A. Sketch the scatterplot with the least-squares line, and sketch the residual plot. Interpret your sketch.
Calculate the least-squares regression line.
To find the least-squares regression line, we need to calculate the slope (\(m\)) and intercept (\(b\)) using the formulas:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
\[
b = \frac{(\sum y) - m(\sum x)}{n}
\]
where \(x\) is the percentage of urban population and \(y\) is the robbery rate.
Using the data provided:
\[
\begin{align_}
\sum x &= 691, \\
\sum y &= 1433, \\
\sum xy &= 108,679, \\
\sum x^2 &= 52,091, \\
n &= 10.
\end{align_}
\]
Calculate \(m\):
\[
m = \frac{10(108,679) - (691)(1433)}{10(52,091) - (691)^2} = \frac{1,086,790 - 990,703}{520,910 - 477,481} = \frac{96,087}{43,429} \approx 2.21
\]
Calculate \(b\):
\[
b = \frac{1433 - 2.21 \times 691}{10} = \frac{1433 - 1,527.11}{10} = \frac{-94.11}{10} \approx -9.41
\]
The least-squares regression line is \(y = 2.21x - 9.41\).
Plot the scatterplot and the regression line.
The scatterplot is plotted with the percentage of urban population on the x-axis and the robbery rate on the y-axis. The regression line \(y = 2.21x - 9.41\) is drawn through the data points.
Sketch the residual plot.
The residual plot is created by plotting the residuals (the differences between the observed and predicted robbery rates) on the y-axis against the percentage of urban population on the x-axis.
Interpret the scatterplot and residual plot.
The scatterplot shows a positive linear relationship between the percentage of urban population and the robbery rate. The residual plot does not show any obvious pattern, indicating that the linear model is appropriate for the data.
\(\boxed{\text{Scatterplot and residual plot interpreted.}}\)
B. Write and interpret the correlation coefficient.
Calculate the correlation coefficient (\(r\)).
The correlation coefficient \(r\) is calculated using the formula:
\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n(\sum x^2) - (\sum x)^2][n(\sum y^2) - (\sum y)^2]}}
\]
Using the data provided:
\[
\begin{align_}
\sum y^2 &= 263,459, \\
\end{align_}
\]
Calculate \(r\):
\[
r = \frac{10(108,679) - (691)(1433)}{\sqrt{[10(52,091) - (691)^2][10(263,459) - (1433)^2]}} = \frac{96,087}{\sqrt{43,429 \times 37,801}} \approx \frac{96,087}{40,563} \approx 0.76
\]
Interpret the correlation coefficient.
The correlation coefficient \(r = 0.76\) indicates a strong positive linear relationship between the percentage of urban population and the robbery rate.
\(\boxed{r = 0.76, \text{ strong positive linear relationship}}\)
C. Write the regression equation and interpret the regression coefficient.
State the regression equation.
The regression equation is \(y = 2.21x - 9.41\).
Interpret the regression coefficient.
The regression coefficient \(2.21\) indicates that for each 1% increase in the urban population, the robbery rate increases by approximately 2.21 per 100,000 people.
\(\boxed{y = 2.21x - 9.41, \text{ each 1% increase in urban population increases robbery rate by 2.21}}\)
D. Show and interpret the coefficient of determination.
Calculate the coefficient of determination (\(r^2\)).
The coefficient of determination \(r^2\) is calculated as:
\[
r^2 = (0.76)^2 = 0.5776
\]
Interpret the coefficient of determination.
The coefficient of determination \(r^2 = 0.5776\) indicates that approximately 57.76% of the variability in the robbery rate can be explained by the percentage of urban population.
\(\boxed{r^2 = 0.5776, \text{ 57.76% of variability explained}}\)
E. State whether you think there is a relationship between the two variables, and justify your answer.
Determine the relationship.
There is a strong positive linear relationship between the percentage of urban population and the robbery rate, as indicated by the correlation coefficient \(r = 0.76\) and the coefficient of determination \(r^2 = 0.5776\).
Justify the relationship.
The strong correlation and the significant percentage of variability explained by the model justify the existence of a relationship between the two variables.
\(\boxed{\text{Yes, there is a strong positive relationship.}}\)
F. Predict the robbery rates in Idaho and Florida. Determine if these are extrapolations or interpolations, and if they are valid predictions.
Predict the robbery rate in Idaho.
For Idaho (\(x = 35\%\)):
\[
y = 2.21 \times 35 - 9.41 = 77.34 - 9.41 = 67.93
\]
Predict the robbery rate in Florida.
For Florida (\(x = 78\%\)):
\[
y = 2.21 \times 78 - 9.41 = 172.38 - 9.41 = 162.97
\]
Determine if these are extrapolations or interpolations.
The prediction for Idaho is an interpolation since 35% is within the range of the data. The prediction for Florida is also an interpolation since 78% is within the range of the data.
Assess the validity of the predictions.
Both predictions are valid as they are interpolations within the range of the data.
\(\boxed{\text{Idaho: 67.93, Florida: 162.97, both are valid interpolations}}\)
\(\boxed{\text{Scatterplot and residual plot interpreted.}}\)
\(\boxed{r = 0.76, \text{ strong positive linear relationship}}\)
\(\boxed{y = 2.21x - 9.41, \text{ each 1% increase in urban population increases robbery rate by 2.21}}\)
\(\boxed{r^2 = 0.5776, \text{ 57.76% of variability explained}}\)
\(\boxed{\text{Yes, there is a strong positive relationship.}}\)
\(\boxed{\text{Idaho: 67.93, Florida: 162.97, both are valid interpolations}}\)
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Answered
