Question

(1 point) The professor of a introductory calculus class has stated that, historically, the distribution of final test grades in the course resemble a Normal distribution with a mean final test mark of μ=62μ=62% and a standard deviation of σ=11σ=11%.

If using/finding zz-values, use three decimals.

(a) What is the probability that a random chosen final test mark in this course will be at least 73%? Answer to four decimals.

(b) In order to pass this course, a student must have a final test mark of at least 50%. What proportion of students will not pass the calculus final test? Use four decimals in your answer.

(c) The top 6% of students writing the final test will receive a letter grade of at least an A in the course. To two decimal places, find the minimum final test mark needed on the calculus final to earn a letter grade of at least an A in the course.

(d) Suppose this professor randomly picked 25 final tests, observing the earned mark on each. What is the probability that 6 of these have a final test grade of less than 50%? Use four decimals in your answer.

Answer 1

Suppose, random variable X denotes final test grade in introductory calculus class.

(a)

Required probability is given by

[Using R-code '1-pnorm(1)']

(b)

Required probability is given by

     [Using R-code 'pnorm(-1.090909)']

(c)

We know,

   [Using R-code 'qnorm(1-0.06)']

Hence, minimum 79.10 final test mark needed to earn letter grade.

(d)

We can model the given situation using Binomial distribution as follows.
Suppose, random variable Y denotes number of students with final test grade less than 50%.
Final test grade of a student is independent of final test grades of other students.
We define getting final test grade less than 50% as success.

Required probability is given by

[Using R-code 'dbinom(6,25,0.1376565)']