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Homicide and Suicide. Preventing suicide is a important issue facing mental health workers. Predicting geographic regions where the risk of suicide is high could help people decide where to increase or improve mental health resources and care. Some psychiatrists have argued that homicide and suicide may have some causes in common. If so, one would expect homicide and suicide rates to be correlated. And if this is true, areas with high rates of homicide might be predicted to have high rates of suicide and therefore be in need of increased mental health resources. Research has had mixed results, including some evidence that there is a positive correlation in certain European countries but not in the United States. Here are data from 2015 for the 11 counties in Ohio with sufficient data for homicides and suicides to allow for estimating rates for both. \(\underline{\text { Rates }}\) are per 100,000 people. DEATH a. Make a scatterplot that shows how suicide rate can be predicted from homicide rate. There is a weak linear relationship, with correlation \(r=-0.0645\). b. Find the least-squares regression line for predicting suicide rate from homicide rate. Add this line to your scatterplot. c. Explain in words what the slope of the regression line tells us. d. Another Ohio county has a homicide rate of \(8.0\) per 100,000 people. What is the county's predicted suicide rate?

Step by step solution

01

Create the Scatterplot

To begin, plot the data points on a scatterplot with the homicide rate on the x-axis and the suicide rate on the y-axis. You should notice that the points do not show a strong pattern, confirming the weak correlation of \( r = -0.0645 \).

02

Find the Least-Squares Regression Line

The least-squares regression line is given by the equation \( y = a + bx \), where \( a \) is the y-intercept and \( b \) is the slope. To calculate these, you use the formulas: \( b = (r)(\frac{s_y}{s_x}) \) and \( a = \bar{y} - b\bar{x} \). Given that \( r = -0.0645 \), you need means (\( \bar{x}, \bar{y} \)) and standard deviations (\( s_x, s_y \)) of homicide and suicide rates to compute \( a \) and \( b \).

03

Compute the Slope and Intercept

Assume that the mean and standard deviation of homicide rates are \( \bar{x} = X \), \( s_x = S_x \), and for suicide rates, \( \bar{y} = Y \), \( s_y = S_y \). Use these values: \( b = (-0.0645)(\frac{S_y}{S_x}) \) and \( a = Y - bX \). After calculating \( b \) and \( a \), the regression equation might look something like \( y = a + bx \).

04

Add the Regression Line to the Scatterplot

Plot the line \( y = a + bx \) on your scatterplot from Step 1 to visualize how the line represents the relationship between homicide and suicide rates according to the least-squares method.

05

Interpret the Slope of the Regression Line

The slope \( b \) tells us the amount by which the suicide rate is expected to change for each one-unit increase in the homicide rate. A negative slope indicates that an increase in homicide rate is expected to result in a decrease in suicide rate, although this correlation is extremely weak.

06

Predict the Suicide Rate for a County with Homicide Rate 8.0

Substitute the homicide rate \( x = 8.0 \) into the regression equation \( y = a + bx \). Calculate \( y \) to find the predicted suicide rate.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Correlation

Understanding the concept of correlation is essential when analyzing relationships between two variables. Correlation measures the strength and direction of a linear relationship between two continuous variables. In this context, we look at the relationship between homicide rates and suicide rates in Ohio counties.

• A correlation coefficient, represented by \( r \), can range from \(-1\) to \(+1\).
• An \( r \) value close to \(+1\) suggests a strong positive correlation, where an increase in one variable tends to be associated with an increase in the other.
• A value close to \(-1\) suggests a strong negative correlation, indicating that as one variable increases, the other tends to decrease.
• An \( r \) value near zero indicates a weak or no linear correlation.

In this exercise, the correlation between homicide and suicide rates was found to be \( r = -0.0645 \), indicating a weak negative linear relationship. This means that generally, changes in homicide rates seem to have almost no predictable effect on suicide rates in these counties. Even though it suggests a slightly negative relationship, it is too weak to be considered significant or reliable.

Regression Line

The regression line is a crucial concept in statistical analysis used to predict the value of one variable based on the value of another. It is represented by a straight line in a scatterplot that best fits the data points. This line helps us make predictions about the dependent variable (y) based on the independent variable (x).

• The least-squares regression line equation is \( y = a + bx \).
• Here, \( a \) is the y-intercept, the value at which the line crosses the y-axis, and \( b \) is the slope.
• To find this line, we use the correlation coefficient and standard deviations of both variables.

Building this regression line allows us to see trends in the dataset and make predictions about what future or other values might be expected. Although the correlation here is weak, providing a regression line can still help contextualize the relationship, albeit with limited predictive power.

Scatterplot

Scatterplots provide a visual way of exploring the relationship between two variables. By plotting each pair of values from the dataset, we can observe patterns, trends, or clusters.

• The x-axis typically represents the independent variable (in this case, homicide rates), while the y-axis represents the dependent variable (suicide rates).
• Each point on the scatterplot corresponds to a pair of values from the data.
• Patterns in these points, such as a general upward or downward trend, indicate correlation and help better understand the data.

In our exercise, constructing a scatterplot helped visually confirm the weak correlation (\( r = -0.0645 \)) between homicide and suicide rates, emphasizing the lack of a clear linear pattern. This aids in understanding why predictions might be difficult or unreliable given the weak relationship.

Slope Interpretation

Interpreting the slope of a regression line gives insight into how changes in the independent variable are expected to influence the dependent variable. In a regression line expressed as \( y = a + bx \), the slope \( b \) plays a significant role.

• The slope indicates the magnitude and direction of change expected in the dependent variable for each one-unit increase in the independent variable.
• A positive slope suggests that as the independent variable increases, the dependent variable is also expected to increase.
• A negative slope implies the opposite: as the independent variable rises, the dependent variable is anticipated to decrease.

In this exercise, a weak negative slope was identified. Although its magnitude is minimal, this implies that for every one-unit increase in homicide rates, there's a negligible expected decrease in suicide rates. This observation underscores how the available data shows an almost nonexistent relationship likely offering little practical predictive guidance.