Standard Normal Distribution
Density Curve
A smooth curve that describes the overall
pattern of the distribution
• The total area under the curve = 1
• The curve will never go below the x-axis
• The area between two values is equal to the
probability that the event will occur between
those values
Uniform Distribution
A continuous random variable has uniform distribution if its values are
spread evenly over the range of possibilities. The graph of a uniform
distribution results in a rectangular shape.
Uniform Distribution
Kim has an interview right after class.
If the class runs longer than 51.5
minutes, she will be late. Given the
previous uniform distribution, find
the probability that a randomly
selected class will last longer than
51.5 minutes.
There is a 25% Chance the class will
last longer than 51.5 minutes.
Uniform Distribution
Find the probability that a
time greater than 50.5
minutes is selected.
Uniform Distribution
Find the probability that a
time between 51.5
minutes and 51.6 minutes
is selected.
Standard Normal Distribution
If a continuous random variable has a distribution with a graph that is
symmetric and bell-shaped , as below, and it can be described by the
following formula , then we say it has a normal distribution.
𝑦=
1 𝑥−𝜇
−2( 𝜎 )2
𝑒
𝜎 2𝜋
Standard Normal Distribution
The standard normal distribution is a normal probability distribution with
𝜇 = 0 𝑎𝑛𝑑 𝜎 = 1. The total area under its density curve is equal to 1.
Standard Normal Distribution
The standard normal distribution is a normal probability distributions with
𝜇 = 0 𝑎𝑛𝑑 𝜎 = 1. The total area under its density curve is equal to 1.
All x-values are z-scores. A z-score refers to how many standard deviations
above or below the mean a particular value is.
The area under the curve is equal to its probability
• Use Table A-2 (back cover of text) This table gives the area underneath the
curve to the left of the z-score.
• Use Ti-83/84 [2nd][Vars] [2: normal cdf(] enter two z scores separate by
a comma.
Standard Normal Distribution
The precision Scientific Instrument company
manufactures thermometers that are suppose to give
readings of 0° at the freezing point of water. Tests on
a large sample reveal that at the freezing point of
water the readings of the thermometers are normally
distributed with a mean of 0° and a standard
deviation of 1.00°C. A thermometer is randomly
selected. Find the probability of getting a reading less
than 1.75.
Standard Normal Distribution
Assume that the readings on the thermometers are
normally distributed with a mean of 0° and a
standard deviation of 1.00°C. A thermometer is
randomly selected. Find the probability of getting a
reading greater than 1.96
Standard Normal Distribution
Assume that the readings on the thermometers are
normally distributed with a mean of 0° and a
standard deviation of 1.00°C. A thermometer is
randomly selected. Find the probability of getting a
reading less than −1.23°.
Standard Normal Distribution
Assume that the readings on the thermometers are
normally distributed with a mean of 0° and a
standard deviation of 1.00°C. A thermometer is
randomly selected. Find the probability of getting a
reading between −2.00° and 1.50°.
Standard Normal Distribution
Find the indicated area under the curve of the standard normal
distribution. About ____% of the area is between z = -2 and z = 2 (or within
2 standard deviations of the mean)
Standard Normal Distribution
Finding z scores from Known Areas
1. Draw a bell-shaped curve and identify the region under the curve that
corresponds to the given probability. If that region is not a cumulative
from the left, work instead with a know region that is cumulative from
the left.
2. Using the cumulative area from the left,
– locate the closest probability using table A-2.
– To find a z score corresponding using Ti-83/84
[2nd ][Vars][invNorm.] enter area cumulative from the left.
Standard Normal Distribution
Assume that the readings on the thermometers are
normally distributed with a mean of 0° Find the
temperature corresponding to the 95th percentile.
That is find the temperature separating the top 5%
from the bottom 95%.
Standard Normal Distribution
Assume that the readings on the thermometers are
normally distributed with a mean of 0° Find the
temperatures separating the bottom 2.5% and the
top 2.5%.
Homework!
• 6.1 : 6- 39 eoo, 41- 49 odd.