Scores on the 1995 SAT verbal aptitude test among Kentucky high school seniors were normally distributed with an average of 420 and the SD of 80.
Scores on the 1995 SAT quantitative aptitude test among Kentucky high school seniors were normally distributed with an average of 440 and SD of 60.
The scatter diagram is football shaped, and the correlation coefficient is r=0.8
For those students who scored 500 on the verbal aptitude test, estimate the average quantitative aptitude test score.
I know that the answer for this question is 488 but I don't know what formula to use and how to set it up.
Well, according to the given information, we know that the verbal aptitude test scores follow a normal distribution with a mean of 420 and a standard deviation of 80. Similarly, the quantitative aptitude test scores also follow a normal distribution with a mean of 440 and a standard deviation of 60.
To estimate the average quantitative aptitude test score for the students who scored 500 on the verbal aptitude test, we can use the concept of the correlation coefficient.
The formula to estimate the average quantitative aptitude test score can be obtained using the equation of the regression line:
Y = a + bx
Where Y is the dependent variable (quantitative aptitude test score), x is the independent variable (verbal aptitude test score), a is the intercept, and b is the slope.
Since r = 0.8, we can calculate the slope (b) using the formula:
b = r * (SDy/SDx)
Given that SDy (standard deviation of quantitative aptitude test scores) is 60 and SDx (standard deviation of verbal aptitude test scores) is 80 and r = 0.8, we can substitute these values into the formula:
b = 0.8 * (60/80) = 0.6
To find the intercept (a), we can use the formula:
a = mean of Y - b * mean of X
Mean of Y (quantitative aptitude test scores) is 440, mean of X (verbal aptitude test scores) is 420, and b is 0.6. Substituting these values into the formula, we get:
a = 440 - (0.6 * 420) = 440 - 252 = 188
Now, we have the values of a and b, so we can substitute them into the equation Y = a + bx:
Y = 188 + 0.6x
Since we want to estimate the average quantitative aptitude test score for those who scored 500 on the verbal aptitude test, we can substitute x = 500 into the equation:
Y = 188 + 0.6(500) = 188 + 300 = 488
Therefore, the estimated average quantitative aptitude test score for students who scored 500 on the verbal aptitude test is 488.
To find the average quantitative aptitude test score for students who scored 500 on the verbal aptitude test, you can use the concept of regression.
Regression is a statistical measure used to estimate the relationship between two variables. In this case, we want to estimate the relationship between the verbal aptitude test scores and quantitative aptitude test scores.
The formula to estimate the average quantitative aptitude test score based on the verbal aptitude test score is:
ŷ = b₁x + b₀
Where:
- ŷ is the estimated quantitative aptitude test score
- b₁ is the slope of the regression line
- x is the verbal aptitude test score
- b₀ is the y-intercept of the regression line
To find these values, we need to calculate the regression equation. The slope of the regression line (b₁) can be found using the following formula:
b₁ = r * (SDy / SDx)
Where:
- r is the correlation coefficient between the two variables (0.8)
- SDy is the standard deviation of the quantitative aptitude test scores (60)
- SDx is the standard deviation of the verbal aptitude test scores (80)
Let's calculate the slope (b₁):
b₁ = 0.8 * (60 / 80) = 0.6
Now, we can calculate the y-intercept (b₀) using the formula:
b₀ = ȳ - b₁ * x̄
Where:
- ȳ is the average quantitative aptitude test score (440)
- x̄ is the average verbal aptitude test score (420)
Let's calculate the y-intercept (b₀):
b₀ = 440 - 0.6 * 420 = 440 - 252 = 188
Now that we have the slope (b₁) and the y-intercept (b₀), we can estimate the average quantitative aptitude test score (ŷ) for students who scored 500 on the verbal aptitude test by substituting x = 500 into the regression equation:
ŷ = 0.6 * 500 + 188
ŷ = 300 + 188
ŷ = 488
Therefore, the estimated average quantitative aptitude test score for students who scored 500 on the verbal aptitude test is 488.
To estimate the average quantitative aptitude test score for students who scored 500 on the verbal aptitude test, you can use the concept of regression. Given the scatter diagram is football-shaped and the correlation coefficient (r) is given as 0.8, we can assume a linear relationship between the two variables.
The formula to calculate the estimated value (Y) of one variable (quantitative aptitude test score) based on the known value (X) of the other variable (verbal aptitude test score) is:
Y = a + bX
In this case, Y represents the quantitative aptitude test score, X represents the verbal aptitude test score, a represents the y-intercept, and b represents the slope of the regression line.
To find the values of a and b, we need additional information. However, since the formula you mentioned already provided the answer (488), it implies that the slope (b) in this case represents the conversion factor from the verbal aptitude test score to the quantitative aptitude test score.
So, you need to divide the average difference between the two tests (20) by the regression coefficient (0.8) to obtain the slope:
Slope (b) = 20 / 0.8 = 25
Now that you have the slope (b), you can substitute it into the regression formula to estimate the average quantitative aptitude test score (Y) when the verbal aptitude test score (X) is 500:
Y = a + bX
Y = a + 25(500)
Y = a + 12,500
Since the average quantitative aptitude test score is 440, you can solve for the y-intercept (a):
440 = a + 12,500
a = 440 - 12,500
a = -12,060
Now you have the values of a and b:
a = -12,060
b = 25
Substituting these values into the regression formula, you can estimate the average quantitative aptitude test score for students who scored 500 on the verbal aptitude test:
Y = a + bX
Y = -12,060 + 25(500)
Y = -12,060 + 12,500
Y = 440
Therefore, the estimated average quantitative aptitude test score for students who scored 500 on the verbal aptitude test is 440.