Download Normal Distribution

Lesson 11.1 Normal Distributions (Day 2)
Learning Goal: (S-ID.A.4)
I can use the mean and standard deviation of a data set to fit it to a normal distribution and to
estimate population percentages.
Essential Question: How does the mean and standard deviation of a data set help to
estimate population percentages?
Warm-Up
Finding Areas Under Normal Curves
Normal Distribution
Symmetric
Empirical Rule: (68, 95, 99.7)
68% of data within 1 standard deviation of the
mean
95% of data within 2 standard deviation of the 99.7% of data within 3 standard deviation of the
mean
mean
Finding a Normal Probability
The areas under a normal curve can
be interpreted as probabilities in a
normal distribution.
Examples:
a) P(x ≤ μ-σ )
c) P(x ≤ μ+2σ)
A normal distribution has mean of μ and standard deviation of σ. An xvalue is randomly selected from the distributions.
b) P(μ-2σ ≤ x ≤ μ)
d) P(μ+σ ≤ x ≤ μ+3σ)
Example:
The scores for a state's peace officer standards
and training test are normally distributed with a
mean of 55 and a standard deviation of 12. The
test scores range from 0 to 100.
About what percent of the people taking the test have scores between 43 and 67?
Between 31 and 55?
An agency in the state will only hire applicants with test scores of 67 or greater. About what
percent of the people have test scores that make them eligible to be hired by the agency?
79 or greater?
What if we needed the percentage of people that scored between 45 and 62?
The Standard Normal Distribution
Any normally distributed data can be transformed to a standard normal distribution that has
a mean of 0 and a standard deviation of 1.
This can be done by calculating the z-scores which represent the number of standard
deviations the x-value is from the mean.
z=
x-μ
σ
To find the probabilities you use a standard normal table which shows the percentage of the
data that is below the data value.
Find P(z ≤ -1.24) =
Find P(z ≥ -1.24) =
Find P(-1.24 ≤ z ≤ -0.78) =
Your Turn
Find P(z ≤ 1.43) =
Find P(z ≥ 1.43) =
Find P(1.35 ≤ z ≤ 0.38) =
Find P(z ≤ 0) =
Explain why this makes sense.
Using a z-score and the Standard Normal Table
Examples
A study finds that the weight of infants at birth are normally distributed with a mean of 3270
grams and a standard deviation of 600 grams. An infant is randomly chosen. What is the
probability that the infant weighs 4170 grams or less?
Step 1: Find the z-score(s) for the data values.
Step 2: Use the Standard Normal Table to find the probability.
What is the probability that the infant weighs 3990 grams or more?
Group Consensus
Suppose the heights (in inches) of adult females (ages 20–29) in the United States are
normally distributed with a mean of 64.1 inches and a standard deviation of 2.75 inches.
Find the percent of women who are less than or equal to 61.6 inches tall.
Find the percent of women who are at least 66 inches tall.
A woman is randomly selected. What is the probability that the woman is between 61.35
inches and 66.85 inches tall?
Which of the above problems can be answered without computing a z-score? Why?
Exit Question:
What is a z-score and how is it used?
Practice to Strengthen Understanding
Hmwk #22 BI p600 2-28 even