Question

The data in the accompanying table represent the population of a certain country every 10 years for the years 1900-2000. An ecologist is interested in finding an equation that describes the population of the country over time. Complete parts (a) through (f) below.

Year, x

1900

1910

1920

1930

1940

1950

1960

1970

1980

1990

2000

Population, y

76,212

92,228

104,021

123.202

132,164

151,325

179,323

203,302

226,542

248,709

281,421

(A) Determine the least-squares regression equation, treating year as the explanatory variable. Choose the correct answer below.

a. y = -3,782,357x + 2,0242

b. y = 1,236,375x - 3,782,357

c. y = 2,0242x - 1,547,901

d. y = 2,0242x – 3,782,357

(B) A normal probability plot of the residuals indicates that the residuals are approximately normally distributed. Test whether a linear relation exists between year and population. Use the a=0.01 level of significance. State the null and alternative hypotheses. Determine the P-value of this hypothesis test. State whether to reject or not reject the null and why.

(C) Draw a scatter diagram, treating year as the explanatory variable.

(D) Plot the residuals against the explanatory variable, year. Does a linear model seem appropriate based on the scatter diagram and residual? plot?

(E) Which of the following is the moral?

a)The moral is that inferential procedures may indicate that a linear relation between the two variables exists even though diagnostic tools? (such as residual? plots) indicate that a linear model is inappropriate.

b)The moral is that inferential procedures may indicate that a nonlinear relation between the two variables exists even though diagnostic tools? (such as residual? plots) indicate that a linear model is appropriate.

c)The moral is that explanatory variables may indicate that a linear relation between the two variables does not exist even though diagnostic tools? (such as residual plots) indicate that a linear model is appropriate.

Answer 1

using excel data analysis tool for regression, following o/p is obtained

Regression Statistics
Multiple R	0.9907
R Square	0.9816
Adjusted R Square	0.9795
Standard Error	9701.9621
Observations	11
ANOVA
	df	SS	MS	F	Significance F
Regression	1	45082213871.65	45082213872	478.9455	0.0000
Residual	9	847152623.08	94128069
Total	10	45929366494.73
	Coefficients	Standard Error	t Stat	P-value	Lower 95%
Intercept	-3782357	180407.6473	-20.9656	0.0000	-4368652
X	2024	92.5046	21.8848	0.0000	1723.822

a)

answer option d)

Y = 2024   *x  -3782357  

b)

Ho:   ß1=   0          
H1:   ß1╪   0          
n=   11              
alpha=   0.01      

SSxx =    Σ(x-x̅)² =    11000.00  
       
estimated std error of slope =Se(ß1) =                s/√Sxx =    92.5046

                  
t stat =    ß1 /Se(ß1) =        21.885  

                  
  
p-value =    0.0000              
decision :    p-value<α , reject Ho

reject Ho and conclude  that linear relations exists between X and y

c)

d)

e)

  The moral is that inferential procedures may indicate that a linear relation between the two variables exists even though diagnostic tools (such as residual plots) indicate that a linear model is inappropriate