Question

A researcher is investigating whether population impacts homicide rate. He uses demographic data from Detroit, MI to compare homicide rates and the number of the population that are white males.

a. Use your calculator to construct a scatter plot of the data. What should the independent variable be? Why?

b. Use your calculator’s regression function to find the equation of the least-squares regression line. Add this to your

scatter plot.

c. Discuss what the following mean in context.

i. The slope of the regression equation

ii. The y-intercept of the regression equation

iii. The correlation \(r\)

iv. The coefficient of determination \(r2\).

Short answer

Part a.

Part b. \(y=546826.41-3753.69x\)

Part c. Answer is explained in the explanation part.

Step-by-step solution

Part a. Step 1. Given information

Given table,

Part a. Step 2. Explanation

Here, you just have to draw the scatter graph. Get the Homicide rate per \(100,000\) people as the \(x-\)axis and Population Size as the \(y-\)axis. Then, mark the points given in the table.

The result scatter graph as follows.

Part b. Step 1. Explanation

The correlation coefficient, or Pearson product-moment correlation coefficient (PMCC) is a numerical value between \(-1\) and \(1\) that expresses the strength of the linear relationship between two variables. When \(r\) is closer to \(1\) it indicates a strong positive relationship. A value of \(0\) indicates that there is no relationship. Values close to \(-1\) signal a strong negative relationship between the two variables.

Simple linear regressionis a way to describe a relationship between two variables through an equation of a straight line, calledline of best fit, that most closely models this relationship.

Here we use the following formula to derive the equation for the line of best fit:

\(y=a+bx\)

where,

\(b=\sum_{i-1}^{n}\frac{x_{i}y_{i}-n\bar{x}\bar{y}}{x^{2}_{i}-n\bar{x}^{2}}\) and \(a=\bar{y}-b\bar{x}\)

Mean of \(x=\bar{x}=\frac{\sum x_{i}}{n}=25.13\)

Mean of \(y=\bar{y}=\frac{\sum y_{i}}{n}=452507.54\)

Then we can get \(a\) using the equation \(a=\bar{y}-\bar{bx}\)

\(a=\bar{y}-\bar{bx}=546826.41\)

\(b\) can be taken using the equation \(b=\sum_{i-1}^{n}\frac{x_{i}y_{i}-n\bar{x}\bar{y}}{x^{2}_{i}-n\bar{x}^{2}}\)

\(b=-3753.69)\)

Then the regression linear equation can be obtained as \(y=546826.41-3753.69x\). When this equation plotted in the previous scatter graph, then it is as follows.

Conclusion: \(y=193.88-1.495x\)

Part c. Step 1. Explanation

The slope of a regression line (b) represents the rate of change in \(y\) as \(x\)changes. Because \(y\) is dependent \on\(x\), the slope describes the predicted values of \(y\) given \(x\). When using the ordinary least squares method, one of the most common linear regressions, slope, is found by calculating \(b\) as the covariance of \(x\) and \(y\), divided by the sum of squares (variance) of \(x\).

\(b=\sum_{i-1}^{n}\frac{x_{i}y_{i}-n\bar{xy}}{x_{i}^{2}-n\bar{x}^{2}}\)

The slope must be calculated before the \(y-\)intercept when using a linear regression, as the intercept is calculated using the slope. The slope of a regression line is used with a t-statistic to test the significance of a linear relationship between \(x\) and \(y\).

The intercept indicates the location where it intersects an axis. The slope and the intercept define the linear relationship between two variables, and can be used to estimate an average rate of change. The greater the magnitude of the slope, the steeper the line and the greater the rate of change.

The correlation coefficient, or Pearson product-moment correlation coefficient (PMCC) is a numerical value between \(-1\) and \(1\) that expresses the strength of the linear relationship between two variables. When \(r\) is closer to \(1\) it indicates a strong positive relationship. A value of \(0\) indicates that there is no relationship. Values close to \(-1\) signal a strong negative relationship between the two variables.

Correlation coefficient \((r): -0.95257479022621\)