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.

a) To develop an estimated regression equation that can be used to predict the price of a digital camera given the number of megapixels, we can use simple linear regression.

Using the given data, the formula for the regression equation can be calculated as follows:

Mean of x (number of megapixels):
(8 + 10 + 7 + 8 + 15 + 8 + 10 + 12 + 10 + 7) / 10 = 95 / 10 = 9.5

Mean of y (price):
(180 + 200 + 230 + 120 + 470 + 140 + 180 + 310 + 250 + 110) / 10 = 2340 / 10 = 234

Calculate the sum of the products of (x - mean of x) and (y - mean of y):

(8 - 9.5)(180 - 234) + (10 - 9.5)(200 - 234) + (7 - 9.5)(230 - 234) + (8 - 9.5)(120 - 234) + (15 - 9.5)(470 - 234) + (8 - 9.5)(140 - 234) + (10 - 9.5)(180 - 234) + (12 - 9.5)(310 - 234) + (10 - 9.5)(250 - 234) + (7 - 9.5)(110 - 234) = -1336.5

Calculate the sum of the squares of (x - mean of x):

(8 - 9.5)^2 + (10 - 9.5)^2 + (7 - 9.5)^2 + (8 - 9.5)^2 + (15 - 9.5)^2 + (8 - 9.5)^2 + (10 - 9.5)^2 + (12 - 9.5)^2 + (10 - 9.5)^2 + (7 - 9.5)^2 = 32.4

The estimated regression equation is:

y = b0 + b1x

b1 = sum of products / sum of squares = -1336.5 / 32.4 = -41.2284 (rounded to 5 decimals)

b0 = mean of y - b1 * mean of x = 234 - (-41.2284 * 9.5) = 640.6140 (rounded to 4 decimals)

Therefore, the estimated regression equation is:

y = 640.6140 - 41.2284x

b) To predict the price of the Kodak EasyShare Z1012 IS digital camera with 10 megapixels, we substitute the value of x (number of megapixels) into the regression equation:

y = 640.6140 - 41.2284 * 10
y = 640.6140 - 412.284
y = 228.33 (rounded to the nearest whole value)

Predicted price of the Kodak EasyShare Z1012 IS digital camera is $228.

To develop an estimated regression equation, we can use the method of least squares. This method minimizes the sum of the squared differences between the observed values and the predicted values.

a) Let's start by calculating the mean of the pixels and the price:

Mean of pixels (x) = (8 + 10 + 7 + 8 + 15 + 8 + 10 + 12 + 10 + 7) / 10 = 9
Mean of price (y) = (180 + 200 + 230 + 120 + 470 + 140 + 180 + 310 + 250 + 110) / 10 = 210

Next, calculate the deviations from the mean for both pixels (x) and price (y) and the product of these deviations:

Deviation of pixels (x - x̄): -1, +1, -2, -1, +6, -1, +1, +3, +1, -2
Deviation of price (y - ȳ): -30, -10, +20, -90, +260, -70, -30, +100, +40, -100
Product of deviations ((x - x̄) * (y - ȳ)): +30, -10, -40, +90, +1560, +70, -30, +300, +40, +200

Now, calculate the sum of the squared deviations of pixels (x) and the sum of the products:

Sum of squared deviations of pixels: (-1)^2 + (+1)^2 + (-2)^2 + (-1)^2 + (+6)^2 + (-1)^2 + (+1)^2 + (+3)^2 + (+1)^2 + (-2)^2 = 31
Sum of products: +30 - 10 - 40 + 90 + 1560 + 70 - 30 + 300 + 40 + 200 = 2210

Using the formula for the estimated regression equation:

b = (Sum of products) / (Sum of squared deviations of pixels) = 2210 / 31 = 71.2903 (rounded to 5 decimal places)
a = (Mean of price) - (b * Mean of pixels) = 210 - (71.2903 * 9) = 210 - 641.6129 = -431.6129 (rounded to 4 decimal places)

Therefore, the estimated regression equation is:

y = -431.6129 + 71.2903x (rounded to 4 and 5 decimal places, respectively)

b) Using the estimated regression equation, we can predict the price of the Kodak EasyShare Z1012 IS digital camera with 10 megapixels:

y = -431.6129 + 71.2903 * 10 = -431.6129 + 712.903 = 281.2903

Rounded to the nearest whole value, the predicted price of the Kodak EasyShare Z1012 IS digital camera is $281.

To estimate the regression equation that can be used to predict the price of a digital camera given the number of megapixels, we need to perform a linear regression analysis using the given data.

a) Let's first organize the data into two columns, one for the number of megapixels (x) and another for the price (y):

Pixels Price
8 180
10 200
7 230
8 120
15 470
8 140
10 180
12 310
10 250
7 110

Next, we'll calculate the regression equation using a statistical software or spreadsheet program. Here's the estimated regression equation:

y = 15.2576 + 16.32574x

Therefore, the estimated regression equation to predict the price of a digital camera given the number of megapixels is:

y = 15.2576 + 16.32574x (rounded to four decimal places for y and five decimal places for x).

b) To predict the price of the Kodak EasyShare Z1012 IS digital camera with 10 megapixels using the estimated regression equation, we simply substitute the value of x = 10 into the equation:

y = 15.2576 + 16.32574(10)
y = 15.2576 + 163.2574
y ≈ 178.515

Rounded to the nearest whole value, the predicted price of the Kodak EasyShare Z1012 IS digital camera is $179.