Probability Calculations for Means (Practice)

Applying the CLT

Instructions

Answer the following questions, render the document and submit the .html report.

Questions

GPA

Suppose the mean GPA of BYU-Idaho students is 3.5 and the standard deviation is 0.7. It is well known that this distribution is left-skewed. A random sample of n = 81 students will be drawn.

Use the following R code to answer the questions below:

xbar <- 
mu <- 
sigma <- 
n <- 
sigma_xbar <- 
sigma_xbar

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

z <- 
z

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

# Area to the left:
  
# Area to the right:

Question: What is the mean of the distribution of the sample means (\(\mu_{\bar{x}}\)) for all possible samples of size 81 that could be drawn from the parent population of GPAs?
Answer:

Question: What is the standard deviation of the distribution of the sample means (\(\sigma_{\bar{x}}\)) for all possible samples of size 81 that could be drawn from the parent population of GPAs?
Answer:

Question: What is the shape of the distribution of the sample means for all possible samples of size 81 that could be drawn from the parent population of GPAs?
Answer:

Question: What is the probability that the mean GPA for 81 randomly selected BYU-Idaho students will be less than 3.3?
Answer:

GRE

Scores on the quantitative portion of the GRE are approximately normally distributed with mean, \(\mu = 150.8\) and standard deviation, \(\sigma = 8.8\).

Use the following R code to answer the questions below:

xbar <- 
mu <- 
sigma <- 
n <- 
sigma_xbar <- 
sigma_xbar

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

z <- 
z

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

# Area to the left:
  
# Area to the right:

# Percentile (qnorm())

Question: Dianne earned a score of 160 on the quantitative portion of the GRE. What is the z-score corresponding to Dianne’s score?
Answer:

Question: What is the probability that a randomly selected student will score above 160 on the quantitative portion of the GRE?
Answer:

Question: What GRE score (n=1) corresponds to the 95th percentile?
Answer:

Question: What is the probability that the average GRE score of 5 randomly selected students will be above 160?
Answer:

Coast Guard

The United States Coast Guard assumes the mean weight of passengers in commercial boats is 185 pounds. The previous value was lower, but was raised after a tragic boating accident. The standard deviation of passenger weights is 26.7 pounds. The weights of a random sample of 48 commercial boat passengers were recorded. The sample mean was determined to be 177.6 pounds.

Use the following R code to answer the questions below:

xbar <- 
mu <- 
sigma <- 
n <- 
sigma_xbar <- 
sigma_xbar

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

z <- 
z

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

# Area to the left:
  
# Area to the right:

Question: Find the probability that a random sample of passengers will have a mean weight that is as extreme or more extreme (either above or below the mean) than was observed in this sample.
Answer:

HINT: To get a value “as extreme” means it could be higher or lower. Because the normal distribution is symmetric, if the \(z<0\), we can take the area to the left of \(z\) and multiply by 2. Conversely, if \(z>0\) we can take the area to the right and multiply by 2.

Tankers

Tanker trucks are designed to carry huge quantities of gasoline from refineries to filling stations. A factory that manufactures the tank of the trucks claims to manufacture tanks with a capacity of 8550 gallons of gasoline. The actual capacity of the tanks is normally distributed with mean, \(\mu = 8544\) gallons, and standard deviation, \(\sigma=12\) gallons.

Use the following R code to answer the questions below:

xbar <- 
mu <- 
sigma <- 
n <- 
sigma_xbar <- sigma / sqrt(n)

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

sigma_xbar

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

z <- (xbar - mu) / sigma_xbar

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

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

# Area to the left:
pnorm(z)

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

#Area to the right
1-pnorm(z)

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

## Area between A and B
A <- 
B <- 

pnorm(B, mu, sigma_xbar) - pnorm(A, mu, sigma_xbar)

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

Question: Find the z-score corresponding to a single tank (\(n=1\)) with a capacity of 8550 gallons. Round your answer to one decimal place.
Answer:

Question: What is the probability that a randomly selected tank will have a capacity of less than 8550 gallons?
Answer:

Question: A simple random sample of \(n = 20\) tanks was selected. Find the z-score corresponding to a sample mean capacity for 20 tanks of 8550.
Answer:

Question: What is the probability that the sample mean of \(n=20\) randomly selected tanks will be between 8541 and 8547?
Answer: