To predict the second test score for a student who scored 43 on the first test using regression analysis, follow these steps:
1. Collect Data:
We are given the first and second test scores for 12 students:
- First test scores, [tex]\(x\)[/tex]: [44, 93, 70, 88, 81, 51, 76, 58, 42, 81, 67, 85]
- Second test scores, [tex]\(y\)[/tex]: [52, 100, 69, 89, 79, 62, 80, 59, 50, 88, 69, 86]
2. Compute the Linear Regression Equation:
The equation of a line (regression line) is generally given by:
[tex]\[
y = mx + c
\][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\(c\)[/tex] is the y-intercept.
Using linear regression techniques (which can be computed using statistical tools or software), we find the slope [tex]\(m\)[/tex] and the y-intercept [tex]\(c\)[/tex]:
- Slope [tex]\(m = 0.900491\)[/tex]
- Intercept [tex]\(c = 10.297416\)[/tex]
3. Predict the Second Test Score:
We need to predict the second test score for a student who scored 43 on the first test. Substitute [tex]\(x = 43\)[/tex] into the regression equation:
[tex]\[
y = m \cdot x + c
\][/tex]
Plugging the values we get:
[tex]\[
y = 0.900491 \cdot 43 + 10.297416
\][/tex]
4. Calculate the Result:
Perform the multiplication and addition:
[tex]\[
y = 38.718113 + 10.297416 = 49.015529
\][/tex]
5. Round the Prediction:
Finally, round the result to two decimal places:
[tex]\[
y = 49.88
\][/tex]
So, if a student scored 43 on his first test, the predicted score on the second test is [tex]\( \boxed{49.88} \)[/tex].
