The Normal Distribution

STAT 240 - Fall 2025

Robert Sholl

Motivation

Sums of Samples

Sample Statistics

Example Data

Example Data

Normal Distributions

General Normal Distribution

General Normal Distribution

General Normal Distribution

General Normal Distribution

General Normal Distribution

General Normal Distribution

Standard Normal Distribution

A special case of the normal distribution with mean \(\mu=0\) and standard deviation \(\sigma=1\).

\[ X \sim N(\mu, \sigma^2) \]


\[ Z = \frac{X-\mu}{\sigma} \sim N(0,1) \]

z-table

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
-0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483
-0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859
-0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517

In Practice

z Problems

  1. Find \(P(Z<-1.96)\).

z Problems

0.04 0.05 0.06 0.07 0.08
-2.1 0.0162 0.0158 0.0154 0.0150 0.0146
-2.0 0.0207 0.0202 0.0197 0.0192 0.0188
-1.9 0.0262 0.0256 0.0250 0.0244 0.0239

z Problems

  1. Find \(P(Z>1.96)\).

z Problems

0.04 0.05 0.06 0.07 0.08
1.9 0.9738 0.9744 0.9750 0.9756 0.9761
2.0 0.9793 0.9798 0.9803 0.9808 0.9812
2.1 0.9838 0.9842 0.9846 0.9850 0.9854

z Problems

  1. Find \(P(-1.96<Z<1.96)\).

z Problems

  1. Find \(z_0\) where \(P(-z_0<Z<z_0) = 0.99\).

z Problems

0.06 0.07 0.08 0.09
-2.6 0.0039 0.0038 0.0037 0.0036
-2.5 0.0052 0.0051 0.0049 0.0048
-2.4 0.0069 0.0068 0.0066 0.0064

Earthquakes

Earthquakes

The population mean of earthquake magnitudes is \(\mu = 7\) and the population standard deviation is \(\sigma = 0.75\)

  1. Calculate the probability that an earthquake worse than the Kamchatka earthquake (magnitude 8.8) occurs.

Earthquakes

The population mean of earthquake magnitudes is \(\mu = 7\) and the population standard deviation is \(\sigma = 0.75\)

  1. Find the range that contains the center \(50\%\) of all earthquakes.

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