While browsing through the magazine rack at a bookstore, a statistician decides

Question

   While browsing through the magazine rack at a bookstore, a statistician decides to examine the relationship between the price of a magazine and the percentage of the magazine space that contains advertisements. The data collected for eight magazines are given in the following table. Here price is the dependent variable.

Percentage containing ads. 37   43   58   49
Price ($) 5.50   6.95   4.95   5.75
Percentage containing ads 70   28   65   32
Price ($)   3.95   8.25   5.50   6.75

a. Find the standard deviation of errors.

b. Compute the coefficient of determination. What percentage of
the variation in price is explained by the least squares regression of price on the percentage of magazine space containing ads? What percentage of this variation is not explained?

Explanation / Answer

(A) Here standard deviation of errors will be square root of mean squaare error.

Where MSE = SSE/dF(Residual)

First we will to calculate the regression line y^ = a + bx

Here the regression line is calculated by the values of x,y, xy, x2 & y2

Here,

a = [(y) (x2 ) - (x) (xy)]/ [ n (x2 ) - (x)2 ]

a = [47.6 * 19916 - 382 * 2152.2] / [8 * 19916 - 3822]

a = 125861.2/13404 = 9.390

b =[ n(xy) - (x)((y)]/ [ n (x2 ) - (x)2 ]

b = [8 * 2152.2 - 382 * 47.6] / [8 * 19916 - 3822]

b = -965.6/13404 = -0.072

y^ = 0.072x + 9.389

Here now we will calculate the sum of resiudals

Here MSE = 3.6800/(n-2) = 3.6800/6 =  0.61333

Standard errro = sqrt(0.6133) = 0.7832

(b) coefficient of determination

R2 = [nxy - xy]2 / [n(x2 - (x)2] [n(y)2 - y2]

R2 = [8 * 2152.2 - 382 * 47.6]2 / [(8 * 19916 - 3822) * ( 8 * 295.595 - 47.62)

R2 = 0.7026

so 70.26%  variation in price is explained by the least squares regression of price on the percentage of magazine space containing ads and 29.74% is not explained by the least squares regression of price on the percentage of magazine space containing ads

Percentage containing ads(x) Price(y) x^2 xy y^2 37 5.5 1369 203.5 30.25 43 6.95 1849 298.85 48.3025 58 4.95 3364 287.1 24.5025 49 5.75 2401 281.75 33.0625 70 3.95 4900 276.5 15.6025 28 8.25 784 231 68.0625 65 5.5 4225 357.5 30.25 32 6.75 1024 216 45.5625 Sum 382 47.6 19916 2152.2 295.595

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