in years.

Solutions

Expert Solution

X Y X * Y X2 Y2
36 256 9216 1296 65536
20 142 2840 400 20164
29 212 6148 841 44944
33 255 8415 1089 65025
41 331 13571 1681 109561
27 173 4671 729 29929
30 209 6270 900 43681
45 297 13365 2025 88209
34 237 8058 1156 56169
22 182 4004 484 33124
Total 317 2294 76558 10601 556342

Part a)

a) A scatter diagram has a horizontal axis labeled "Age (years)" with values from 0 to 50 and a vertical axis labeled "Price ($)" with values from 0 to 350. The scatter diagram has 10 points. A pattern goes up and right from (20, 142) to (45, 331). The points are scattered moderately from the pattern.

Part b)

b) The scatter diagram indicates a positive linear relationship between age and price.  

Part c)

Equation of regression line is Ŷ = a + bX


b = 6.952
a =( Σ Y - ( b * Σ X) ) / n
a =( 2294 - ( 6.952 * 317 ) ) / 10
a = 9.022
Equation of regression line becomes Ŷ = 9.0216 + 6.952 X

d) For every additional year of age, the price of the wine increases by the amount of the slope


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