Answer :
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?
Answer:
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[tex]_{ class-size[/tex] × 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[tex]_{ class-size[/tex] = ^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[tex]_{ class-size[/tex] × 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