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The table below shows the number of SAT prep classes five students attended and the scores they received on the test.

| Number of Prep Classes Attended (x) | 3 | 1 | 6 | 7 | 6 |
|-------------------------------------|----|----|----|----|----|
| Math SAT Score (y) | 500| 410| 620| 720| 500|

1. State the linear regression equation for this data set, rounding all values to the nearest hundredth.
2. State the correlation coefficient, rounded to the nearest hundredth.
3. Explain what this correlation coefficient indicates about the linear fit of the data.

Sagot :

GinnyAnswer
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.
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