To estimate the parameters $\beta_0$ (intercept) and $\beta_1$ (slope) for the regression model:

$\text{Second Test Grade} = \beta_0 + \beta_1 (\text{First Test Grade}) + \epsilon_i$

we would typically use statistical software (such as R, Python, or Excel) to run a linear regression and get these estimates. Here's a step-by-step breakdown of how you would approach this:

Steps to Perform:

1. Input the data: You would input the pairs of first and second test scores for all 21 students.
2. Run linear regression: The software will compute the estimates for $\beta_0$ and $\beta_1$ based on the formula above, using the least squares method to minimize the residuals (the errors between the predicted and actual second test scores).
3. Output: The output will provide you with the estimated $\beta_0$ (intercept) and $\beta_1$ (slope) as well as their standard errors, t-values, and p-values.

Since I don't have access to statistical software directly, I'll outline how you can do this in popular software:

Option 1: Using R

rCopy code
# Enter the data
first_test <- c(50, 91, 55, 58, 79, 70, 97, 51, 51, 59, 69, 56, 56, 41, 49, 58, 58, 99, 87, 95, 97)
second_test <- c(70, 58, 74, 65, 60, 60, 48, 73, 73, 66, 67, 70, 73, 76, 72, 68, 73, 53, 58, 50, 55)

# Fit the linear regression model
model <- lm(second_test ~ first_test)

# Output the regression summary
summary(model)

Option 2: Using Python (with libraries such as statsmodels or scikit-learn)

pythonCopy code
import numpy as np
import statsmodels.api as sm

# Enter the data
first_test = np.array([50, 91, 55, 58, 79, 70, 97, 51, 51, 59, 69, 56, 56, 41, 49, 58, 58, 99, 87, 95, 97])
second_test = np.array([70, 58, 74, 65, 60, 60, 48, 73, 73, 66, 67, 70, 73, 76, 72, 68, 73, 53, 58, 50, 55])

# Add a constant for the intercept term
X = sm.add_constant(first_test)

# Fit the linear regression model
model = sm.OLS(second_test, X).fit()

# Output the regression summary
print(model.summary())

Both methods will give you the estimates for $\beta_0$ and $\beta_1$. Here's what the result typically looks like:

• $\beta_1$ (slope) tells you how much the second test grade changes for each unit change in the first test grade.
• $\beta_0$ (intercept) tells you the expected second test grade when the first test grade is 0.

Once you run the regression, it will give you the necessary coefficients.

Would you like help interpreting the results once you run this, or do you need assistance with another part of the process?