Question

North Carolina State University posts the grade distribution for its courses online. Students in one section of English 210 in the Fall 2008 semester received 33% A's, 24% B's, 18% C's, 16% D's, and 9% F's.

A.) Using A=4, B=3, C=2, D=1, and F=0, take X to be the grade of a randomly chose English 210 student. Use the definition of mean, and standard deviation for discrete random variables to find the mean and standard deviation of grades in this course.

B.) English 210 is a large course. We can take the grade of an SRS of 50 students to be independent of each other. If x̅ is the average of these 50 grades, what are the mean and standard deviation of x?

C.) What is the probability P(X≥3) that a randomly chosen English 210 students gets a B or better? What is the approximate probability P(x̅ ≥3) that the grade point average for 50 randomly chosen English 210 students is a B or better?

D.) Give the 95% confidence interval of μ

Answer 1

Let X be the grade of a randomly chosen English 210 student. X can take the values 4,3,2,1, 0 (Corresponding to the letter grades A,B,C,D,E,F)

We also know that  in the Fall 2008 semester received 33% A's, 24% B's, 18% C's, 16% D's, and 9% F's.

The probability mass function of X is

x	P(x)
0	0.09
1	0.16
2	0.18
3	0.24
4	0.33

a) The mean of X is

The expected value of is

the variance of X is

the standard deviation of X is

ans: the mean of grades in this course is and standard deviation of grades in this course is  

b) Let be the average grade of a 50 randomly selected English 210 students. Using the central limit theorem we know that is approximately normally distributed if the sample size n>30.

Here since n=50, we can say that has normal distribution with mean

and standard deviation (or standard error of mean)

ans: If is the average of these 50 grades, the mean of is and standard deviation of is

c) the probability P(X≥3) that a randomly chosen English 210 students gets a B or better is

ans: the probability P(X≥3) that a randomly chosen English 210 students gets a B or better is 0.57

The approximate probability P(x̅ ≥3) that the grade point average for 50 randomly chosen English 210 students is a B or better is

ans:

The approximate probability P(x̅ ≥3) that the grade point average for 50 randomly chosen English 210 students is a B or better is 0.0096

D) The 95% confidence interval implies that the level of significance is

The critical value of Z is calculated using

This is same as

Now using the standard normal tables we can get that for z=1.96, P(Z<1.96) = 0.5+0.4750=0.9750

Hence the critical value is

the 95% confidence interval of μ is

ans: the 95% confidence interval of μ is [2.19,2.93]