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)
# Probability that a randomly selected person has an IQ score greater than 65
1-pnorm(65 , mean= 100, sd = 15)
## [1] 0.9901847

The probability of selecting a person whith IQ < 65 is 99%

  1. P(x < 150)
# Probability that a randomly selected person has an IQ score less than 150
pnorm(150 , mean = 100, sd = 15 )
## [1] 0.9995709

The probability of selecting person with IQ < 150 is 99%

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?
# Found minimum qualifying IQ.
qnorm(0.95, mean = 100,sd= 15)
## [1] 124.6728

the minimum qualifying IQ is 125

  1. If one person is randomly selected, what is the probability that their IQ score is greater than 125?
# Probability of one person's IQ > 125 
1-pnorm(125, 100, 15)
## [1] 0.04779035

The probability of selecting one person with IQ greater than 125 is 4%

Question 3

  1. Still using the mean and standard deviation from question 1, what is the z-score for an IQ of 140? \[z = \frac{(x - \mu)}{\sigma} \]
# Find the Z-score for an IQ of 140 with mu = 100, sd = 15
(140 - 100)/ 15
## [1] 2.666667

The z-score for an IQ of 140 with mu = 100 and sd = 15 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?

Yes, IQ af 140 is unusual because the standard deviation is 2.67 which is more then two.

  1. What is the probability of getting an IQ greater than 140?
# probability of IQ > 140.
1-pnorm(140, 100, 15)
## [1] 0.003830381

The probability of getting an IQ greater than 140 is 0%

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?
# Expected average (mean)
0.2*15
## [1] 3

The expected average is three correct anwers.

  1. What is the probability that you get every question correct?
# Getting 15/15 correct.
dbinom (15, 15, 0.2)
## [1] 3.2768e-11

The probability of getting 100% correct is 0.0000000000327

  1. What is the probability that you get every question incorrect?
# Getting 0/15 correct.
dbinom(0 , 15, 0.2)
## [1] 0.03518437

The propability of getting everything incorrect is 3%

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%?
# Getting exactly 60% correct.
(0.6 * 15)
## [1] 9

To get 60% 0f the test correct. there will have to be 9 answers right.

  1. If a grade of 60% or lower is considered failing, then what is the probability of you failing?
# Probability of getting 60% correct or lower 
pbinom(9, 15, .2)
## [1] 0.9998868

To have a test grade of 60% or lower, the propability of failing this test is 99%.

  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?
# Probability of passing the quiz with 80% or more,ie x = 12 or more correct.
1- pbinom(q = 11,size =  15,prob = .2)
## [1] 1.011253e-06

the probability of maintaining a passing grade with 80% and more is 0.00000101%

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?
# probability of getting all five employees to come to work.
dbinom( 5, 5, .8)
## [1] 0.32768

the probability to have at least 5 employees comeing to work, from 5 employees scheduled is 33%

  1. Suppose you schedule 7 employees, what is the probability that at least 5 come to work?
# probability of gettig at least 5 employees to come to work. 
1-pbinom( 4, 7 , .8)
## [1] 0.851968

the probability to have at least 5 employees comeing to work, from 7 employees scheduled is 85%.

  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.
# Employees schedule, 99% confident Gtting 5 servers at least.  
1-pbinom(4, 10, .99)
## [1] 1
1-pbinom(4 , 9, .99)
## [1] 1
1-pbinom(4, 8, .8)
## [1] 0.9437184
1-pbinom(4, 9 , .8)
## [1] 0.9804186
1-pbinom(4, 10, .8)
## [1] 0.9936306

We have to schedule 10 employees in order to be 99% cofident that at least 5 people show up.