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23.2 The Normal Distribution
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Normal Distributions
•All Normal curves are symmetric, single-peaked, and bellshaped
•A Specific Normal curve is described by giving its mean µ
and standard deviation σ.
Two Normal curves, showing the mean µ and
standard deviation σ.
Normal Distributions
• We abbreviate the Normal distribution with mean µ and
standard deviation σ as N(µ,σ).
• Any particular Normal distribution is completely specified by
two numbers: its mean µ and standard deviation σ.
• The mean of a Normal distribution is the center of the
symmetric Normal curve.
• The standard deviation is the distance from the center to the
change-of-curvature points on either side.
Normal Distributions are Useful…
•
Normal distributions are good descriptions for some
distributions of real data.
•
Normal distributions are good approximations of the
results of many kinds of chance outcomes.
•
Many statistical inference procedures are based on
Normal distributions.
The 68-95-99.7 Rule
Although there are many Normal curves, they all have
properties in common.
The 68-95-99.7 Rule (“The Empirical Rule”)
In the Normal distribution with mean µ and standard deviation σ:
•Approximately 68% of the observations fall within σ of µ.
•Approximately 95% of the observations fall within 2σ of µ.
•Approximately 99.7% of the observations fall within 3σ of µ.
Explain 1- Example 1
Suppose the heights (in inches) of men (ages 20–29) years old in the
United States are normally distributed with a mean of 69.3 inches and
a standard deviation of 2.92 inches. Find each of the following.
A. The percent of men who are between 63.46 inches and 75.14 inches
tall.
Suppose the heights (in inches) of men (ages 20–29) years old in the
United States are normally distributed with a mean of 69.3 inches and
a standard deviation of 2.92 inches. Find each of the following.
Do Your Turn 5 and 6
Importance of Standardizing
•There are infinitely many different Normal
distributions; all with unique standard deviations and
means.
•In order to more effectively compare different Normal
distributions we “standardize”.
•Standardizing allows us to compare apples to apples.
The Standardized Normal Distribution
All Normal distributions are the same if we measure in
units of size σ from the mean µ as center.
The standardized Normal distribution is the Normal distribution
with mean 0 and standard deviation 1.
Example 2
• Suppose the heights (in inches) of women (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 no more than 65 inches tall and the probability that a randomly
chosen woman is between 60 inches and 63 inches tall.
Do Your Turn 7-8 & Elaborate 9-11
Using the Standard Normal Table
Using the Standard Normal Table, find the following:
Z-Score
P-value
-2.23
1.65
.52
.79
.23
Let’s Practice
In the 2008 Wimbledon tennis tournament,
Rafael Nadal averaged 115 miles per hour (mph)
on his first serves. Assume that the distribution
of his first serve speeds is Normal with a mean of
115 mph and a standard deviation of 6.2 mph.
About what proportion of his first serves would
you expect to be less than 120 mph? Greater
than?
1. Draw and label an Normal curve with the mean and
standard deviation.
2. Calculate the z- score
x= variable
µ= mean
σ= standard deviation
3. Determine the p-value by looking up the z-score in the
Standard Normal table.
P(z < 0.81) = .7910
Z
.00
.01
.02
0.7
.7580
.7611
.7642
0.8
.7881
.7910
.7939
0.9
.8159
.8186
.8212
4. Conclude in context.
We expect that 79.1% of Nadal’s first serves will be less
than 120 mph.
We expect that 20.9% of Nadal’s first serves will be greater
than 120 mps.
Let’s Practice
When Tiger Woods hits his driver, the
distance the ball travels can be
described by N(304, 8). What percent of
Tiger’s drives travel between 305 and
325 yards?
Normal Distribution Calculations
When Tiger Woods hits his driver, the distance the ball
travels can be described by N(304, 8). What percent of
Tiger’s drives travel between 305 and 325 yards?
Step 1: Draw Distribution
Normal Distribution Calculations
When Tiger Woods hits his driver, the distance the ball
travels can be described by N(304, 8). What percent of
Tiger’s drives travel between 305 and 325 yards?
Step 2: Z- Scores
Normal Distribution Calculations
Step 3: P-values
Using Table A, we can find the area to the left of z=2.63 and
the area to the left of z=0.13.
Normal Distribution Calculations
Step 4: Conclude In Context
Finding Areas Under the Standard Normal
Curve
Find the proportion of observations from the standard
Normal distribution that are between -1.25 and 0.81.
Step 1: Look up area to
the left of 0.81 using
table A.
Finding Areas Under the Standard Normal
Curve
Find the proportion of observations from the standard
Normal distribution that are between -1.25 and 0.81.
Step 2: Find the area to the left of -1.25
Finding Areas Under the Standard Normal
Curve
Find the proportion of observations from the standard
Normal distribution that are between -1.25 and 0.81.
Step 3: Subtract.
Finding Areas Under the Standard Normal
Curve
Can you find the same proportion using a different
approach?
1 - (0.1056 + 0.2090)
= 1 – 0.3146
= 0.6854
11-6 Confidence
Intervals and
Hypothesis Testing
Confidence Intervals
• Inferential statistics are used to draw conclusions or statistical
inferences about a population using a sample.
• Example: In a recent poll, 1514 teens who owned a portable media
player had an average of 1033 songs. The poll had the following
disclaimer: “For results based on the total sample of national teens,
one can say with 95% confidence that the margin of sampling error is
±31 songs.
Confidence Intervals
• Is an estimate of a population parameter stated as a range with a
specific degree of certainty.
• A 95% confidence interval for a population means that we are 95%
sure that the mean will fall within the range of z-values.
• Example: We were 95% confident that the population mean is within 31 songs
of 1033.
Maximum Error of Estimate
Maximum Error Estimate Example
ORAL HYGIENE A poll of 422 randomly selected adults showed that they
brushed their teeth an average of 11.4 times per week with a standard
deviation of 1.6. Use a 99% confidence interval to find the maximum error of
estimate for the number of times per week adults brush their teeth.
Maximum Error of Estimate Cont.
Maximum Error of Estimate
Example
READING A poll of 385 randomly selected adults showed that they read for
recreation an average of 39.3 minutes per week with a standard deviation of 9.6
minutes. Use a 95% confidence interval to find the maximum error of estimate for
the number of minutes per week adults read for recreation.
A.
0.49
B.
0.80
C.
0.96
D.
1.26
Confidence Interval for the Population Mean
Confidence Interval
PACKAGING A poll of 156 randomly selected members of a golf course showed
that they play an average of 4.6 times every summer with a standard deviation
of 1.1. Determine a 95% confidence interval for the population mean.
Confidence Interval for Population Mean