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 <- 3.3
mu <- 3.5
sigma <- 0.7
n <- 81
sigma_xbar <- sigma / sqrt(n)
sigma_xbar

[1] 0.07777778

z <- (xbar-mu) / sigma_xbar
z

[1] -2.571429

# Area to the left:
pnorm(z)  

[1] 0.005063995

# Area to the right:
1-pnorm(z)

[1] 0.994936

Question: What is the mean of the distribution of the sample means (${\mu }_{\overline{x}}$) for all possible samples of size 81 that could be drawn from the parent population of GPAs?
Answer: 3.5

Question: What is the standard deviation of the distribution of the sample means (${\sigma }_{\overline{x}}$) for all possible samples of size 81 that could be drawn from the parent population of GPAs?
Answer: 0.07777778

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: Normal because CLT (n > 30)

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

z <- 
z

# 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:

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 <- 8550
mu <- 8544
sigma <- 12
n <- 1
sigma_xbar <- sigma / sqrt(n)
sigma_xbar

[1] 12

z <- (xbar - mu) / sigma_xbar
z

[1] 0.5

# Area to the left:
pnorm(z)

[1] 0.6914625

#Area to the right
1-pnorm(z)

[1] 0.3085375

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

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

Error: 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: 0.5

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: