Probability question

Question

Past records show that at a given college 20% of the students who began as psychology majors either changed their major or dropped out the school. An incoming class has 110 beginning psychology majors. What is the probability that as many as 30 of these students leave the psychology program?

Answer 1

let x be number of students

want to find P(x >= 30)

use normal approximation to binomial distribution.

P(x >= 30) = P(x > 29.5) using continuity correction

normalized z-stat: z = (x - mean)/SD

mean = 110 * 0.2 = 22

standard deviation (SD) = sqrt(110 * 0.2 * 0.8) = 4.2

z = (x - 22)/4.2

apply normalized z stat to limit of inequality P(x > 29.5):

P(x > 29.5) = P(z > (29.5 - 22)/4.2) = P(z > 1.79) = P(z < -1.79) = 0.0367

Answer 2

Binary Distribution hits again!

The college is stating that as a class, students entering as psychology majors leave the psychology program either by transfer to another major or dropping out of school at an independent rate p of 20% (0.2).

Since the drops are independent and the rate is constant the distribution of drops follow a binomial distribution.

Calculating the odds that 30 or less students will drop (out of this cohort of 110) follows the cumulative distribution function for p = .2 ,n = 110 and k (max number of drops) = 30.

P(x<=30) = .975 (rounded to 3 significant digits)

Further research: For those interested the following link brings you to a good article on the binomial distributions and appropriate formulae for the mdf and the cdf.

https://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm

A really detailed step by step description of how to do this can be found at the following site

http://statisticshelper.com/binomial-probability-calculator#answer

(note - select "at most X successes").