Question

Suppose the scores of a certain high school diploma test follow a normal distribution in the population with a mean of 195 and standard deviation of 30.

1. About ______ percent of the students have a score between 135 and 195.

2. About ______ percent of the students have a score between 225 and 255.

3. The middle 95% of the students have a score between ________  and ________   .

4. Recently class A just had a Math exam, but class B had a Verbal exam.

- Joe in class A has a math score of 160, and all the math scores in class A have a mean of 140 and a standard deviation of 10.

- Eric in class B has a verbal score of 80, and all the verbal scores in class B have a mean of 50 and a standard deviation of 12.

Let’s assume students in classes A and B have very similar academic background, and both classes are hugh classes with lots of students. Then roughly speaking, relative to their respective classmates, who did better in the recent exam, Joe or Eric?

(A) Joe’s math score 160 is better

(B) Eric’s verbal score 80 is better

(C) They are about the same

(D) We also need the variance of the two data sets to compare Joe’s and Eric’s scores

5. A sample consists of 26 scores. What is the degrees of freedom for the sample standard deviation?​

Answer 1

(1) Mean is 195 and standard deviation is 30

Using empirical formula, we know that 47.5% area lies to the -2 standard deviation of the mean

we have to find the % of students who have score between 135 and 195

So, % of students between -2 standard deviation of mean and mean = 47.5 + 0 = 47.5%

(2) 225 is one standard deviation above mean and 255 is two standard deviation above mean

or we can write 195 + (1*30) = 225 and 195+(2*30) = 255

Using empirical rule, % of area between and is 13.5%

So, answer is 13.5%

(3) Middle 95% of the students will fall within 2 standard deviation above and below the mean

so, we have

and

So, middle 95% fall within 135 and 255 using empirical rule

(4) Joe math score is 2 standard deviation more than the mean score whereas the Eric math score is greater than 2 standard deviation by the mean math score.

Joe: 140 + 2*10 = 160

Eric: 50 +2*12 = 74 but given score is 80, so it is higher

So, Eric maths score is higher as compared to Joe, OPTION B is correct answer.

(5) We know that degree of freedom = n-1

where n = 26

So, required degree of freedom = 26-1 = 25