The editor of a major academic book publisher claims that a large part of the co

ID: 3375723 • Letter: T

Question

The editor of a major academic book publisher claims that a large part of the cost of books is the cost of paper. This implies that larger books will cost more money. As an experiment to analyse the claim, a university student visits the bookstore and records the number of pages (pages)and the selling price (price) of 12 randomly selected books.

These data are listed below.

(d) It is believed that increases in number pages increases the selling price. Test the belief at the 1% level of significance

(e) Estimate the following regression models and interpret the regression coefficients:

Book Number of Pages Selling Price($) 1 844 55 2 727 50 3 360 35 4 915 60 5 295 30 6 706 50 7 410 40 8 905 53 9 1058 65 10 865 54 11 677 42 12 912 58 In price) = 81 + B2 In pages) +e

Explanation / Answer

Solution

Let y = ln(price) and x = ln(pages).

Then, the postulated model is: y = ?1 + ?2.x + e ………………………………………….(1)

Back-up Theory

Least square estimates of ?1, ?2 and other related concepts are detailed below:

Estimated Regression of Y on X is given by: Y = ?1cap + ?2capX, ………………………….(2)

where

?2cap = Sxy/Sxx and ?1cap = Ybar – ?2cap.Xbar..……………………………………………..(3)

Mean X = Xbar = (1/n)sum of xi ………………………………………….……………….(4)

Mean Y = Ybar = (1/n)sum of yi ………………………………………….……………….(5)

Sxx = sum of (xi – Xbar)2 …………………………………………………..………………………………..(6)

Syy = sum of (yi – Ybar)2 …………………………………………………..………………………………..(7)

Sxy = sum of {(xi – Xbar)(yi – Ybar)} …………………………………………………………………….………(8)

All the above sums are over i = 1, 2, …., n

n = sample size ………………………………………………………………………………(9)

Estimate of ?2 is given by s2 = (Syy – ?2cap2Sxx)/(n - 2)…………………………………..(10)

Standard Error of ?2cap is sb, where sb2 = s2/Sxx

100(1 - ?)% Confidence Interval (CI) for ?2 = ?2cap ± {SE(?1cap) x tn – 2, ?/2}

NOTE: Answers to the point are given below. Detailed calculations follow at the end.

Part (a)

Estimated regression equation is: ln(price) = 0.3695 + 0.5382ln(pages) ANSWER 1

Part (b)

Estimate of ?2 represents the change in y per unit change in x. In the present context, when ln(pages) increases/decreases by 1 unit,

ln(price) will increase/decrease by 0.5382 ANSWER 2

Estimate of ?1 represents the value of y when x = 0. In the present context,

when ln(pages) = 0, ln(price) will be 0.3695 ANSWER 3

Part (c)

95% CI for ?2 is [0.4517, 0.6258] ANSWER

Part (d)

Since the above CI does not contain zero and is strictly positive, it can be inferred that ?2 is positive. This in turn implies that

increases in number pages increases the selling price ANSWER

Details of Calculations

6.73815

4.007333

6.58893

3.912023

5.8861

3.555348

6.81892

4.094345

5.68698

3.401197

6.55962

3.91202

6.01616

3.68888

6.807935

3.970292

6.964136

4.174387

6.76273

3.988984

6.517671

3.73767

6.81564

4.060443

xbar

6.51

ybar

3.87524371

Sxx

1.91887259

Syy

0.59462634

Sxy

1.03278222

?2cap

0.53822345

?1cap

0.36948196

0.06225651

s (?2cap)

0.04494296

tn-2,?/2

1.94809946

CI ?2LB

0.45067009

CI ?2UB

0.62577681

DONE