Question

question 7 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 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! 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. find the least - squares regression line for predicting suicide rate from homicide rate, suicide rate = a + b×(homicide rate). add this line to your scatterplot. give your answers to three decimal places. a = b = -0.049

Explanation:

Step1: Calculate means

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

Step2: Use the formula for $b$

The formula for the slope $b$ of the least - squares regression line is $b = r\frac{s_y}{s_x}$. We are given $r=- 0.0645$. To find $s_x$ and $s_y$:
The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}$.
For $x$:
$\sum_{i=1}^{11}(x_i - \bar{x})^2=(4.0 - 8.064)^2+(10.8 - 8.064)^2+\cdots+(7.1 - 8.064)^2$
$=(-4.064)^2+(2.736)^2+\cdots+(-0.964)^2$
$=16.516+7.486+\cdots+0.929$
$=71.978$
$s_x=\sqrt{\frac{71.978}{10}}\approx2.683$
For $y$:
$\sum_{i = 1}^{11}(y_i-\bar{y})^2=(11.2 - 13.909)^2+(15.3 - 13.909)^2+\cdots+(17.9 - 13.909)^2$
$=(-2.709)^2+(1.391)^2+\cdots+(4.0)^2$
$=7.349+1.935+\cdots+16$
$=58.509$
$s_y=\sqrt{\frac{58.509}{10}}\approx2.419$
$b=r\frac{s_y}{s_x}=-0.0645\times\frac{2.419}{2.683}\approx - 0.049$

Step3: Use the formula for $a$

The formula for the intercept $a$ of the least - squares regression line is $a=\bar{y}-b\bar{x}$.
$a = 13.909-(-0.049)\times8.064$
$=13.909 + 0.395$
$=14.304$

Answer:

$a = 14.304$
$b=-0.049$