Question

In: Statistics and Probability

The data in the accompanying table represent the population of a certain country every 10 years...

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 population, y year, x population, y
1900 73212 1960 179323
1910 92228 1970 203302
1920 104021 1980 226542
1930 123202 1990 248709
1940 132164 2000 281421
1950 151325

Determine the P-value of this hypothesis test.

P-value = __

c) draw a scatter diagram, treating year as the explanatory variable.

d) plot the residuals against the explanatory variable, year.

e) does a linear model seem appropriate based on the scatter diagram and residual plot?

f) what is the moral?

Solutions

Expert Solution

Here are all the calculations of the regression equation:

Year y A = (x-x-bar) B = (y-ybar) C= A*B A^2 D = y-y-pred D^2
1900 73212 -50 -91828.8 4591441 2500 -10075.3 101512036
1910 92228 -40 -72812.8 2912513 1600 -8710.49 75872651.9
1920 104021 -30 -61019.8 1830595 900 -122.664 15046.3677
1930 123202 -20 -41838.8 836776.4 400 1077.164 1160281.5
1940 132164 -10 -32876.8 328768.2 100 12495.99 156149789
1950 151325 0 -13715.8 0 0 13715.82 188123668
1960 179323 10 14282.18 142821.8 100 6098.645 37193476.4
1970 203302 20 38261.18 765223.6 400 2500.473 6252363.86
1980 226542 30 61501.18 1845035 900 -358.7 128665.69
1990 248709 40 83668.18 3346727 1600 -2144.87 4600479.02
2000 281421 50 116380.2 5819009 2500 -14476 209555892
AVERAGE 1950 165040.8 SUM 22418910 11000 780564350
b1 2038.083 SSE 9312.861
b0 -3809221 Standard Error of b1 88.79464

Null Hypothesis: Coefficient of variable 'year' is zero i.e b1 = 0

The formulas used are:

Hence, we will run a t-statistic test on the coefficient

t = Coefficient/Standard Error

t = 2038.083/88.79

t = 22.952

Hence, the p-value at 0.05 level is 2.69E-0.9 (p<0.001) which is significant.

c) Scatter diagram:

d) Residuals plot:

e) From the scatter plot and residual plot, it seems that the data points are randomly dispersed in the residual plot and the scatter plot shows a near linear relationship.

f) Hence, there is a significant relationship between both the variables and year could be used to predict the population of the country over time.


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