empirical rule

empirical rule

reference : OpenIntro Statistics, 3/e, p.136

Load library

library(openintro)

Suppose that the random variable \(Z\) has a standard normal probability distribution, \[Z \sim N(0, 1)\]

Use pnorm to calculate \(P(Z \le 1), P(Z \le 2), P(Z \le 3)\), and then enter those values in the space below.

The first answer is already provided to indicate a nice way to answer questions such as these. Compare what was written on the web page with what was written on the Rmd file.

• \(P(Z \le 1)\) = 0.841
• \(P(Z \le 2)\) = 0.977
• \(P(Z \le 3)\) = 0.999

Use pnorm to calculate \(P(-1 \le Z \le 1), P(-2 \le Z \le 2), P(-3 \le Z \le 3)\), and then enter those values in the space below.

• \(P(-1 \le Z \le 1)\) = 0.683
• \(P(-2 \le Z \le 2)\) = 0.954
• \(P(-3 \le Z \le 3)\) = 0.997

Relate your calculations to the following figure from the OpenIntro Statistics textbook. The three last results make up what is known as the empirical rule. Summarize that rule in a sentence or two.

• By three standard deviations on each side of the mean in a normal distribution, 99.7% of the data should be included in that area.
• The probability that you will randomly select an entry of a data set in a normal distribution is 68% at + or - 1SD, 95% at + or - 2SD, and 99.7% at + or - 3SD.