Purpose

This project will demonstrate your understanding of the normal and binomial probability distributions in R and RStudio.

Question 1

Assume IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. If a person is randomly selected, find each of the requested probabilities. Here, x, denotes the IQ of the randomly selected person.

  1. P(x > 65)
pnorm( 65, mean = 100, sd = 15, lower.tail = FALSE)
## [1] 0.9901847
# There is a 0.99 probality that a randomly selected IQ score is higher than 65.
  1. P(x < 150)
pnorm( 150, 100, 15, lower.tail = TRUE)
## [1] 0.9995709
# There is a 0.999 chance that a randomly selected IQ score is lower than 150.

,

Question 2

Assume the same mean and standard deviation of IQ scores that was described in question 2.

  1. A high school offers a special program for gifted students. In order to qualify, students must have IQ scores in the top 5%. What is the minimum qualifying IQ?
qnorm( 0.95, 100, 15, lower.tail = TRUE)
## [1] 124.6728
# An IQ score of 125 is the minimum needed to be in the top 5%.
  1. If one person is randomly selected, what is the probability that their IQ score is greater than 125?
pnorm(125, 100, 15, lower.tail = FALSE )
## [1] 0.04779035
# There is a 0.047 probabiltiythat a randomly selected IQ is greater than 125.

Question 3

  1. Still using the mean and standard deviation from question 1, what is the z-score for an IQ of 140?
(140 - 100)/ 15
## [1] 2.666667
# The z-score for an IQ of 140 is 2.67.
  1. We mentioned in week 6 that a data value is considered “unusual” if it lies more than two standard deviations from the mean. Is an IQ of 140 considered unusual?
# An IQ of 140 is considered unusual, because it has a z-score of 2.67. Which is more than two standard deviations away from the mean.
  1. What is the probability of getting an IQ greater than 140?
pnorm( 140, 100, 15, lower.tail = FALSE)
## [1] 0.003830381
# There is a 0.0038 probability that one have an IQ greater than 140.

Question 4

You are taking a 15-question multiple choice quiz and each question has 5 options (a,b,c,d,e) and you randomly guess every question.

  1. How many questions do you expect to answer correct on average?
# mean = nxp

15*0.20
## [1] 3
# One would expect to get 3 questions right if one guessed on all of them.
  1. What is the probability that you get every question correct?
dbinom(15, 15, 0.03)
## [1] 1.434891e-23
# There is a 1.434891e^-23 chance that by guessing one will get every question right.
  1. What is the probability that you get every question incorrect?
dbinom(0, 15, 0.20)
## [1] 0.03518437
# There is a 0.035 probalitly of getting every question wrong.

Question 5

Consider still the 15-question multiple choice quiz that each question has 5 options (a,b,c,d,e) and you randomly guess every question.

  1. How many questions does one need to answer correctly in order score exactly a 60%?
#(60/100) = (x/15)
(60/100)
## [1] 0.6
# 0.6= x/15
0.6*15
## [1] 9
#9=x

# One needs to get six answers correct to recieve a sixty percent.
  1. If a grade of 60% or lower is considered failing, then what is the probability of you failing?
pbinom(9, 15, 0.2)
## [1] 0.9998868
#There is a 0.9998 probability of one failing while guessing on every question.
  1. If you need a grade of 80% or higher on this quiz to maintain a passing grade, what is the probability of you maintaining that passing grade?
pbinom(12,15,0.2)
## [1] 0.9999999
1- pbinom(12,15,0.2)
## [1] 5.704909e-08
# There is a 5.704909e-08 probablity that one will keep their grade at a passing rate.

Question 6

Suppose you own a catering company. You hire local college students as servers. Not being the most reliable employees, there is an 80% chance that any one server will actually show up for a scheduled event. For a wedding scheduled on Saturday, you need at least 5 servers.

  1. Suppose you schedule 5 employees, what is the probability that all 5 come to work?
dbinom(5,5,0.8)
## [1] 0.32768
# There is a 0.33 probability that five employees show up.
  1. Suppose you schedule 7 employees, what is the probability that at least 5 come to work?
pbinom(4, 7, 0.8, lower.tail = FALSE)
## [1] 0.851968
# If eight workers are schedueled, then there is a 0.85 probability that five workers show up.
  1. It is really important that you have at least 5 servers show up! How many employees should you schedule in order to be 99% confident that at least 5 show up? Hint: there is no single formula for the answer here, perhaps use some kind of trial and error method, but show all work in an r chunk below.
pbinom(4, 9, 0.8, lower.tail = FALSE)
## [1] 0.9804186
pbinom(4, 10, 0.8, lower.tail = FALSE)
## [1] 0.9936306
# One needs to scheduele ten employees to have a 99% chance that five will show up.