The number of megapixels in a digital camera is one of the most important factors in determining picture quality. But, do digital cameras with more megapixels cost more? The following data show the number of megapixels and the price ($) for 10 digital cameras (Consumer Reports, March 2009).
Pixels Price
8 180
10 200
7 230
8 120
15 470
8 140
10 180
12 310
10 250
7 110
a)Use these data to develop an estimated regression equation that can be used to predict the price of a digital camera given the number of megapixels.
y = ? (to 4 decimals) + ? x (to 5 decimals)
b) The Kodak EasyShare Z1012 IS digital camera has 10 megapixels. Predict the price of this camera using the estimated regression equation developed in part (a).
$ Round your answer to nearest whole value.
To develop an estimated regression equation to predict the price of a digital camera given the number of megapixels, we can use linear regression.
First, we need to organize the given data:
Pixels (x) Price (y)
8 180
10 200
7 230
8 120
15 470
8 140
10 180
12 310
10 250
7 110
a) To find the estimated regression equation, you can follow these steps:
1. Calculate the mean (average) of the pixels and price data.
Mean of x (pixels) = (8 + 10 + 7 + 8 + 15 + 8 + 10 + 12 + 10 + 7) / 10 = 9.5
Mean of y (price) = (180 + 200 + 230 + 120 + 470 + 140 + 180 + 310 + 250 + 110) / 10 = 212
2. Calculate the deviations of each data point from the mean.
Deviations of x: -1.5, 0.5, -2.5, -1.5, 5.5, -1.5, 0.5, 2.5, 0.5, -2.5
Deviations of y: -32, -12, 18, -92, 258, -72, -32, 98, 38, -102
3. Calculate the product of deviations of x and y.
Product of deviations: (-1.5)(-32), (0.5)(-12), (-2.5)(18), (-1.5)(-92), (5.5)(258),
(-1.5)(-72), (0.5)(-32), (2.5)(98), (0.5)(38), (-2.5)(-102)
-1.5 * -32 = 48, 0.5 * -12 = -6, -2.5 * 18 = -45, -1.5 * -92 = 138, 5.5 * 258 = 1419,
-1.5 * -72 = 108, 0.5 * -32 = -16, 2.5 * 98 = 245, 0.5 * 38 = 19, -2.5 * -102 = 255
4. Calculate the sum of the squared deviations of x.
Sum of squared deviations of x: (-1.5)^2 + (0.5)^2 + (-2.5)^2 + (-1.5)^2 + (5.5)^2 +
(-1.5)^2 + (0.5)^2 + (2.5)^2 + (0.5)^2 + (-2.5)^2
2.25 + 0.25 + 6.25 + 2.25 + 30.25 + 2.25 + 0.25 + 6.25 + 0.25 + 6.25 = 56.5
5. Calculate the sum of the product of deviations of x and y.
Sum of product of deviations: 48 - 6 - 45 + 138 + 1419 + 108 - 16 + 245 + 19 + 255 = 2616
6. Calculate the slope (b) of the regression line.
b = [Sum of product of deviations] / [Sum of squared deviations of x]
= 2616 / 56.5 = 46.25 (rounded to 5 decimal places)
7. Calculate the intercept (a) of the regression line.
a = Mean of y - (b * Mean of x)
= 212 - (46.25 * 9.5) = 212 - 438.125 = -226.125 (rounded to 4 decimal places)
Therefore, the estimated regression equation that can be used to predict the price of a digital camera given the number of megapixels is:
y = -226.125 + 46.25x
b) To predict the price of the Kodak EasyShare Z1012 IS digital camera with 10 megapixels using the estimated regression equation, substitute x = 10 into the equation:
y = -226.125 + 46.25 * 10
= -226.125 + 462.5
≈ 236.375
The predicted price of the Kodak EasyShare Z1012 IS digital camera is approximately $236 (rounded to the nearest whole value).