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Sagot :
To find the linear regression equation for the given data set and analyze the correlation coefficient, follow these steps:
### Step 1: Define Variables
The number of prep classes attended and the corresponding SAT scores for five students are:
- [tex]\( x = [3, 1, 6, 7, 6] \)[/tex]
- [tex]\( y = [500, 410, 620, 720, 500] \)[/tex]
### Step 2: Compute Linear Regression
Linear regression involves finding the best-fitting line for the given data points. The equation of a line is typically written as:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\( m \)[/tex] is the slope of the line
- [tex]\( b \)[/tex] is the y-intercept
Given our data:
- The calculated slope ([tex]\( m \)[/tex]) is approximately [tex]\( 40.48 \)[/tex].
- The calculated y-intercept ([tex]\( b \)[/tex]) is approximately [tex]\( 363.81 \)[/tex].
Thus, the linear regression equation, rounded to the nearest hundredth, is:
[tex]\[ y = 40.48x + 363.81 \][/tex]
### Step 3: State the Correlation Coefficient
The correlation coefficient ([tex]\( r \)[/tex]) measures the strength and direction of the linear relationship between the two variables. For our data set, the correlation coefficient is:
[tex]\[ r \approx 0.84 \][/tex]
### Step 4: Interpret the Correlation Coefficient
The correlation coefficient can be categorized as follows:
- Strong correlation: [tex]\( |r| > 0.8 \)[/tex]
- Moderate correlation: [tex]\( 0.5 < |r| \leq 0.8 \)[/tex]
- Weak correlation: [tex]\( |r| \leq 0.5 \)[/tex]
Given that the correlation coefficient is [tex]\( 0.84 \)[/tex], which is more than 0.8, this indicates a strong positive linear relationship between the number of prep classes attended and the SAT scores.
### Summary
The linear regression equation for the given data set, rounded to the nearest hundredth, is:
[tex]\[ y = 40.48x + 363.81 \][/tex]
The correlation coefficient, rounded to the nearest hundredth, is:
[tex]\[ r \approx 0.84 \][/tex]
This correlation coefficient indicates a strong linear fit of the data.
### Step 1: Define Variables
The number of prep classes attended and the corresponding SAT scores for five students are:
- [tex]\( x = [3, 1, 6, 7, 6] \)[/tex]
- [tex]\( y = [500, 410, 620, 720, 500] \)[/tex]
### Step 2: Compute Linear Regression
Linear regression involves finding the best-fitting line for the given data points. The equation of a line is typically written as:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\( m \)[/tex] is the slope of the line
- [tex]\( b \)[/tex] is the y-intercept
Given our data:
- The calculated slope ([tex]\( m \)[/tex]) is approximately [tex]\( 40.48 \)[/tex].
- The calculated y-intercept ([tex]\( b \)[/tex]) is approximately [tex]\( 363.81 \)[/tex].
Thus, the linear regression equation, rounded to the nearest hundredth, is:
[tex]\[ y = 40.48x + 363.81 \][/tex]
### Step 3: State the Correlation Coefficient
The correlation coefficient ([tex]\( r \)[/tex]) measures the strength and direction of the linear relationship between the two variables. For our data set, the correlation coefficient is:
[tex]\[ r \approx 0.84 \][/tex]
### Step 4: Interpret the Correlation Coefficient
The correlation coefficient can be categorized as follows:
- Strong correlation: [tex]\( |r| > 0.8 \)[/tex]
- Moderate correlation: [tex]\( 0.5 < |r| \leq 0.8 \)[/tex]
- Weak correlation: [tex]\( |r| \leq 0.5 \)[/tex]
Given that the correlation coefficient is [tex]\( 0.84 \)[/tex], which is more than 0.8, this indicates a strong positive linear relationship between the number of prep classes attended and the SAT scores.
### Summary
The linear regression equation for the given data set, rounded to the nearest hundredth, is:
[tex]\[ y = 40.48x + 363.81 \][/tex]
The correlation coefficient, rounded to the nearest hundredth, is:
[tex]\[ r \approx 0.84 \][/tex]
This correlation coefficient indicates a strong linear fit of the data.
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