Suppose that a researcher using data on class size (CS) and average test scores from 100 thirdgrade classes, estimates the following OLS regression. Test̂Score = 520.4 − 5.82 × CS n = 100, R 2 = 0.08, SER = 11.5 a. A class has 22 students. What is the regression’s prediction for this classroom’s average test score?
b. Last year, a class had 19 students, and this year it has 23 students. What is the regression’s prediction for the change in the classroom average test score?
c. The sample average class size across the 100 classrooms is 21.4. What is the sample average of the test scores across the 100 classrooms?
d. What is the sample standard deviation of test scores across the 100 classrooms?

This question is incomplete, the complete question is;

Suppose that a researcher using data on class size (CS) and average test scores from 100 third grade classes, estimates the following OLS regression. Test-Score = 520.4 - 5.82 × CS, n = 100, R² = 0.08, SER = 11.5

a. A class has 22 students. What is the regression's prediction for this classroom's average test score?

b. Last year, a class had 19 students, and this year it has 23 students. What is the regression's prediction for the change in the classroom average test score?

c. The sample average class size across the 100 classrooms is 21.4. What is the sample average of the test scores across the 100 classrooms?

d. What is the sample standard deviation of test scores across the 100 classrooms?

a)

the regression's prediction for this classroom's average test score is 392.36

b)

the regression's prediction for the change in the classroom average test score is -23.28

c)

the sample average of the test scores across the 100 classrooms is 395.852

d)

the sample standard deviation of test scores across the 100 classrooms is 11.92887

Step-by-step explanation:

Given the data in the question;

Test-Score = 520.4 - 5.82 × CS, n = 100, R² = 0.08, SER = 11.5 -----1

the general formula for the average test score is as follows;

Test score = ^β₀ + ( ^β₁ × CS ) -------- 2

the general for change in test score ;

ΔTest Score = β × ΔClass size  -------- 3

General formula for the sum of squared residuals SSR

SSR = ( n - 2 ) SER² ----- 4

General formula for total sum of squares TSS

TSS = SRR / 1 - R² -------- 5

General formula for sample standard deviation;

Sy = √(TSS / (n-1) )  ------ 6

now, from the given formula;

^β₀ = 520.4

aslo, β = ^β₁ =  - 5.82

so

a) A class has 22 students. What is the regression's prediction for this classroom's average test score?

given that class size CS is 22, to get  the regression's prediction for this classroom's average test score, we make use of formula 2 above;

Test score = ^β₀ + ( ^β₁ × CS )

so we substitute

Test score = 520.4 + ( -5.82  × 22 )

Test score = 520.4 + ( - 128.04 )

Test score = 520.4 - 128.04

Test score = 392.36

Therefore,  the regression's prediction for this classroom's average test score is 392.36

b) Last year, a class had 19 students, and this year it has 23 students. What is the regression's prediction for the change in the classroom average test score.

we make use of formula 3 above

ΔTest Score = β × ΔClass size

we substitute

ΔTest Score = -5.82 × ( 23 - 19 )

ΔTest Score = -5.82 × 4

ΔTest Score = -23.28

Therefore,  the regression's prediction for the change in the classroom average test score is -23.28

c) The sample average class size across the 100 classrooms is 21.4. What is the sample average of the test scores across the 100 classrooms?

we make use of formula 2 above;

Test score = ^β₀ + ( ^β₁ × CS )

we substitute

Test score = 520.4 + ( -5.82 × 21.4 )

Test score = 520.4 + ( -124.548 )

Test score = 520.4 - 124.548

Test score = 395.852

Therefore, the sample average of the test scores across the 100 classrooms is 395.852

d) What is the sample standard deviation of test scores across the 100 classrooms.

first we make use of formula 4 above; to calculate the sum of squared residuals SSR

SSR = ( n - 2 ) SER²

we substitute

SSR = ( 100 - 2 ) (11.5)²

SSR = 98 × 132.25

SSR = 12,960.5

Also, for total sum of squares TSS, we use formula 5

TSS = SRR / 1 - R²

we that R² = 0.08; from the given formula

so we substitute

TSS = 12,960.5 / 1 - 0.08

TSS = 12,960.5 / 0.92

TSS = 14087.5

so, the sample standard deviation will be;

from formula 6 above

Sy = √(TSS / (n-1) )

we substitute

Sy = √(14087.5 / (100-1) )

Sy = √(14087.5 / 99)

Sy = √142.297979

Sy = 11.92887

Therefore, the sample standard deviation of test scores across the 100 classrooms is 11.92887

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step-by-step explanation: