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. 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 county homicide rate suicide rate butler 4.0 11.2 clark 10.8 15.3 cuyahoga 12.2 11.4 franklin 8.7 12.3 hamilton 10.2 11.0 lorain 3.3 14.3 lucas 6.0 12.6 mahoning 11.7 15.2 montgomery 8.9 15.7 stark 5.8 16.1 summit 7.1 17.9 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.
First, calculate the mean of $x$: $\bar{x}=\frac{4.0 + 10.8+12.2 + 8.7+10.2+3.3+6.0+11.7+8.9+5.8+7.1}{11}=\frac{88.7}{11}\approx8.064$
Calculate 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 $b_1$

The formula for the slope of the regression line $b_1=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}$
$\sum_{i = 1}^{11}(x_i-\bar{x})(y_i - \bar{y})=(4.0 - 8.064)(11.2-13.909)+(10.8 - 8.064)(15.3 - 13.909)+(12.2-8.064)(11.4 - 13.909)+(8.7-8.064)(12.3 - 13.909)+(10.2-8.064)(11.0 - 13.909)+(3.3-8.064)(14.3 - 13.909)+(6.0-8.064)(12.6 - 13.909)+(11.7-8.064)(15.2 - 13.909)+(8.9-8.064)(15.7 - 13.909)+(5.8-8.064)(16.1 - 13.909)+(7.1-8.064)(17.9 - 13.909)$
$=(-4.064)(-2.709)+(2.736)(1.391)+(4.136)(-2.509)+(0.636)(-1.609)+(2.136)(-2.909)+(-4.764)(0.391)+(-2.064)(-1.309)+(3.636)(1.291)+(0.836)(1.791)+(-2.264)(2.191)+(-0.964)(3.991)$
$=10.919+3.806-10.377 - 1.023-6.214-1.863 + 2.702+4.694+1.497-4.960-3.848$
$=4.023$

$\sum_{i = 1}^{11}(x_i-\bar{x})^2=(4.0 - 8.064)^2+(10.8 - 8.064)^2+(12.2-8.064)^2+(8.7-8.064)^2+(10.2-8.064)^2+(3.3-8.064)^2+(6.0-8.064)^2+(11.7-8.064)^2+(8.9-8.064)^2+(5.8-8.064)^2+(7.1-8.064)^2$
$=(-4.064)^2+(2.736)^2+(4.136)^2+(0.636)^2+(2.136)^2+(-4.764)^2+(-2.064)^2+(3.636)^2+(0.836)^2+(-2.264)^2+(-0.964)^2$
$=16.516+7.486+17.106+0.404+4.562+22.695+4.259+13.22+0.699+5.125+0.929$
$=92.997$

$b_1=\frac{4.023}{92.997}\approx0.043$

Step3: Calculate the intercept $b_0$

The formula for the intercept $b_0=\bar{y}-b_1\bar{x}$
$b_0 = 13.909-0.043\times8.064$
$b_0=13.909 - 0.347$
$b_0 = 13.562$

Step4: Predict the suicide - rate

The regression equation is $\hat{y}=b_0 + b_1x$
When $x = 8.0$, $\hat{y}=13.562+0.043\times8.0$
$\hat{y}=13.562 + 0.344$
$\hat{y}=13.906$

Answer:

$13.906$