SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard
Normal Distribution.
Do Now:
Scott was one of 50 junior boys to take the PSAT at his
school. He scored 64 on the Critical Reading test. This
placed Scott at the 68th percentile within the group of
boys. Looking at all 50 boys’ Critical Reading scores, the
mean was 58.2 and the standard deviation was 9.4.
Calculate and compare Scott’s z-score among the group
of test takers
SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard
Normal Distribution.
Normal curves -The distributions they describe are called
Normal distributions
SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard
Normal Distribution.
DEFINITION: The 68–95–99.7 rule
In the Normal distribution with mean μ and standard deviation σ:
Approximately 68% of the observations fall within 1σ of the mean μ.
Approximately 95% of the observations fall within 2σ of μ.
Approximately 99.7% of the observations fall within 3σ of μ
SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard
Normal Distribution.
SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard
Normal Distribution.
Example 1: Using the 68–95–99.7 rule
Suppose we know that a distribution is exactly Normal for the scores of a New
York State 6th grade vocabulary exam with mean μ = 6.84 and standard
deviation σ = 1.55.
(a) Sketch a Normal curve for this distribution of test scores. Label the
points that are one, two, and three standard deviations from the mean.
(b) What percent of the NYS vocabulary scores are less than 3.74? Show your
work.
(c) What percent of the scores are between 5.29 and 9.94? Show your work.
SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard
Normal Distribution.
Example 2:
The distribution of heights of young women aged 18 to 24 is a normal distribution
with a mean µ = 64.5 and σ = 2.5).
(a) Sketch a Normal curve for the distribution of young women’s heights. Label the
points one, two, and three standard deviations from the mean.
(b) What percent of young women have heights greater than 67 inches?
(c) What percent of young women have heights between 62 and 72 inches?
SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard
Normal Distribution.
Example 3:
Suppose we know that the average (μ) high school relationship is 42 days with a
standard deviation (σ) of 2 days.
(a) Assuming that the distribution of days is approximately Normal, make an accurate
sketch of the distribution with the horizontal axis marked in days.
(a) Between which standard deviations does the interval from 38-42 lie?
(b) What percentage represents this interval?
(c) What does this percentage represent in the context of this problem?
SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard
Normal Distribution.
Do Now:
Suppose we know that the time it takes for college students to commute to their
classroom from their dorms is normally distributed with an average time of μ = 8
minutes and a standard deviation of σ = 2 minutes.
(a) Sketch a Normal curve for the distribution of college students commute time in
minutes.
(b) Determine the interval that represents the middle 68% of the data.
(a) Interpret the values and percentage less than -1σ .
SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard
Normal Distribution.
Let’s look again at the Normal curve for the distribution in the Do Now:
What if we wanted to find the percentage of values less than -1.21σ? Can we
find this using our Normal curve?
SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard
Normal Distribution.
The standard Normal table (z-table)
**The z-table is a table of areas under the standard Normal curve. The table entry for
each value z is the area under the curve to the left of z.
SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard
Normal Distribution.
Example:
(a)Find the proportion of observations less than -1.25
(b)Find the percent of observations more than 0.81
(c)Find the percent of observations between -1.25 and 0.81
(d)Find the z value of all observations less than 34%
SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard
Normal Distribution.
1. Use the z-table to find the proportion of observations from a
standard Normal distribution that fall in each of the following regions. In
each case, sketch a standard Normal curve and shade the area
representing the region.
(a) z < 1.39
(b) z > −2.15
(c) −0.56 < z < 1.81
2. Use the z-table to find the value z from the standard Normal
distribution that satisfies each of the following conditions.
(a) The 20th percentile
(b) 45% of all observations are greater than z
SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard
Normal Distribution.
Exit Ticket
A study of elite distance runners found a mean body weight of 63.1 kilograms (kg),
with a standard deviation of 4.8 kg.
(a) Assuming that the distribution of weights is approximately Normal, make an
accurate sketch of the weight distribution with the horizontal axis marked in
kilograms.
(b) Use the 68–95–99.7 rule to find the proportion of runners whose body weight is
between 48.7 and 67.9 kg.
(c) What proportion of runners have body weights below 60 kg?
(d) What proportion of runners have body weights above 70 kg?
SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard
Normal Distribution.
Do Now:
(a) Find the proportion of observations less than -1.09
(b) Find the percent of observations more than 2.41
(c) Find the percent of observations between 0.92 and 2.11
(d) Find the z value of all observations more than 81%
SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard
Normal Distribution.
How to Solve Problems Involving Normal Distributions
1. State the information you know.
2. Sketch your distribution and the area you are looking
for.
3. Standardize your value in terms of a Normal standard
variable (find your z-value) and use the z-table
4. Write your conclusion in the context of the problem.
SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard
Normal Distribution.
Example 1: On the driving range, Tiger Woods practices his
swing with a particular club by hitting many, many balls. When
Tiger hits his driver, the distance the ball travels follows a
Normal distribution with mean 304 yards and standard
deviation 8 yards. What percent of Tiger’s drives travel at least
290 yards?
SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard
Normal Distribution.
Example 1(b): What distance would a ball have to travel to be
at the 80th percentile of Tiger Woods’s drive lengths?
SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard
Normal Distribution.
YOU TRY!! High levels of cholesterol in the blood increase the risk of heart
disease. For 14-year-old boys, the distribution of blood cholesterol is approximately
Normal with mean μ = 170 milligrams of cholesterol per deciliter of
blood (mg/dl) and standard deviation σ = 30 mg/dl.
(a) People with cholesterol levels between 200 and 240 mg/dl are at considerable
risk for heart disease. What percent of 14-year-old boys have blood cholesterol
between 200 and 240 mg/dl?
(b) What is the first quartile of the distribution of blood cholesterol?