Project #3 - Normal and Binomial Distributions

MAT143H - Introduction to Statistics Honors

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(mu=100,sigma=15)

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

#Proportion of the population with IQ greater than 65
pnorm(65, mean = 100, sd = 15, lower.tail = FALSE)

## [1] 0.9901847

The proportion of the population with an IQ greater than 65 is approximately 0.99018. This means about 99.02% of people have an IQ above 65.

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

# IQ score for top 5% of the population 
qnorm(.95, mean = 100, sd = 15)

## [1] 124.6728

The IQ score that represents the top 5% of the population is approximately 124.67. This means a person must score about 125 or higher to be in the top 5% of IQ scores.

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.

#Usual range 
lower <- 100 - (2*15)
upper <- 100 + (2*15)

lower

## [1] 70

upper

## [1] 130

The usual range is from 70 to 130.

2. What proportion of the population falls outside this range?

#Probability outside this range
pnorm(q= 70, mean = 100, sd = 15) + pnorm(q= 130, mean = 100, sd = 15, lower.tail = FALSE)

## [1] 0.04550026

The proportion of the population that falls outside of this range is about 4.55%.

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.

# Z-scores
zalex <- (1650 - 1500)/ 300
ztaylor <- (27 - 21)/ 5

zalex

## [1] 0.5

ztaylor

## [1] 1.2

2. Which student performed better relative to other test-takers? Taylor performed better relative to other test-takers because their Z-score was higher.

3. Explain why comparing the raw scores alone would be misleading. Comparing the raw scores would be misleading because the SAT and ACT have different scales of measurement. Using the Z-scores allows for a standardization of the scores, making them comparable.

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 answer 
15 * (1/5)

## [1] 3

The expected correct number is 3.

2. What is the probability that you get every question correct?

#Probability of getting every question correct 
dbinom(x = 15, size = 15, prob = 1/5)

## [1] 3.2768e-11

The probability of getting all 15 questions correct is small.

3. What is the probability that you get every question incorrect?

#Probability of getting every question incorrect 
dbinom(x= 0, size = 15, prob = 1/5)

## [1] 0.03518437

The probability of getting all 15 questions incorrect is small.

4. What is the probability of getting exactly 10 questions correct?

#Probability of getting exactly 10 questions correct 
dbinom(x = 10, size = 15, prob = 1/5)

## [1] 0.000100764

The probability of getting exactly 10 questions correct is low.

5. What is the probability of getting 10 or more correct answers?

#Probability of getting 10 or more questions correct 
pbinom(q = 9, size = 15, prob = 1/5, lower.tail = FALSE)

## [1] 0.0001132257

The probability of getting 10 or more questions correct is unlikely.

6. 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.)

Based on the probability above, getting 10 or more correct is extremely unlikely. Because of its unlikeness, I wouldn’t believe the student’s claim of guessing.

7. 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?

#Needs at least an 80%
pbinom(q = 11, size = 15, prob = 1/5, lower.tail = FALSE)

## [1] 1.011253e-06

The probability of maintaining a passing grade of at least 12 correct is low, meaning it’s unlikely to pass by randomly guessing.

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 fewer than 8 employees show up
pbinom(q= 7, size = 10, prob = 0.85)

## [1] 0.1798035

2. What is the probability the company has enough workers for this shift?

#Probability that at least 8 employees show up 
pbinom(q= 7, size = 10, prob = 0.85, lower.tail = FALSE)

## [1] 0.8201965

3. Explain what this probability means in the context of scheduling workers.

This probability represents the likelihood that at least 8 employees show up, meaning the company can operate normally.

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

Since the probability of having at least 8 workers is below 95%, the company should schedule more employees to increase the likelihood of having enough workers.

Question 6

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

#Simulate 10,000 ACT scores from a normal distribution 
set.seed(1)
scores <- rnorm(n=10000, mean = 21, sd = 5)

2. Find what percent of your simulated ACT scores were above 30

#Calculate the proportion of simulated scores greater than 30
mean(scores > 30)

## [1] 0.0375

3. Now compute the theoretical probability of getting an ACT score above 30 using pnorm().

#Calculate the theoretical probability of scoring above 30 
pnorm(q= 30, mean = 21, sd = 5, lower.tail = FALSE)

## [1] 0.03593032

4. Compare the two values. Why are they similar but not identical?

The values are similar because the simulation approximates the probability, but there are differences that occur due to randomness.

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

Example: Basketball Free Throws

A basketball player makes free throws with a probability of 0.8. The player takes 10 shots.

The Parameters n = 10, p = 0.8

What is the probability the basketball player makes exactly 8 shots?

#Probability of making exactly 8 out of 10 free throw shots 
dbinom(x = 8, size = 10, prob = 0.8)

## [1] 0.3019899

The probability that the basketball player will make exactly 8 out of 10 free throw shots is about 0.3019899. This means there is about a 30.2% chance the player makes exactly 8 out of 10 shots.