Question

A statistical analysis is made of the midterm and final scores in a large course, with the following results:
Average midterm score = 65, SD = 10,
Average final score = 65, SD = 12, r = 0.6
The scatter diagram is football shaped.
a. About what percentage of the class final scores above 70?
b. A student midterm was 75. Predict his final score
c. Suppose the percentile rank of midterm score was 95%, predict his percentile rank on the final score
d. Of those whose midterm score was 70, about what percentage of final scores over 80?

Answer 1

Let X shows the midterm score and Y shows the final score.

(a)

The z-score for Y = 70 is

The percentage of the class final scores above 70 is

P(Y > 70) = P(z > 0.42) = 1 - P(z <= 0.42) = 1 - 0.6628 = 0.3372

(b)

Here we have

Slope of the regression equation is

Y intercept of equation is

Equation of regression line is

The predicted value for x = 75 is

(c)

First we need z-score that has 0.95 area to its left. z-score 1.645 has 0.95 area to its left. So z score corresponding to percentile rank of midterm is

So z-score for percentile rank of final score is

So percentile rank for final score is equal to area left to z-score 0.987. So required percentile rank is

P(z < 0.987) = 0.8382 = 83.82%

(d)

Here conditional distribution of Y given X=70 follows normal distribution with mean

and standard deviation:

So z-score for Y = 80 | X=70 is

So required probability is

P(Y > 80 | X = 70) = P(z > 1.1875) = 1 - P(z <= 1.1875) = 0.1175

Answer: 11.75%