Question

question 9 of 39 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. rates are per 100,000 people.

county\thomicide rate\tsuicide rate
butler\t4.0\t11.2
clark\t10.8\t15.3
cuyahoga\t12.2\t11.4
franklin\t8.7\t12.3
hamilton\t10.2\t11.0
lorain\t3.3\t14.3
lucas\t6.0\t12.6
mahoning\t11.7\t15.2
montgomery\t8.9\t15.7
stark\t5.8\t16.1
summit\t7.1\t17.9

another ohio county has a homicide rate of 8.0 per 100,000 people. what is the county’s predicted suicide rate? give your answer to three decimal places.
predicted suicide rate: per 100,000 people

to access the data, click the link for your preferred software format.
csv excel (xls) excel (xlsx) jmp mac - text minitab14 - 18 minitab18+ pc - text r spss ti crunchit!

Explanation:

Step1: Calculate means

Let \(x\) be the homicide - rate and \(y\) be the suicide - rate.
The mean of \(x\), \(\bar{x}=\frac{4.0 + 10.8+12.2 + 8.7+10.2+3.3+6.0+11.7+8.9+5.8+7.1}{11}=\frac{88.7}{11}\approx8.064\)
The mean of \(y\), \(\bar{y}=\frac{11.2 + 15.3+11.4 + 12.3+11.0+14.3+12.6+15.2+15.7+16.1+17.9}{11}=\frac{153}{11}\approx13.909\)

Step2: Calculate the slope \(b_1\)

\[

$$\begin{align*} b_1&=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}\\ \sum_{i = 1}^{11}(x_i-\bar{x})(y_i - \bar{y})&=(4.0 - 8.064)(11.2-13.909)+(10.8 - 8.064)(15.3 - 13.909)+\cdots+(7.1 - 8.064)(17.9 - 13.909)\\ &=(- 4.064)(-2.709)+(2.736)(1.391)+\cdots+(-0.964)(4.0)\\ &\approx51.97\\ \sum_{i = 1}^{11}(x_i-\bar{x})^2&=(4.0 - 8.064)^2+(10.8 - 8.064)^2+\cdots+(7.1 - 8.064)^2\\ &=(-4.064)^2+(2.736)^2+\cdots+(-0.964)^2\\ &\approx77.79 \end{align*}$$

\]
\(b_1=\frac{51.97}{77.79}\approx0.668\)

Step3: Calculate the intercept \(b_0\)

\(b_0=\bar{y}-b_1\bar{x}=13.909-0.668\times8.064 = 13.909 - 5.397=8.512\)

Step4: Predict the suicide - rate

The regression equation is \(\hat{y}=b_0 + b_1x\).
When \(x = 8.0\), \(\hat{y}=8.512+0.668\times8.0=8.512 + 5.344 = 13.856\)

Answer:

13.856