Normal Distribution Exercises

Harold Nelson

March 18, 2024

Exercises on the Normal Distribution

Example of MyNormProb usage

v = MyNormProb(lb=100,ub=110,mean=100,sd=10,
   MyLabel = "Non-Standard Normal RV")

v
## [1] 0.3413447

The Probability Between Two Values Using MyNormProb()

par(mfrow=c(3,1))
MyNormProb(ub=1)
## [1] 0.8413131
MyNormProb(ub = -1)
## [1] 0.1586236
MyNormProb(lb=-1,ub=1)

## [1] 0.6826895
pnorm(1) - pnorm(-1)
## [1] 0.6826895
par(mfrow=c(1,1))

Exercise 1

Find the probability that a single random number drawn from a set of numbers with a standard normal distribution is < .7. Do this three ways.

  1. Use pnorm()
  2. Use MyNormProb() to get a picture.
  3. Create a set of 1,000,000 numbers with this distribution and find the fraction that meet the requirement.

Exercise 1 Answer

# Method 1.
pnorm(.7)
## [1] 0.7580363
# Method 2
MyNormProb(ub=.7)

## [1] 0.7580047
# Method 3
x = rnorm(1000000)
mean(x<.7)
## [1] 0.758577

Exercise 2

Find the probability that a single random number drawn from a set of numbers with a standard normal distribution is > .7. Do this three ways.

  1. Use pnorm()
  2. Use MyNormProb() to get a picture.
  3. Create a set of 1,000,000 numbers with this distribution and find the fraction that meet the requirement.

Exercise 2 Answer

# Method 1
1 - pnorm(.7)
## [1] 0.2419637
# Method 2
MyNormProb(lb=.7)

## [1] 0.241932
# Method 3
x = rnorm(1000000)
mean(x>.7)
## [1] 0.242511

Exercise 3

Find the probability that a single random number drawn from a set of numbers with a standard normal distribution is > .3 and < .9. Do this three ways.

  1. Use pnorm()
  2. Use MyNormProb() to get a picture.
  3. Create a set of 1,000,000 numbers with this distribution and find the fraction that meet the requirement.

Exercise 3 Answers

pnorm(.9) - pnorm(.3)
## [1] 0.1980285
# Method 2
MyNormProb(lb=.3,ub=.9)

## [1] 0.1980285
# Method 3
x = rnorm(1000000)
mean(x>.3 & x < .9)
## [1] 0.198377

Exercise 4

Find the probability that a single random number drawn from a set of numbers with a normal distribution having a mean of 100 and a standard deviation of 10 is > 90 and < 115. Do this three ways.

  1. Use pnorm()
  2. Use MyNormProb() to get a picture.
  3. Create a set of 1,000,000 numbers with this distribution and find the fraction that meet the requirement.

Exercise 4 Answers

pnorm(115,mean=100,sd=10) - pnorm(90,mean=100,sd=10)
## [1] 0.7745375
# Method 2
MyNormProb(lb=90,ub=115,mean=100,sd=10,
           MyLabel = "Non-Standard Normal RV (100,10)")

## [1] 0.7745375
# Method 3
x = rnorm(1000000,mean=100,sd=10)
mean(x> 90 & x < 115)
## [1] 0.774297

Exercise 5

90% of a standard normal distribution lies to the left of x. What is the value of x?

Exercise 5 Answer

qnorm(.9)
## [1] 1.281552

Exercise 6

10% of a standard normal distribution lies to the right of x. What is the value of x?

Exercise 6 Answer

qnorm(1 - .1)
## [1] 1.281552

Exercise 7

A bank’s loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. Find P60, the score which separates the lower 60% from the top 40%.

Exercise 7 Answer

qnorm(.6,mean=200,sd=50)
## [1] 212.6674

Exercise 8

The weights of certain machine components are normally distributed with a mean of 8.5 g and a standard deviation of 0.09 g. Find the two weights that separate the top 3% and the bottom 3%. These weights could serve as limits used to identify which components should be rejected.

Exercise 8 Answer

Top = qnorm(1-.03,mean = 8.5, sd = .09)
Top
## [1] 8.669271
Bottom = qnorm(.03,mean = 8.5, sd = .09)
Bottom
## [1] 8.330729

Exercise 9

The weights of certain machine components are normally distributed with a mean of 8.5 g and a standard deviation of 0.09 g. What is the probability that a randomly selected component weighs more than 8.6 g.

Exercise 9 Answer

1 - pnorm(8.6,mean=8.5,sd=.09)
## [1] 0.1332603