Normal Probability Practice

The Normal Distribution

Recall that the normal distribution is a probability distribution defined by its center (\(\mu\)) and its spread (\(\sigma\)).

Normal Probability Calculator

The code below can serve as a template for any calculations involving the normal distribution. Feel free to copy and paste this into the subsequent code chunks.

NOTE: R saves any objects that are named. This means that if you run mu <- 12, R keeps track of mu until you change it. When doing multiple problems that all use mu, be sure to run the whole code chunk, which will overwrite the names from previous examples.

# Value
x <- 

# Population Mean
mu <- 

# Population Standard deviation
sigma <- 

# Z-score:  number of standard deviations above the mean
z <- (x-mu) / sigma

Error in eval(expr, envir, enclos): object 'x' not found

Error in eval(expr, envir, enclos): object 'z' not found

# Probability of observing a z-score LESS THAN the one we observed (left tail)
pnorm(z)

Error in eval(expr, envir, enclos): object 'z' not found

# Probability of observing a z-score GREATER THAN the one we observed (Right Tail)
1-pnorm(z)

Error in eval(expr, envir, enclos): object 'z' not found

# For calculating the area between 2 numbers
A <- 
B <- 
pnorm(B, mu, sigma) - pnorm(A, mu, sigma)

Error in eval(expr, envir, enclos): object 'B' not found

# Find the X_bar associated with a spceific percentile
qnorm(.25, mu, sigma)

Error in eval(expr, envir, enclos): object 'mu' not found

Brisket Competition

Competition was tight at a Saint Louis BBQ competition. Brisket scores were normally distributed with an average score of 5.941 with a standard deviation of 0.04.

Question: What’s the probability of getting a score GREATER than 6?
Answer:

Question: What’s the probability of getting a score LESS than 4?
Answer:

Question: What is the 99th percentile of this Brisket Competition?
Answer:

Prbability of a “False Start”

“At high level meets, the time between the gun and first kick against the starting block is measured electronically, via sensors built in the gun and the blocks. A reaction time less than 0.1 s is considered a false start. The 0.2-second interval accounts for the sum of the time it takes for the sound of the starter’s pistol to reach the runners’ ears, and the time they take to react to it.” (https://en.wikipedia.org/wiki/100_metres) Let’s suppose that reaction times are normally distributed with a mean of 0.2 seconds and a standard deviation of 0.03.

What’s the probability of a false start? (meaning a reaction time LESS than 0.1)

Law School Admissions Test (LSAT)

The scores on the LSAT are approximately normal with mean of 150.7 and standard deviation of 10.2. (Source: www.lsat.org.) Queen’s School of Business in Kingston, Ontario requires a minimum LSAT score of 157 for admission.

Question: What is the z-score for a score of 157?
Answer:

Question: What is the probability that a randomly selected person who took the LSAT would score above 157?
Answer:

Question: What is the probability that a randomly selected test taker will have a score between 155 and 170?
Answer:

Question: What is the 97.4th percentile of LSAT scores?
Answer: