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STA 2023
Sections 5.1 and 5.2
Finding 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:
The mean, median, and mode are equal.
The normal curve is bell-shaped and is symmetric
about the mean.
The total area under the normal curve is equal to 1.
The normal curve approaches, but never touches, the
x-axis as it extends farther and farther away from the
mean.
1.
2.
3.
4.
1.
The x-axis is a horizontal asymptote to the curve
The graph contains points of inflection located 1
standard deviation away from the mean.
5.

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.


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(-10000,2.13) = .9834
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