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.
orchestra answered 2 years agoRelated Solutions
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
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...
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. LOADING... Click the icon to view the data table. (a)
Determine the least-squares regression equation, treating year as
the explanatory variable. Choose the correct answer below. A.
ModifyingAbove y with caretequals2 comma 022xnegative 1 comma 542
comma...
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.
Year, x Population, y
1900 79,212
1910 92,228
1920 104,021
1930 123,202
1940 132,164
1950 151,325
1960 179,323
1970 203,302
1980 226,542
1990 248,709
2000 281,421
(a) Determine the least-squares regression equation, treating
year as the explanatory...
The data in the accompanying table represent the rate of return of a certain company stock...
The data in the accompanying table represent the rate of return
of a certain company stock for 11 months, compared with the rate of
return of a certain index of 500 stocks. Both are in percent. Test
if there is a positive relationship at the α=0.01 level of
significance.
Month
Rates of return of the index, x
Rates of return of the company stock, y
Apr-07
-5.94
-2.36
May-07
4.78
3.91
Jun-07
-2.02
-0.74
Jul-07
-6.13
-3.08
Aug-07
-4.27
-2.59...
The data in the accompanying table represent the rate of return of a certain company stock...
The data in the accompanying table represent the rate of return
of a certain company stock for 11 months, compared with the rate
of return of a certain index of 500 stocks. Both are in percent.
Complete parts (a) through (d)
below.
Month Rates of return Rates of return of the index-x the company
stock-y
Apr-07 4.33 3.38
May-07 3.25 5.09
Jun-07 -1.78 0.54
Jul-07 -3.20 2.88
Aug-07 1.29 2.69
Sept-07 3.58 7.41
Oct-07 1.48 -4.83
Nov-07 -4.40 -2.38
Dec-07...
The data in the accompanying table represent the rate of return of a certain company stock...
The data in the accompanying table represent the rate of return
of a certain company stock for 11 months, compared with the rate
of return of a certain index of 500 stocks. Both are in percent.
Complete parts (a) through (d) below.
Month Rates_of_return_of_the_index_-_x
Rates_of_return_of_the_company_stock_-_y
Apr-07 4.33 3.38
May-07 3.35 5.09
Jun-07 -1.78 0.54
Jul-07 -3.20 2.88
Aug-07 1.29 2.69
Sept-07 3.58 7.41
Oct-07 1.48 -4.83
Nov-07 -4.40 -2.38
Dec-07 -0.86 2.37
Jan-08 -6.12 -4.27
Feb-08 -3.48 -3.77
(a) Treating...
The data in the accompanying table represent the rate of return of a certain company stock...
The data in the accompanying table represent the rate of return
of a certain company stock for 11 months, compared with the rate
of return of a certain index of 500 stocks. Both are in percent.
Complete parts (a) through (d) below. LOADING... Click the icon
to view the data table.
(a) Treating the rate of return of the index as the
explanatory variable, x, use technology to determine the estimates
of beta po and beta po1.
The estimate of...
The data in the accompanying table represent the rate of return of a certain company stock...
The data in the accompanying table represent the rate of return
of a certain company stock for 11 months, compared with the rate
of return of a certain index of 500 stocks. Both are in percent.
Complete parts (a) through (d) below.
Month Rates_of_return_of_the_index_-_x
Rates_of_return_of_the_company_stock_-_y
Apr-07 4.33 3.28
May-07 3.35 5.09
Jun-07 -1.78 0.54
Jul-07 -3.20 2.88
Aug-07 1.29 2.69
Sept-07 3.58 7.41
Oct-07 1.48 -4.83
Nov-07 -4.40 -2.38
Dec-07 -0.86 2.37
Jan-08 -6.12 -4.27
Feb-08 -3.48 -3.77
(a) Treating...
The data in the accompanying table represent the rate of return of a certain company stock...
The data in the accompanying table represent the rate of return
of a certain company stock for 11 months, compared with the rate
of return of a certain index of 500 stocks. Both are in percent.
Complete parts (a) through (d) below. LOADING... Click the icon
to view the data table. (a) Treating the rate of return of the
index as the explanatory variable, x, use technology to determine
the estimates of beta 0 and beta 1. The estimate of...
The data in the accompanying table represent the rate of return of a certain company stock...
The data in the accompanying table represent the rate of return
of a certain company stock for 11 months, compared with the rate
of return of a certain index of 500 stocks. Both are in percent.
Complete parts (a) through (d) below.
Month Rates_of_return_of_the_index_-_x
Rates_of_return_of_the_company_stock_-_y
Apr-18 4.23 3.38
May-18 3.25 5.09
Jun-18 -1.78 0.54
Jul-18 -3.20 2.88
Aug-18 1.29 2.69
Sept-18 3.58 7.41
Oct-18 1.48 -4.83
Nov-18 -4.40 -2.38
Dec-18 -0.86 2.37
Jan-19 -6.12 -4.27
Feb-19 -3.48 -3.77
(a) Treating...
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