the correlation between two tests given to 100 students in the junior clas is .70 the mean for the first test was 125 with standard deviation 20 the mean of the second test is 80 with standard devation 10 a student took the first test and scored 100. what is your best estimate of this student's score on the second test?
To estimate the student's score on the second test, we can use the concept of regression.
Regression is a statistical measure used to establish a relationship between two variables. In this case, we want to determine how the second test scores are related to the first test scores.
The given information tells us that there is a correlation coefficient of 0.70 between the two tests. The correlation coefficient quantifies the strength and direction of the linear relationship between two variables. In this case, a correlation coefficient of 0.70 indicates a moderately positive linear relationship between the first and second test scores.
To estimate the student's score on the second test, we will use the equation of the regression line, y = a + bx. In this equation, "y" represents the second test score, "x" represents the first test score, "a" represents the y-intercept, and "b" represents the slope of the regression line.
First, we need to calculate the values of "a" and "b" using the given information:
Step 1: Calculate the mean and standard deviation for the first and second tests:
- Mean of the first test (x̄₁) = 125
- Standard deviation of the first test (σ₁) = 20
- Mean of the second test (x̄₂) = 80
- Standard deviation of the second test (σ₂) = 10
Step 2: Calculate the slope (b):
b = (correlation coefficient * σ₂) / σ₁
= (0.70 * 10) / 20
= 0.35
Step 3: Calculate the y-intercept (a):
a = x̄₂ - (b * x̄₁)
= 80 - (0.35 * 125)
= 80 - 43.75
= 36.25
Now that we have estimated the values of "a" and "b," we can substitute the first test score (x = 100) into the regression equation to find the estimate of the second test score (y).
y = a + bx
= 36.25 + (0.35 * 100)
= 36.25 + 35
= 71.25
Therefore, the best estimate of the student's score on the second test is 71.25.