STA 2023Sections 5.1 and 5.2Finding Probabilities for Normal Distributions.

Properties of a Normal Distribution A normal distribution is a continuous

probability distribution for a random variable x. The graph of a normal distribution is called the normal curve.

Properties:1. The mean, median, and mode are equal.2. The normal curve is bell-shaped and is

symmetric about the mean.3. The total area under the normal curve is equal

to 1.4. The normal curve approaches, but never

touches, the x-axis as it extends farther and farther away from the mean.

1. The x-axis is a horizontal asymptote to the curve

5. The graph contains points of inflection located 1 standard deviation away from the mean.

These are points where the graph changes the way it curves.

The Standard Normal Distribution

Finding Areas Under the Standard Normal Curve To find areas under the Standard Normal

Curve, we will be using the calculator. Press 2nd, Vars, then 2:normalcdf(

Syntax: normalcdf(lower bound, upper bound) The lower bound and the upper bound

correspond to the area of the standard normal curve that we are finding.

Always draw the standard normal curve and shade in the area you are looking for so that you clearly find your lower and upper bound.

If a tail is use as a bound for the area, use -10000 for the lower bound or 10000 for the upper bound.

Example 1: Find the area under the standard normal curve to the left of z = 2.13. Answer: normalcdf(-10000,2.13) = .9834

Example 2: Find the area under the standard normal curve to the right of z = -1.16. Answer: normalcdf(-1.16,10000) = .8770

Example 3: Find the area under the standard normal curve between z = -2.17 and z = -1.35. Answer: normalcdf(-2.17, -1.35) = .0735

Example 4: Find the area under the standard normal curve to the left of z = -0.82 or to the right of z = 1.17. Answer: normalcdf(-10000,-0.82)+normalcdf(1.17,10000)

= .3271

Probability and Normal Distributions

We can find the probability of any normal distribution by converting the data into the standard normal distribution using the z-score formula.

The area under the standard normal curve is equal to the probability of an event happening in the normal distribution.

Example 6: The monthly utility bills in a city are normally distributed, with a mean of $100 and a standard deviation of $12. A utility bill is randomly selected.

Find the probability that the utility bill is less than $70. Answer: .0062

Find the probability that the utility bill is between $90 and $120. Answer: .7493

If we look at a group of 150 utility bills, how many of those bills will be between $90 and $120? Answer: 112 utility bills.

Find the probability that the utility bill is more than $140. Answer: .0004