https://file.aiquickdraw.com/mathbot/file/17274792711582c0h99in.jpg

Let's go through each of the tasks presented in the image step by step:

### 1. Prediction of Average Test Score for a Classroom of 25 Students
We are given the regression equation:
\[
\hat{TestScore} = 473.564 + (-5.2962) \times CS
\]
Where \( CS \) is the class size. For a classroom of 25 students:
\[
\hat{TestScore} = 473.564 + (-5.2962) \times 25
\]
\[
\hat{TestScore} = 473.564 - 132.405
\]
\[
\hat{TestScore} = 341.159 \approx 341.16
\]

Thus, the predicted average test score for a class of 25 students is **341.16** (rounded to two decimal places).

---

### 2. Change in Average Test Score from 22 to 26 Students
The change in test score can be calculated using the same regression formula. The difference between the two predictions is:
\[
\Delta \hat{TestScore} = \hat{TestScore}_{26} - \hat{TestScore}_{22}
\]
First, let's calculate the test score predictions for both class sizes:

For 22 students:
\[
\hat{TestScore}_{22} = 473.564 + (-5.2962) \times 22 = 473.564 - 116.5164 = 357.0476
\]

For 26 students:
\[
\hat{TestScore}_{26} = 473.564 + (-5.2962) \times 26 = 473.564 - 137.7012 = 335.8628
\]

Now, the change in test score is:
\[
\Delta \hat{TestScore} = 335.8628 - 357.0476 = -21.1848 \approx -21.19
\]

Thus, the predicted change in the classroom average test score is **-21.19**.

---

### 3. Sample Average Class Size Across 102 Classrooms
It is given that the sample average class size across 102 classrooms is **19.47**.

---

### 4. Sample Average of the Test Scores
The hint suggests reviewing the formulas for the OLS estimators to compute this value. In OLS regression, the intercept \(\hat{\beta_0} = 473.564\) and the slope \(\hat{\beta_1} = -5.2962\) indicate that for a class size of \(19.47\), the expected test score is:

\[
\hat{TestScore} = 473.564 + (-5.2962) \times 19.47
\]
\[
\hat{TestScore} = 473.564 - 103.0735 = 370.4905
\]

Thus, the sample average of the test scores across the 102 classrooms is approximately **370.49**.

---

Let me know if you'd like further clarification on any of these steps or concepts! Here are five follow-up questions to explore:

1. How would the predicted test score change if the class size was 30 students instead of 25?
2. What does the negative slope in the regression equation tell us about the relationship between class size and test scores?
3. How does the value of \( R^2 = 0.09 \) influence the interpretation of the regression results?
4. What does the Standard Error of Regression (SER = 10.5) tell us about the accuracy of the predictions?
5. How can we calculate confidence intervals for the predicted test scores based on the given regression?

**Tip:** Understanding how to interpret the coefficients in regression models can provide insights into how different factors impact the dependent variable, in this case, the test scores.

Suppose that a researcher, using data on class size (CS) and average test scores from 102 third-grade classes, estimates the OLS regression: TestScore = 473.564 + (-5.2962) * CS. A classroom has 25 students, and the regression's prediction for that classroom's average test score is 341.16. The change in the classroom's average test score when class size increases from 22 to 26 is -21.19. The sample average class size across the 102 classrooms is 19.47. What is the sample average of the test scores across the 102 classrooms?

Learn how to calculate the average test scores using an OLS regression equation, including how class size impacts test scores. The regression estimates test scores for a class of 25 students at 341.16 and predicts a change of -21.19 when class size increases from 22 to 26 students. Suitable for advanced high school or college students.

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