Download PowerPoint 5.1, 5.2 - Southeast Missouri State University

5.1 What is Normal?
LEARNING GOAL
Understand what is meant by a normal
distribution and be able to identify situations in
which a normal distribution is likely to arise.
When exploring a distribution:
1.
2.
3.
4.
Always plot your data: make a graph
Look for overall pattern—shape, outliers,
center, spread (SOCS)
Choose either five number summary or
mean and standard deviation to describe
center and spread.
Sometimes the overall pattern of a large
number of observations is so regular that we
can describe it by smooth curve.
What is the shape of this distribution?
Density Curve
•This distribution can be approximated by a
smooth curve called a density curve.
•This density curve is used in place of
histogram to picture the overall shape of the
distribution of the data.
•Histograms show the counts or relative
frequency of observations in each class by the
heights of the bars.
•The density curve shows the proportion of
observations in any region by areas under the
curve.
•A scale is chosen so that the total area under
the density curve is 1.
•What is the shape of this distribution?
•The histogram can be approximated by a
density curve.
Normal Distribution
Symmetric, bell-shaped distribution with a
single peak
 It’s peak corresponds to the mean,
median and mode of the distribution.
 It’s variation can be characterized by the
standard deviation of the distribution.
 The normal distribution is a density
curve.

Relative Frequencies and the
Normal Distribution
With a histogram, the height of a bar
indicates the relative frequency.
 With a density curve, the area under the
curve corresponding to a range of values
on the horizontal axis is the relative
frequency of those values.
 Because the total relative frequency must
be 1, the total area under the density
curve must equal 1, or 100%.

•What is the shape of both distributions?
•What is the mean? Median? Mode?
•What is the area under the curve?
•How do the standard deviations compare?
When Can We Expect a
Normal Distribution?
Conditions for a Normal Distribution
 Most data values are clustered near the
mean giving a well-defined single peak.
 Symmetric distribution with data values
spread evenly around the mean.
 Tapering tails because larger deviations
from the mean become increasingly rare.
 Individual data values result from many
factors, such as genetic and environmental
factors.
Select the distribution that appears to
be the most normal.
a.
b.
c.
d.
From each data set, state whether you would expect it to
be normally distributed. Explain your reasoning.
Numbers resulting from rolling a single
die.
 Weights of adult Golden Retriever dogs.
 Measured braking reaction times of 18
year old drivers.
 ACT scores of all students that took the
test in 2008.

5.2 Properties of the
Normal Distribution
LEARNING GOAL
Know how to interpret the normal distribution in
terms of the 68-95-99.7 rule, standard scores, and
percentiles.
Is it time to replace your TV?
Consumer Reports conducted a survey in
which participants were asked how long they
owned their last TV set before they replaced it.
It was found that the mean time is 8.2 years and
the standard deviation is 1.1 years.
 Is it reasonable to assume this is a normal
distribution?
 Is it unusual to replace a TV in 7 years?
 Is it unusual to have a TV for 14 years before
replacing it?

Notation
Sample
Mean denoted by x
Standard deviation denoted by s

Population
 Mean denoted by 
 Standard deviation denoted by 


Which are parameters? Which are
statistics?
68-95-99.7 Rule
for a Normal Distribution
 About
68% of the data points fall
within 1 standard deviation of the
mean.
 About 95% of the data points fall
within 2 standard deviations of the
mean.
 About 99.7% of the data points fall
within 3 standard deviations of the
mean.
68-95-99.7 Rule
95%
68%
-1sd
+1sd
-2 sd
+2 sd
99.7%
-3 sd
+3 sd
Health and Nutrition Examination Study of
1976-1980
(HANES)
 Heights
of adults, aged 18-24
◦ women
 mean: 65.0 inches
 standard deviation: 2.5 inches
◦ men
 mean: 70.0 inches
 standard deviation: 2.8 inches
Look at examples with this data and the empirical rule.
Unusual values
95% of all values are within 2 standard
deviations of the mean, so 5% of all values
are more than 2 standard deviations away
from the mean.
 Unusual values are values that are more
than 2 standard deviations away from the
mean.

Is it time to replace your TV?
Consumer Reports conducted a survey in
which participants were asked how long they
owned their last TV set before they replaced it.
It was found that the mean time is 8.2 years and
the standard deviation is 1.1 years. Assuming a
normal distribution:
 Is it unusual to replace a TV in 7 years?
 Is it unusual to have a TV for 14 years before
replacing it?

Standard Normal Distribution
• The standard normal distribution is a normal
distribution with a mean of 0 and a standard
deviation of 1.
• The area under the curve is still 1.
• Every normal distribution can be transformed into
a standard normal distribution.
• To do this we find a standard score with the
following formula:
.
Standard Scores
z is the standardized score
 x is the data value
  is the population mean
  is the population standard deviation

z
x

The standard score tells us how many
standard deviations an observed value is
above or below the mean.
ACT scores—2002


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In 2008, the mean composite score in Missouri was
21.4 with a standard deviation of approximately 5.
The qualifying score for a Bright Flight scholarship is 31.
What percentile does this represent?
The Regents Scholarship at Southeast requires an ACT
score of 27. What percent of students taking the ACT
qualify for the Regents Scholarship?
If you wanted to give a scholarship to the top 5% of the
scores, what ACT score would you require?
Regular admission (option 2) to Southeast requires an
18 on the ACT and 2.5 high school GPA. What percent
of students taking the ACT qualify for admission to
Southeast?
American Express Card
American Express charges merchants higher fees than any other
credit or debit card, according to the USA Today article
“American Express fees take flak” (12/23/2004). The company
believes they can do this because they claim the customers using
the American Express card spend more. The average annual
charges per card in 2003 were $9600 according to data from
American Express and The Neilson Report. Assume that the
annual charges per card are approximately normally distributed
with a standard deviation of $2100.
 What percent of American Express card users charge less than
$4000?
 What percent charge more than $16000?
 What percent charge between $5000 and $10,000 dollars?
 10% of the card users charge less than what amount?
 20% of the card users charge more than what amount?