To calculate the correlation coefficient for the given data and round it to the nearest thousandth, follow these steps:
### Step 1: Gather the Data
List the given [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
3 & 37.8 \\
\hline
5 & 75.8 \\
\hline
7 & 62.3 \\
\hline
11 & 52.8 \\
\hline
13 & 85.5 \\
\hline
16 & 41.2 \\
\hline
19 & 75.1 \\
\hline
\end{array}
\][/tex]
### Step 2: Calculate the Means
Compute the mean (average) of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
\bar{x} = \frac{\sum x}{n} = \frac{3 + 5 + 7 + 11 + 13 + 16 + 19}{7} = \frac{74}{7} \approx 10.571
\][/tex]
[tex]\[
\bar{y} = \frac{\sum y}{n} = \frac{37.8 + 75.8 + 62.3 + 52.8 + 85.5 + 41.2 + 75.1}{7} = \frac{430.5}{7} \approx 61.5
\][/tex]
### Step 3: Compute the Covariance
First, find the products of deviations from the mean for each pair [tex]\((x_i, y_i)\)[/tex]:
[tex]\[
(x_i - \bar{x})(y_i - \bar{y})
\][/tex]
Then sum these products:
[tex]\[
\sum (x_i - \bar{x})(y_i - \bar{y}) \approx \sum ([(3 - 10.571)(37.8 - 61.5), (5 - 10.571)(75.8 - 61.5), \dots])
\][/tex]
### Step 4: Compute the Standard Deviations
Calculate the standard deviations of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
s_x = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\][/tex]
[tex]\[
s_y = \sqrt{\frac{\sum (y_i - \bar{y})^2}{n-1}}
\][/tex]
### Step 5: Compute the Correlation Coefficient
Using the formula for the Pearson correlation coefficient:
[tex]\[
r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{(n-1)s_x s_y}
\][/tex]
### Step 6: Round the Result
Finally, after calculating the correlation coefficient, round it to the nearest thousandth.
The correlation coefficient for the given data is approximately 0.241 when rounded to the nearest thousandth.
