A statistics professor would like to build a model relating student scores on the first test to the scores on the second test. The test scores from a random sample of 21
students who have previously taken the course are given in the table.

Test Scores
Student	First Test Grade	Second Test Grade
1	78
79
2	92
83
3	95
84
4	77
73
5	72
72
6	40
57
7	64
70
8	65
74
9	44
64
10	100
82
11	53
66
12	61
68
13	80
75
14	40
62
15	88
83
16	59
69
17	78
80
18	93
82
19	48
67
20	87
81
21	71
71
Step 2 of 2 :  Interpret the coefficient of the first test grade in the model.

To interpret the coefficient of the first test grade in a regression model, we need to assume a simple linear regression equation:  

\[
Y = \beta_0 + \beta_1 X + \epsilon
\]

where:  
- \( Y \) is the second test grade (dependent variable),  
- \( X \) is the first test grade (independent variable),  
- \( \beta_0 \) is the y-intercept,  
- \( \beta_1 \) is the coefficient of the first test grade,  
- \( \epsilon \) is the error term.

The coefficient \( \beta_1 \) represents the expected change in the second test grade for each additional point scored on the first test. In simpler terms:  

- If \( \beta_1 \) is **positive**, students who score higher on the first test tend to score higher on the second test.  
- If \( \beta_1 \) is **negative**, students who score higher on the first test tend to score lower on the second test.  
- If \( \beta_1 \) is **zero**, there is no relationship between the first and second test scores.  

Would you like me to calculate the regression model for you using the given data?  

### Related Questions:
1. How do you calculate the coefficient \( \beta_1 \) using least squares regression?  
2. What does the intercept (\( \beta_0 \)) represent in this context?  
3. How do you determine if the relationship between the test scores is statistically significant?  
4. What is the correlation coefficient between the two test scores?  
5. How do you use regression analysis to predict a student's second test score given their first test score?  

**Tip:** Always check the R-squared value to understand how well your model explains the variability in the data!

A statistics professor would like to build a model relating student scores on the first test to the scores on the second test. The test scores from a random sample of 21 students who have previously taken the course are given in the table.

Learn how to interpret the coefficient of the first test grade in a regression model that relates students' scores on two tests. The model uses simple linear regression to predict second test scores based on the first test. Suitable for high school students studying statistics and regression analysis.

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