To estimate the parameters of the linear regression model:

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

I will compute the least squares regression estimates using the provided data. Let me calculate this now. numpy as np import statsmodels.api as sm

Given data

first_test_scores = np.array([61, 43, 50, 95, 46, 67, 58, 69, 96, 53, 72, 67, 46, 80, 63, 44, 55, 77, 93, 86, 52]) second_test_scores = np.array([71, 80, 73, 54, 81, 73, 75, 71, 55, 72, 63, 73, 73, 67, 66, 76, 75, 68, 61, 60, 79])

Add a constant for the intercept

X = sm.add_constant(first_test_scores)

Fit the regression model

model = sm.OLS(second_test_scores, X).fit()

Extract coefficients

beta_0 = model.params[0] beta_1 = model.params[1]

Round to 4 decimal places

beta_0_rounded = round(beta_0, 4) beta_1_rounded = round(beta_1, 4)

(beta_0_rounded, beta_1_rounded)The estimated regression model is:

$\text{Second Test Grade} = 96.35 - 0.4059 (\text{First Test Grade}) + \varepsilon_i$

Let me know if you need further explanation!

Related Questions:

1. What does the negative slope indicate about the relationship between the two test scores?
2. How can we interpret the intercept in this regression model?
3. What statistical assumptions are required for this model to be valid?
4. How can we check the goodness-of-fit of this regression model?
5. What other variables might improve the predictive power of this model?

Tip:

Always check the residuals of your regression model to ensure that the assumptions of linear regression (such as homoscedasticity and normality) are not violated.