Practice 7 Working with the Normal Distribution

7.1 Directions

In this practice exercise, you will work with the normal distributionin R.

7.2 A closer look at the code

In this practice, you will learn about the pnorm() and qnorm commands.

7.2.1 Let’s learn to use the pnorm() command

The r code window below will calculate and plot the probability that a normally distributed random variable is less than value. You can change the mean, standard deviation, and value.

Experiment with the settings of mean, standard deviation, and value.

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You can get help on any command in R by typing ? in font of the name of the command. Do so in the R code window below.

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The pnorm() command has four arguments that we need to be concerned about.

  • q which holds the value to find the probability of being less than or greater than
  • mean which holds the mean of the normal distribution
  • sd which holds the standard deviation of the normal distribution
  • lower.tail which is set to true for a less than calculation or false for a greater than calculation.

Here are a few examples from the voice thread.

# P(X <=  70) X ~ N(75,5)
pnorm(q=70, mean=75, sd=5, lower.tail=T)
## [1] 0.1586553
# P(X >= 85) X ~ N(75,5)
pnorm(q=85, mean=75, sd=5, lower.tail=F)
## [1] 0.02275013
# P(Z >= 1) where Z ~ N(0,1)
pnorm(q=1, mean=0, sd=1, lower.tail=F)
## [1] 0.1586553

7.2.2 Let’s learn to use the qnorm() command

The r code window below will calculate a value, X, such that the probability of drawing a value less than or equal to X is the probability you specify. Then the probability density function is plotted. You can change the mean, standard deviation, and value.

Experiment with the settings of mean, standard deviation, and value.

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The qnorm() command has four arguments that we need to be concerned about.

  • p which holds the value for which you wish to find a probability.
  • mean which holds the mean of the normal distribution
  • sd which holds the standard deviation of the normal distribution
  • lower.tail which is set to true for a less than calculation or false for a greater than calculation.

Here is an examples from the voice thread.

# Find the value of Z such that P(X <= Z) = 25% where x ~ N(75,5)
qnorm(p=0.25, mean=75, sd=5, lower.tail=T)
## [1] 71.62755

7.3 R code used in the VoiceThread

# P(X <= 70) X ~ N(75,5)
pnorm(q=70, mean=75, sd=5, lower.tail=T)

# P(X >= 85) X ~ N(75,5)
pnorm(q=85, mean=75, sd=5, lower.tail=F)

# P(Z >= 1) where Z ~ N(0,1)
pnorm(q=1, mean=0, sd=1, lower.tail=F)

# Find the value of X such that P(X <= 70) = 25%  where X ~ N(75,5)
qnorm(p=0.25, mean=75, sd=5, lower.tail=T)

7.4 Now you try

Use R to complete the following activities (this is just for practice you do not need to turn anything in).

  1. Find the value of probability that Z is greater than 1 where Z ~ N(0,1).
  2. Find the value of X such that P(Z <= X) = 25% where X ~ N(75,5).
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