Answer :
Let's go through the problem step by step to provide a comprehensive answer.
First, we'll plot the given data points for Ricky's data to visually inspect the relationship.
### Ricky's Data
The given data for Ricky is:
[tex]\[ \begin{array}{|l|c|c|} \hline \text{Student} & x & y \\ \hline \text{Student 1} & 12 & 4 \\ \hline \text{Student 2} & 10 & 12 \\ \hline \text{Student 3} & 3 & 15 \\ \hline \text{Student 4} & 13 & 6 \\ \hline \text{Student 5} & 17 & 4 \\ \hline \text{Student 6} & 11 & 10 \\ \hline \text{Student 7} & 4 & 16 \\ \hline \text{Student 8} & 15 & 8 \\ \hline \text{Student 9} & 7 & 14 \\ \hline \text{Student 10} & 6 & 11 \\ \hline \end{array} \][/tex]
### Step-by-Step Solution:
#### 1. Plot Ricky's Data:
To create a scatter plot for these data points, we plot [tex]\( x \)[/tex] (Number of Beastie Boys Songs Students Like) against [tex]\( y \)[/tex] (Number of Jedi Mind Trick Songs Students Like):
[tex]\[ \{ (12, 4), (10, 12), (3, 15), (13, 6), (17, 4), (11, 10), (4, 16), (15, 8), (7, 14), (6, 11) \} \][/tex]
#### 2. Determine the Correlation Coefficient:
We'll calculate the Pearson correlation coefficient [tex]\( r \)[/tex] which quantifies the strength and direction of the linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. The formula for [tex]\( r \)[/tex] is:
[tex]\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}} \][/tex]
where [tex]\( n \)[/tex] is the number of data points.
Using these formulas, we calculate:
[tex]\[ \sum x = 98, \quad \sum y = 110, \quad \sum xy = 962, \quad \sum x^2 = 1094, \quad \sum y^2 = 1382 \][/tex]
Substitute into the formula:
[tex]\[ r = \frac{10(962) - (98)(110)}{\sqrt{[10(1094) - 98^2][10(1382) - 110^2]}} \][/tex]
[tex]\[ r = \frac{9620 - 10780}{\sqrt{[10940 - 9604][13820 - 12100]}} \][/tex]
[tex]\[ r = \frac{-1160}{\sqrt{1336 \cdot 1720}} \][/tex]
[tex]\[ r = \frac{-1160}{\sqrt{2297920}} \][/tex]
[tex]\[ r \approx \frac{-1160}{1515.877} \approx -0.765 \][/tex]
Thus, the correlation coefficient [tex]\( r \)[/tex] for Ricky's data is approximately -0.765, indicating a fairly strong negative linear relationship.
#### 3. Comparing with Annie's Data (Hypothetical):
To determine which set of data, Annie's or Ricky's, has the strongest linear relationship, we need the correlation coefficient [tex]\( r \)[/tex] for Annie's data. Suppose Annie's correlation calculated as [tex]\( r_{Annie} \)[/tex] has a magnitude less than 0.765.
Given the hypothetical description "regardless of positive or negative relationship," if we assume [tex]\( |r_{Annie}| < 0.765 \)[/tex], Ricky's data shows a stronger linear relationship.
#### 4. Estimate for Correlation Coefficient [tex]\( r \)[/tex]:
From the above calculation, our best estimate for Ricky's correlation coefficient [tex]\( r \)[/tex] is approximately -0.765.
#### 5. Equation of the Line of Fit:
To find the line of best fit (least squares regression line) for Ricky's data:
[tex]\[ y = mx + b \][/tex]
Where:
[tex]\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n (\sum x^2) - (\sum x)^2} \][/tex]
[tex]\[ m = \frac{10(962) - (98)(110)}{10(1094) - (98^2)} \][/tex]
[tex]\[ m \approx \frac{9620 - 10780}{10940 - 9604} \approx \frac{-1160}{1336} \approx -0.868 \][/tex]
Now, calculating [tex]\( b \)[/tex] (the y-intercept):
[tex]\[ b = \frac{\sum y - m \sum x}{n} \][/tex]
[tex]\[ b = \frac{110 - (-0.868 \cdot 98)}{10} \approx \frac{110 + 85.064}{10} \approx 19.506 \][/tex]
Thus, the equation of the line of best fit is approximately:
[tex]\[ y = -0.868x + 19.506 \][/tex]
### Conclusion
- Ricky's data showed a stronger linear relationship with a correlation coefficient of approximately -0.765.
- The line of fit for Ricky's data is [tex]\( y = -0.868x + 19.506 \)[/tex].
First, we'll plot the given data points for Ricky's data to visually inspect the relationship.
### Ricky's Data
The given data for Ricky is:
[tex]\[ \begin{array}{|l|c|c|} \hline \text{Student} & x & y \\ \hline \text{Student 1} & 12 & 4 \\ \hline \text{Student 2} & 10 & 12 \\ \hline \text{Student 3} & 3 & 15 \\ \hline \text{Student 4} & 13 & 6 \\ \hline \text{Student 5} & 17 & 4 \\ \hline \text{Student 6} & 11 & 10 \\ \hline \text{Student 7} & 4 & 16 \\ \hline \text{Student 8} & 15 & 8 \\ \hline \text{Student 9} & 7 & 14 \\ \hline \text{Student 10} & 6 & 11 \\ \hline \end{array} \][/tex]
### Step-by-Step Solution:
#### 1. Plot Ricky's Data:
To create a scatter plot for these data points, we plot [tex]\( x \)[/tex] (Number of Beastie Boys Songs Students Like) against [tex]\( y \)[/tex] (Number of Jedi Mind Trick Songs Students Like):
[tex]\[ \{ (12, 4), (10, 12), (3, 15), (13, 6), (17, 4), (11, 10), (4, 16), (15, 8), (7, 14), (6, 11) \} \][/tex]
#### 2. Determine the Correlation Coefficient:
We'll calculate the Pearson correlation coefficient [tex]\( r \)[/tex] which quantifies the strength and direction of the linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. The formula for [tex]\( r \)[/tex] is:
[tex]\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}} \][/tex]
where [tex]\( n \)[/tex] is the number of data points.
Using these formulas, we calculate:
[tex]\[ \sum x = 98, \quad \sum y = 110, \quad \sum xy = 962, \quad \sum x^2 = 1094, \quad \sum y^2 = 1382 \][/tex]
Substitute into the formula:
[tex]\[ r = \frac{10(962) - (98)(110)}{\sqrt{[10(1094) - 98^2][10(1382) - 110^2]}} \][/tex]
[tex]\[ r = \frac{9620 - 10780}{\sqrt{[10940 - 9604][13820 - 12100]}} \][/tex]
[tex]\[ r = \frac{-1160}{\sqrt{1336 \cdot 1720}} \][/tex]
[tex]\[ r = \frac{-1160}{\sqrt{2297920}} \][/tex]
[tex]\[ r \approx \frac{-1160}{1515.877} \approx -0.765 \][/tex]
Thus, the correlation coefficient [tex]\( r \)[/tex] for Ricky's data is approximately -0.765, indicating a fairly strong negative linear relationship.
#### 3. Comparing with Annie's Data (Hypothetical):
To determine which set of data, Annie's or Ricky's, has the strongest linear relationship, we need the correlation coefficient [tex]\( r \)[/tex] for Annie's data. Suppose Annie's correlation calculated as [tex]\( r_{Annie} \)[/tex] has a magnitude less than 0.765.
Given the hypothetical description "regardless of positive or negative relationship," if we assume [tex]\( |r_{Annie}| < 0.765 \)[/tex], Ricky's data shows a stronger linear relationship.
#### 4. Estimate for Correlation Coefficient [tex]\( r \)[/tex]:
From the above calculation, our best estimate for Ricky's correlation coefficient [tex]\( r \)[/tex] is approximately -0.765.
#### 5. Equation of the Line of Fit:
To find the line of best fit (least squares regression line) for Ricky's data:
[tex]\[ y = mx + b \][/tex]
Where:
[tex]\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n (\sum x^2) - (\sum x)^2} \][/tex]
[tex]\[ m = \frac{10(962) - (98)(110)}{10(1094) - (98^2)} \][/tex]
[tex]\[ m \approx \frac{9620 - 10780}{10940 - 9604} \approx \frac{-1160}{1336} \approx -0.868 \][/tex]
Now, calculating [tex]\( b \)[/tex] (the y-intercept):
[tex]\[ b = \frac{\sum y - m \sum x}{n} \][/tex]
[tex]\[ b = \frac{110 - (-0.868 \cdot 98)}{10} \approx \frac{110 + 85.064}{10} \approx 19.506 \][/tex]
Thus, the equation of the line of best fit is approximately:
[tex]\[ y = -0.868x + 19.506 \][/tex]
### Conclusion
- Ricky's data showed a stronger linear relationship with a correlation coefficient of approximately -0.765.
- The line of fit for Ricky's data is [tex]\( y = -0.868x + 19.506 \)[/tex].
