In a statistic class, the average score of midterm is 75 and the SD is 8. The average score of final is 65 and the SD is 12. The scatter diagram of the two variables (midterm/final scores) is football shaped. The correlation between the two variables is 0.65. The regression line for estimating final based on midterm is drawn through the scatter.
A student who got 70 in the midterm is estimated to get X1 in the final, and the SD of this estimation is Y1.
What is X1? Y1?
To calculate X1 (the estimated final score for a student who got 70 in the midterm), we can use the equation of the regression line.
The equation of a regression line is given by:
y = a + bx
Where:
y is the estimated final score
a is the y-intercept
b is the slope of the line
x is the midterm score
In this case, the y-intercept was not provided. However, we can calculate it using the average scores of the midterm and final:
y = a + bx
65 = a + 0.65 * 75
Solving for a:
a = 65 - 0.65 * 75
a = 65 - 48.75
a = 16.25
So the equation of the regression line is:
y = 16.25 + 0.65x
To calculate X1, we substitute x = 70 into the equation:
X1 = 16.25 + 0.65 * 70
X1 = 16.25 + 45.5
X1 = 61.75
Therefore, the estimated final score for a student who got 70 in the midterm is approximately 61.75.
To calculate Y1 (the standard deviation of this estimation), we use the formula for the standard error of estimate:
S.E. = sqrt((1 - r^2) * MSE)
Where:
r is the correlation coefficient (0.65)
MSE is the mean squared error, which is equal to the variance of the residuals
The variance of the residuals is given by:
Var(residuals) = (SD of final scores)^2 * (1 - r^2)
Let's calculate Y1 step by step:
Var(residuals) = (12)^2 * (1 - (0.65)^2)
Var(residuals) = 144 * (1 - 0.4225)
Var(residuals) = 144 * 0.5775
Var(residuals) = 83.16
Now, calculate the standard error:
S.E. = sqrt((1 - (0.65)^2) * 83.16)
S.E. = sqrt(0.5775 * 83.16)
S.E. = sqrt(47.94)
S.E. ≈ 6.93
Thus, the standard deviation (Y1) of the estimation for a student who scored 70 in the midterm is approximately 6.93.
To find X1, which is the estimated final score for a student who got 70 on the midterm, we can use the regression line that was drawn through the scatter plot. The regression line allows us to estimate the value of one variable (final score) based on the other variable (midterm score).
The correlation between the midterm and final scores is 0.65. This means that there is a positive relationship between the two variables, and as the midterm score increases, the final score is also likely to increase. The regression line takes into account this relationship and provides an estimate for the final score based on the given midterm score.
To calculate X1, we need to use the equation of the regression line, which can be represented as:
Final score = Intercept + (Regression coefficient * Midterm score)
The coefficient represents how much the final score is expected to change for each unit increase in the midterm score. In this case, the regression line was drawn through the scatter plot, so the coefficient is already determined.
Since you haven't provided the specific values for the intercept and regression coefficient, I can't provide the exact calculation for X1. However, you can use the given information to estimate it. For a student who got a 70 on the midterm, find the corresponding value on the regression line to get X1, the estimated final score.
Now, let's move on to Y1, which represents the standard deviation of this estimation. The standard deviation of this estimation is a measure of how spread out the estimated final scores are from the regression line. It tells us how much variability there is in the estimates.
To calculate Y1, we need to use the equation for the standard deviation of the estimate, which can be represented as:
SD of estimate = SD of Y * sqrt(1/n + (X1 - Xbar)^2 / ∑(Xi - Xbar)^2)
In this equation, SD of Y refers to the standard deviation of the final scores, and Xbar represents the mean of the midterm scores. n refers to the sample size.
Again, without the actual values for the standard deviations and mean, we can't provide the exact calculation for Y1. However, you can plug in the given values to estimate it.