Purpose

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

Question 1

IQ scores are approximately normally distributed with: X ∼ N(μ=100,σ=15)

  1. What proportion of the population has an IQ greater than 65? Interpret the result in context in a complete sentence.
#Proportion of population with an IQ greater than 65

pnorm(q=65, mean=100, sd=15, lower.tail=FALSE)
## [1] 0.9901847

99% of the population has an IQ greater than 65.

  1. What IQ score represents the top 5% of the population? Explain in a sentence what this value means in plain language.
#IQ correlated with top 5% of the population

qnorm (p = 0.95, mean = 100, sd = 15)
## [1] 124.6728

An IQ score of about 124.7 represents the top 5% of the population, or, the 95th percentile. This means that someone with an IQ score of 124.7 has scored higher than 95% of test takers.

Question 2

Recall our definition: A value is considered unusual if it lies more than two standard deviations from the mean.

  1. Find the IQ values that mark the lower and upper bounds of the “usual” range.
#IQ values that mark lower and upper bounds of "usual" range

100-30
## [1] 70
100+30
## [1] 130
  1. What proportion of the population falls outside this range?
#Proportion of population outside of lower and upper bounds

pnorm(q= 70, mean=100, sd=15) + pnorm(q=130, mean=100, sd=15, lower.tail=FALSE)
## [1] 0.04550026

Question 3

Two students took different standardized tests.

Alex took the SAT and scored 1650. Taylor took the ACT and scored 27.

Assume the distributions:
SAT∼N(1500,300) ACT∼N(21,5)

  1. Compute the z-score for each student.
#Alex

(1650-1500)/300
## [1] 0.5
#Taylor

(27-21)/5
## [1] 1.2
  1. Which student performed better relative to other test-takers?

Taylor performed better relative to other ACT test takers because her Z score is higher.

  1. Explain why comparing the raw scores alone would be misleading.

Comparing raw scales alone would be misleading because the SAT and the ACT have different grading scales. This means they also have different means and standard deviations. Although it looks like Alex has a higher score, Taylor has a better score relative to other ACT test takers because her Z score is higher.

Question 4

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

  1. How many questions do you expect to answer correctly on average?
#Average correct answers

15*1/5
## [1] 3
  1. What is the probability that you get every question correct?
#Probability every question is correct

dbinom(x=15, size=15, prob=1/5)
## [1] 3.2768e-11
  1. What is the probability that you get every question incorrect?
#Probability every question is incorrect

dbinom(x=0, size=15, prob=1/5)
## [1] 0.03518437
  1. What is the probability of getting exactly 10 questions correct?
#Probability exactly 10 questions are correct

dbinom(x=10, size=15, prob=1/5)
## [1] 0.000100764
  1. What is the probability of getting 10 or more correct answers?
#Probability of 10 or more questions being correct

pbinom(q=9, size=15, prob=1/5, lower.tail = FALSE)
## [1] 0.0001132257
  1. Suppose a student claims they guessed randomly but got 10 out of 15 correct. Based on your probability above, do you believe this claim? Explain your reasoning. (There is no single correct answer, but your reasoning must use the probability you calculated.)

No, I do not believe this claim. The probability of that happening is extremely small, about 0.0001. This makes it extremely unlikely the student was randomly guessing on the quiz.

  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?
#Scoring 12 or more correct

pbinom(q=11, size=15, prob=1/5, lower.tail = FALSE)
## [1] 1.011253e-06

Question 5

A company schedules 10 employees for a shift. Each employee independently shows up with probability: p = 0.85

Let X = number of employees who show up

The company needs at least 8 workers to operate normally.

  1. What is the probability that fewer than 8 employees show up?
#Probability that less than 8 employees show up

pbinom(q=7, size=10, prob=0.85)
## [1] 0.1798035
  1. What is the probability the company has enough workers for this shift?
#Probability of enough workers showing up

pbinom(q=7, size=10, prob=0.85, lower.tail = FALSE)
## [1] 0.8201965
  1. Explain what this probability means in the context of scheduling workers.

There is an 82% chance the company has enough workers staffed and an 18% chance they do not. The probability of having enough workers is higher, but there’s still a significant chance they may not.

  1. Management wants at least a 95% chance of having enough workers. Should they schedule more than 10 employees? Explain your reasoning.

Yes, management should schedule more than 10 employees to increase their chance of having enough workers. With each employee there is a 85% chance they’ll show up, adding more employees increases the probability of having enough workers each shift.

Question 6

ACT scores are approximately normally distributed where: X ∼ N(21,5) a. Use R to simulate 10,000 ACT scores.

# 10,000 ACT score simulation

ACT <- rnorm(n = 10000, mean = 21, sd = 5)
  1. Find what percent of your simulated ACT scores were above 30
#ACT scores above 30

prop.table(table(ACT>30))
## 
##  FALSE   TRUE 
## 0.9636 0.0364
  1. Now compute the theoretical probability of getting an ACT score above 30 using pnorm().
#Theoretical ACT scores above 30

pnorm(q = 30, mean = 21, sd = 5, lower.tail = FALSE)
## [1] 0.03593032
  1. Compare the two values. Why are they similar but not identical?

The two values are similar because they’re both normally distributing the same situation. They’re not the same because the simulation pulls from a random sample of ACT scores so there can be slight differences.

Question 7

Create your own real-world situation that could be modeled using either a binomial distribution or a normal distribution.

Your problem must include: * A description of the situation * Identification of reasonable parameters (mean, sd OR n, p) * One probability calculation in R * A written interpretation of the result

Examples might include: * basketball free throws * weather events * exam scores * products being defective

The Uniform Bar Exam is a test taken before you can practice law. UBE scores are approximately normally distributed with: X ∼ N(279,33)

What proportion of test takers will score a 300 or higher?

#Proportion of test takers with a 300 or higher

pnorm(q = 300, mean = 279, sd = 33, lower.tail = FALSE)
## [1] 0.2622697

About 26% of test takers should score a 300 or higher on the UBE.