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The data below shows the number of hikers per month, [tex]$x$[/tex], and the corresponding number of injuries suffered, [tex]$y$[/tex], on a certain local hiking trail over a period of six months.

\begin{tabular}{|l|c|c|c|c|c|c|c|}
\hline
Number of hikers, [tex]$x$[/tex] & 138 & 104 & 131 & 135 & 116 & 92 & 103 \\
\hline
Number of injuries, [tex]$y$[/tex] & 15 & 4 & 8 & 10 & 7 & 3 & 4 \\
\hline
\end{tabular}

Determine the regression line equation, [tex]$y=ax+b$[/tex], and the corresponding correlation coefficient, [tex]$r$[/tex]. Round the values of [tex]$a$[/tex], [tex]$b$[/tex], and [tex]$r$[/tex] to the nearest thousandth.



Answer :

GinnyAnswer
Let's determine the regression line equation [tex]\( y = a x + b \)[/tex] and the corresponding correlation coefficient [tex]\( r \)[/tex] using the given data. Here's the step-by-step solution:

1. Given Data:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline x & 138 & 104 & 131 & 135 & 116 & 92 & 103 \\ \hline y & 15 & 4 & 8 & 10 & 7 & 3 & 4 \\ \hline \end{array} \][/tex]

2. Perform Linear Regression Analysis:
- To find the slope [tex]\( a \)[/tex] and the y-intercept [tex]\( b \)[/tex] of the regression line, we'll use the linear regression formulas.
- The correlation coefficient [tex]\( r \)[/tex] quantifies the strength and direction of the linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].

3. Solution:
- After calculating the necessary statistics from the data (formulas include sums of products, means, etc.), we derive:
- The slope [tex]\( a \)[/tex] of the regression line,
- The intercept [tex]\( b \)[/tex], and
- The correlation coefficient [tex]\( r \)[/tex].

4. Values Calculation:
- Slope [tex]\( a \)[/tex]:
[tex]\[ a = 0.214 \][/tex]
- Intercept [tex]\( b \)[/tex]:
[tex]\[ b = -17.709 \][/tex]
- Correlation Coefficient [tex]\( r \)[/tex]:
[tex]\[ r = 0.911 \][/tex]

5. Regression Line Equation:
Plugging the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the regression line equation:
[tex]\[ y = 0.214 x - 17.709 \][/tex]

Therefore, the regression line equation is:
[tex]\[ y = 0.214 x - 17.709 \][/tex]

And the corresponding correlation coefficient is:
[tex]\[ r = 0.911 \][/tex]

These values are rounded to the nearest thousandth as required.

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