Question

Professor Isaac Asimov was one of the most prolific writers of all time. Prior to his death, he wrote nearly 500 books during a 40-year career. In fact, as his career progressed, he became even more productive in terms of the number of books written within a given period of time. The data given in increments of 100:
Number of books, x: 100 200 300 400 500
Time in months, y: 237 350 419 465 507
a. Assume that the number of books x and the time in months y are linearly related. Find the least-squares line relating y to x.
b. Plot the time as a function of the number of books written using a scatterplot, and graph the least-squares line on the same paper. Does this seem to provide a good fit to the data points?
c. Construct the ANOVA table for the linear regression.

Answer 1

Answer:

Step-by-step explanation:

n = sample size = 5

a) Let us determine the sum

$\sum x_i= 100+200+300+400+490=1490$

$\sum x_i^2= 100^2+200^2+300^2+400^2+490^2=540100$

$\sum y_i = 237+350+419+465+507=1978$

$\sum y_i^2= 237^2+350^2+419^2+465^2+507^2=827504$

$\sum x_i y_i=100 \times 237 + 200\times350+300 \times 419 + 400 \times 465 + 490 \times 507=653830$

Now we can determine $S_x_x, S_x_y, S_y_y$

$S_x_x = \sum x_i^2-\frac{(\sum x_1)^2}{n} \\= 540100 - \frac{1490^2}{5} \\= 96080$

$S_x_y = \sum x_i y_i -\frac{(\sum x_i)(\sum y_i) }{n}$

$653830 - \frac{1490 \times 1978 }{5} = 64386$

$S_y_y = \sum y_i^2-\frac{(\sum y_i)^2 }{n} = 82750-\frac{1978^2}{5} \\\\= 45007.2$

The estimate b of the slope β is the ratio of $S_x_y$ and $S_x_x$

$b = \frac{S_x_y}{S_x_x}$

$\frac{64386}{96080} = 0.67$

The mean is the sum of all value divide by number of values

$\bar x= \frac{\sum x_i}{n} \\\\= \frac{100+200+300+400+490}{5} \\\\= \frac{1490}{5} = 298$

$\bar y= \frac{\sum y_i}{n} \\\\= \frac{237+350+419+465+507}{5} \\\\= \frac{1978}{5} = 395.6$

The estimate a of the intercept is

$a = \bar y - b \bar x$

$= 395.6 - 0.69 \times 298\\= 195.9$

General least square equation;

$\bar y = \alpha + \beta x$

replace alpha by a = 3 and beta by b = 0.67 in general least equation

y = a + bx

195.9 + 0.67x

b)

Scatter plot is shown in the attached file

x is on the horizontal axis

y is n the vertical axis

The degree of freedom of regression is 1

because we use one variable s predictor variable

$d_f_R = 1$

The degree of freedom of error is the sample size n decrease by 2

$d_f_E =n-2= 5 - 2=3$

Total df is equal to the sum  of seperate degree of freedom dfR  and dfE

total df = 1 +3  4

$SSR = \frac{(S_x_y)^2}{S_x_x} = \frac{64386^2}{96080} \\\\= 43146.9296$

Total SS =Syy= 45007.2

SSE + Total SS = SSR

= 45007.2 - 43146.9296

= 1860.2705