Answer :
Let's solve the given problem step by step:
### Part (a): Relevant Sums
Given:
[tex]\[ \sum x = 185, \quad \sum y = 781, \quad \sum x^2 = 3711, \quad \sum y^2 = 62910, \quad \sum xy = 13721 \][/tex]
These are the sums we need to calculate the correlation coefficient and other statistics.
### Part (b): Correlation Coefficient
The formula for the correlation coefficient [tex]\( r \)[/tex] is:
[tex]\[ r = \frac{n \sum xy - \sum x \sum y}{\sqrt{n \sum x^2 - (\sum x)^2} \sqrt{n \sum y^2 - (\sum y)^2}} \][/tex]
Plugging in the numbers:
- [tex]\( n = 10 \)[/tex]
- [tex]\( \sum xy = 13721 \)[/tex]
- [tex]\( \sum x = 185 \)[/tex]
- [tex]\( \sum y = 781 \)[/tex]
- [tex]\( \sum x^2 = 3711 \)[/tex]
- [tex]\( \sum y^2 = 62910 \)[/tex]
Calculating each term:
1. Calculate the numerator:
[tex]\[ n \sum xy = 10 \times 13721 = 137210 \][/tex]
[tex]\[ \sum x \sum y = 185 \times 781 = 144485 \][/tex]
[tex]\[ \text{Numerator} = 137210 - 144485 = -7275 \][/tex]
2. Calculate the denominator:
[tex]\[ n \sum x^2 = 10 \times 3711 = 37110 \][/tex]
[tex]\[ (\sum x)^2 = 185^2 = 34225 \][/tex]
[tex]\[ \sqrt{37110 - 34225} = \sqrt{2885} \approx 53.712 \][/tex]
[tex]\[ n \sum y^2 = 10 \times 62910 = 629100 \][/tex]
[tex]\[ (\sum y)^2 = 781^2 = 609961 \][/tex]
[tex]\[ \sqrt{629100 - 609961} = \sqrt{19139} \approx 138.344 \][/tex]
[tex]\[ \text{Denominator} = 53.712 \times 138.344 \approx 7434.109 \][/tex]
3. Finally, the correlation coefficient:
[tex]\[ r = \frac{-7275}{7434.109} \approx -0.980 \][/tex]
### Part (c): Standard Deviations
Given:
[tex]\[ s_y = 14.56365, \quad s_z = 5.661763 \][/tex]
These are the standard deviations of [tex]\( y \)[/tex] and [tex]\( z \)[/tex], respectively.
### Part (d): Slope of the Best-Fit Line
The slope [tex]\( a \)[/tex] of the best-fit line is found using:
[tex]\[ a = r \frac{s_y}{s_z} \][/tex]
Plugging in the numbers:
[tex]\[ a = -0.980 \times \frac{14.56365}{5.661763} \approx -2.521 \][/tex]
### Part (e): [tex]\( y \)[/tex]-Intercept of the Best-Fit Line
The [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] of the best-fit line is found using:
[tex]\[ b = y_{\text{mean}} - a x_{\text{mean}} \][/tex]
First, find [tex]\( y_{\text{mean}} \)[/tex] and [tex]\( x_{\text{mean}} \)[/tex]:
[tex]\[ y_{\text{mean}} = \frac{\sum y}{n} = \frac{781}{10} = 78.1 \][/tex]
[tex]\[ x_{\text{mean}} = \frac{\sum x}{n} = \frac{185}{10} = 18.5 \][/tex]
Then, calculate the [tex]\( y \)[/tex]-intercept:
[tex]\[ b = 78.1 - (-2.521 \times 18.5) \approx 78.1 + 46.635 \approx 124.735 \][/tex]
### Summary:
- The sums are:
[tex]\[ \sum x = 185, \quad \sum y = 781, \quad \sum x^2 = 3711, \quad \sum y^2 = 62910, \quad \sum xy = 13721 \][/tex]
- The correlation coefficient:
[tex]\[ r \approx -0.980 \][/tex]
- Standard deviations:
[tex]\[ s_y = 14.56365, \quad s_z = 5.661763 \][/tex]
- Slope of the best-fit line:
[tex]\[ a \approx -2.521 \][/tex]
- [tex]\( y \)[/tex]-Intercept of the best-fit line:
[tex]\[ b \approx 124.735 \][/tex]
### Part (a): Relevant Sums
Given:
[tex]\[ \sum x = 185, \quad \sum y = 781, \quad \sum x^2 = 3711, \quad \sum y^2 = 62910, \quad \sum xy = 13721 \][/tex]
These are the sums we need to calculate the correlation coefficient and other statistics.
### Part (b): Correlation Coefficient
The formula for the correlation coefficient [tex]\( r \)[/tex] is:
[tex]\[ r = \frac{n \sum xy - \sum x \sum y}{\sqrt{n \sum x^2 - (\sum x)^2} \sqrt{n \sum y^2 - (\sum y)^2}} \][/tex]
Plugging in the numbers:
- [tex]\( n = 10 \)[/tex]
- [tex]\( \sum xy = 13721 \)[/tex]
- [tex]\( \sum x = 185 \)[/tex]
- [tex]\( \sum y = 781 \)[/tex]
- [tex]\( \sum x^2 = 3711 \)[/tex]
- [tex]\( \sum y^2 = 62910 \)[/tex]
Calculating each term:
1. Calculate the numerator:
[tex]\[ n \sum xy = 10 \times 13721 = 137210 \][/tex]
[tex]\[ \sum x \sum y = 185 \times 781 = 144485 \][/tex]
[tex]\[ \text{Numerator} = 137210 - 144485 = -7275 \][/tex]
2. Calculate the denominator:
[tex]\[ n \sum x^2 = 10 \times 3711 = 37110 \][/tex]
[tex]\[ (\sum x)^2 = 185^2 = 34225 \][/tex]
[tex]\[ \sqrt{37110 - 34225} = \sqrt{2885} \approx 53.712 \][/tex]
[tex]\[ n \sum y^2 = 10 \times 62910 = 629100 \][/tex]
[tex]\[ (\sum y)^2 = 781^2 = 609961 \][/tex]
[tex]\[ \sqrt{629100 - 609961} = \sqrt{19139} \approx 138.344 \][/tex]
[tex]\[ \text{Denominator} = 53.712 \times 138.344 \approx 7434.109 \][/tex]
3. Finally, the correlation coefficient:
[tex]\[ r = \frac{-7275}{7434.109} \approx -0.980 \][/tex]
### Part (c): Standard Deviations
Given:
[tex]\[ s_y = 14.56365, \quad s_z = 5.661763 \][/tex]
These are the standard deviations of [tex]\( y \)[/tex] and [tex]\( z \)[/tex], respectively.
### Part (d): Slope of the Best-Fit Line
The slope [tex]\( a \)[/tex] of the best-fit line is found using:
[tex]\[ a = r \frac{s_y}{s_z} \][/tex]
Plugging in the numbers:
[tex]\[ a = -0.980 \times \frac{14.56365}{5.661763} \approx -2.521 \][/tex]
### Part (e): [tex]\( y \)[/tex]-Intercept of the Best-Fit Line
The [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] of the best-fit line is found using:
[tex]\[ b = y_{\text{mean}} - a x_{\text{mean}} \][/tex]
First, find [tex]\( y_{\text{mean}} \)[/tex] and [tex]\( x_{\text{mean}} \)[/tex]:
[tex]\[ y_{\text{mean}} = \frac{\sum y}{n} = \frac{781}{10} = 78.1 \][/tex]
[tex]\[ x_{\text{mean}} = \frac{\sum x}{n} = \frac{185}{10} = 18.5 \][/tex]
Then, calculate the [tex]\( y \)[/tex]-intercept:
[tex]\[ b = 78.1 - (-2.521 \times 18.5) \approx 78.1 + 46.635 \approx 124.735 \][/tex]
### Summary:
- The sums are:
[tex]\[ \sum x = 185, \quad \sum y = 781, \quad \sum x^2 = 3711, \quad \sum y^2 = 62910, \quad \sum xy = 13721 \][/tex]
- The correlation coefficient:
[tex]\[ r \approx -0.980 \][/tex]
- Standard deviations:
[tex]\[ s_y = 14.56365, \quad s_z = 5.661763 \][/tex]
- Slope of the best-fit line:
[tex]\[ a \approx -2.521 \][/tex]
- [tex]\( y \)[/tex]-Intercept of the best-fit line:
[tex]\[ b \approx 124.735 \][/tex]
