Question

Styles 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 (3) below Year, x 1900 1910 1920 1930 1940 1950 Population, y Year, x Population, y 179,323 203,302 79,212 1960 95,228 1970 104,021 1980 123,202 1990 132,164 2000 151,325 226,542 248,709 281,421 (a)Determine the least-squares regression equation, treating year as the explanatory variable. Choose the correct answer below A. y 1,236,351x - 3,733,948 B. y 2,000x - 3, 733,948 C. y 3, 733,948x + 2,000 D. y 2,000x - 3, 733,948 (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 (c) Determine the P-value of this hypothesis test P-value= (Round to three decimal places as needed.) (d) State the appropriate conclusion. Choose the correct answer below A. Do not reject. Upper H 0HO. There is not sufficient evidence to conclude that a linear relation exists between year and population B. Reject Upper H 0HO. There is sufficient evidence to conclude that a linear relation exists between year and population. This is the correct answer C. Do not reject Upper H 0H0. There is sufficient evidence to conclude that a linear relation exists between year and population. D. Reject Upper H 0HO. There is not sufficient evidence to conclude that a linear relation exists between year and population. Your answer is not correct. Draw a scatter diagram, treating year as the explanatory variable. Choose the correct graph below Plot the residuals against the explanatory variable. Show on graph. (e) Does a linear model seem appropriate based on the scatter diagram and residual plot? Yes or No. (f) What is the moral? A. 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. 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 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

Answer 1

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

Regression Statistics
Multiple R	0.9891
R Square	0.9783
Adjusted R Square	0.9759
Standard Error	10410.0972
Observations	11
ANOVA
	df	SS	MS	F	Significance F
Regression	1	43995640108.01	43995640108	405.9757	0.0000
Residual	9	975331113.99	108370124
Total	10	44970971222.00
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-3733948	193575.3942	-19.2894	0.0000	-4171846	-3296050
X	2000	99.2564	20.1488	0.0000	1775.367	2224.434

a)

so, answer is option b)

please note that option b) and option d) are same

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) = Se/√Sxx = 10410/√11000 = 99.2564

                  
t stat =    ß1 /Se(ß1) =        20.149
                  
t-critical value=        3.2498   

  

since, t-stat > t-critical , reject Ho,

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

c)

p-value =    0.000 [using t table]

d)

decision :    p-value<α , reject Ho         

answer : option B)

e)

answer is No

X Residual Plot 20000 10000 0 운-10000 20000 1880 1900 1920 1940 1960 1980 2000 2020

f)

option c)