Purpose

In this project, students will demonstrate their understanding of probability and the normal and binomial distributions.

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 when x is more than 65
pnorm(65, mean = 100, sd = 15, lower.tail = FALSE)
## [1] 0.9901847

When x is more than 65 the probability is 99.02%

  1. P(x < 150)
# Probability when x is less than 150
pnorm(150, mean = 100, sd = 15)
## [1] 0.9995709

When x is less than 150 the probability is 99.96%

Question 2

Assume the same mean and standard deviation of IQ scores that was described in question 1.

  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?
# The minimum qualifying IQ
qnorm(0.05, mean = 100, sd = 15, lower.tail = FALSE)
## [1] 124.6728

The minimum qualifying IQ is approximately 124.67

  1. If one person is randomly selected, what is the probability that their IQ score is greater than 110?
# Probability of having an IQ score greater than 110
pnorm(110, mean = 100, sd = 15, lower.tail = FALSE)
## [1] 0.2524925

When randomly selected the probability of finding someone with an IQ score greater than 110 is 25.25%

Question 3

  1. Still using the mean and standard deviation from question 1, what is the z-score for an IQ of 140?
# z-score of an IQ of 140
(140 - 100)/15
## [1] 2.666667

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?

Yes. With a z-score of 2.67, an IQ of 140 is considered unusual.

  1. What is the probability of getting an IQ greater than 140?
# Probability of getting an IQ greater than 140
pnorm(140, mean = 100, sd = 15, lower.tail = FALSE)
## [1] 0.003830381

The probability of getting and IQ greater than 140 is 0.38%

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 correctly on average?
# Expected average
0.2*15
## [1] 3

The expected average is 3.

  1. What is the probability that you get every question correct?
# Probability of getting every question correctly
dbinom(x = 15, size = 15, prob = 0.2)
## [1] 3.2768e-11
  1. What is the probability that you get every question incorrect?
# Probability of getting every question incorrectly
dbinom(x = 0, size = 15, prob = 0.2)
## [1] 0.03518437

The probability of getting all the questions incorrectly is 3.52%

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%?
# Expected correct answers for a score of exactly 60%
0.6*15
## [1] 9

In order to score exactly 60% someone need to get 9 questions right.

  1. If a grade of 60% or lower is considered failing, then what is the probability of you failing?
# Probabiity of failing
pbinom(q = 9, size = 15, prob = 0.2)
## [1] 0.9998868

For a grade of 60% or lower, the probability of failing is 99.99%

  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 maintaining a passing grade
pbinom(q = 11, size = 15, prob = 0.2, lower.tail = FALSE)
## [1] 1.011253e-06

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(x = 5, size = 5, prob = 0.8)
## [1] 0.32768

The probability of all 5 employees coming to work is 32.77%

  1. Suppose you schedule 7 employees, what is the probability that at least 5 come to work?
pbinom(q = 4, size = 7, prob = 0.8, lower.tail = FALSE)
## [1] 0.851968

If we schedule 7 employees, the probability of at least 5 employees coming to work is 85.20%

  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, so maybe use some kind of trial and error method.
# find the number of employees that should be schedule to work to have 5 of them coming by calcuting with different sizes
pbinom(q = 4, size = 8, prob = 0.8, lower.tail = FALSE)
## [1] 0.9437184
pbinom(q = 4, size = 9, prob = 0.8, lower.tail = FALSE)
## [1] 0.9804186
pbinom(q = 4, size = 10, prob = 0.8, lower.tail = FALSE)
## [1] 0.9936306

10 employees should be scheudled to work if we want 5 emplpoyees to work.

Question 7

  1. Generate a random sample of 10,000 numbers from a normal distribution with mean of 51 and standard deviation of 7. Store that data in object called rand_nums.
# Generate a random sample and store it into an object called rand_nums
rand_nums <- rnorm(n = 10000, mean = 51, sd = 7)
  1. Create a histogram of that random sample.
# Histogram of rand_nums
hist(rand_nums)

Question 8

  1. How many values in your rand_nums vector are below 40?
# Number of values below 40 
table(rand_nums < 40) 
## 
## FALSE  TRUE 
##  9440   560

614 values in rand_nums is below 40

  1. For a theoretical normal distribution, how many of those 10,000 values would you expect to be below 40?
# Expected number of values below 40
pnorm(q = 40, mean = 51, sd = 7)
## [1] 0.05804157
pnorm(q = 40, mean = 51, sd = 7) * 10000
## [1] 580.4157
  1. Is your answer in part a reasonably close to your answer in part b? Yes. 580.42 is not that far from 614