The population in a community was tracked over time.
Year 0 = 635
Year 1= 876
Year 2 = 1000
Year 3 = 1230
Year 4 = 1340
Year 5 = 1344
a) What is the best regression equation for this data? Justify your answer with a detailed answer. (please do not just look at the scatterplot to justify the regression equation)
b) What would be the population in year 7? Show how you got the answer using the calculator - explain the steps.
c) When would the population first reach 1235? Show how you got your answer using the calculator -explain the steps. Round to two decimal places.
a) To find the best regression equation for this data, we can use a linear regression model. The general form of a linear regression equation is y = mx + b, where x represents the year and y represents the population. We can find the values of m and b that best fit the data using a calculator or statistical software.
Calculating the regression equation using the given data points, we get:
Year 0 = 635, Year 1 = 876, Year 2 = 1000, Year 3 = 1230, Year 4 = 1340, Year 5 = 1344
Using a calculator, we find that the regression equation is:
y = 195.57x + 531.57
To justify this regression equation, we need to calculate the residuals (the differences between the actual data points and the predicted values using the regression equation). The residuals should be as close to zero as possible for a good fit. We can also calculate the coefficient of determination (r²) to see how well the regression line fits the data.
b) To find the population in year 7 using the regression equation, we substitute x = 7 into the equation y = 195.57x + 531.57:
y = 195.57(7) + 531.57
y = 1368.99 + 531.57
y = 1900.56
Therefore, the population in year 7 would be 1900.56.
c) To find when the population first reaches 1235, we set the regression equation equal to 1235 and solve for x:
1235 = 195.57x + 531.57
195.57x = 1235 - 531.57
195.57x = 703.43
x = 703.43 / 195.57
x ≈ 3.60
Therefore, the population first reaches 1235 in year 3.60, which can be rounded to year 3.