Transcript The Standardized Normal Distribution

Section 2.2
Normal Distributions
Mrs. Daniel
AP Statistics
Section 2.2
Normal Distributions
After this section, you should be able to…
 DESCRIBE and APPLY the 68-95-99.7 Rule
 DESCRIBE the standard Normal Distribution
 PERFORM Normal distribution calculations
 ASSESS Normality
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 different sizes and shapes of 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.
The distribution of Iowa Test of Basic Skills (ITBS) vocabulary
scores for 7th grade students in Gary, Indiana, is close to
Normal. Suppose the distribution is N(6.84, 1.55) and the
range is between 0 and 12.
a) Sketch the Normal density curve for this distribution.
The distribution of Iowa Test of Basic Skills (ITBS) vocabulary
scores for 7th grade students in Gary, Indiana, is close to
Normal. Suppose the distribution is N(6.84, 1.55) and the
range is between 0 and 12.
a) Sketch the Normal density curve for this distribution.
The distribution of Iowa Test of Basic Skills (ITBS) vocabulary
scores for 7th grade students in Gary, Indiana, is close to
Normal. Suppose the distribution is N(6.84, 1.55).
b) Using the Empirical Rule, what percent of ITBS vocabulary
scores are less than 3.74?
The distribution of Iowa Test of Basic Skills (ITBS) vocabulary
scores for 7th grade students in Gary, Indiana, is close to
Normal. Suppose the distribution is N(6.84, 1.55).
b) Using the Empirical Rule, what percent of ITBS vocabulary
scores are less than 3.74?
The distribution of Iowa Test of Basic Skills (ITBS) vocabulary
scores for 7th grade students in Gary, Indiana, is close to
Normal. Suppose the distribution is N(6.84, 1.55).?
c) Using the Empirical Rule, what percent of the scores are
between 5.29 and 9.94?
The distribution of Iowa Test of Basic Skills (ITBS) vocabulary
scores for 7th grade students in Gary, Indiana, is close to
Normal. Suppose the distribution is N(6.84, 1.55).?
c) Using the Empirical Rule, What percent of the scores are
between 5.29 and 9.94?
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.
• We can compare SAT and ACT scores by
standardizing.
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.
How to Standardize a Variable:
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.
4. Conclude in context.
The Standard Normal Table
Because all Normal distributions are the same when we
standardize, we can find areas under any Normal curve
from a single table.
The Standard Normal table is a table of the areas under
the standard normal curve. The table entry for each value
“z” is area under the curve to the LEFT of z.
The area to left is called the “p-value”
• Probability
• Percent
Using the Standard Normal Table
Row: Ones and tenths digits
Column: Hundredths digit
Practice: What is the p-value for a z-score of -2.33?
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?
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
Step 2: Z- Scores
325 - 304
When x = 325, z =
 2.63
8
305 - 304
When x = 305, z =
 0.13
8
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.
0.9957 – 0.5517 = 0.4440.
Step 4: Conclude In Context
About 44% of Tiger’s drives travel between 305 and 325 yards.
Normal Calculations on Calculator
NormalCDF
NormalPDF
InvNorm
Calculates
Example
Probability of
What percent of students
obtaining a value
scored between 70 and
BETWEEN two values 95 on the test?
Probability of
obtaining PRECISELY
or EXACTLY a specific
x-value
X-value given
probability or
percentile
What is the probability
that Suzy scored a 75 on
the test?
Tommy scored a 92 on
the test; what proportion
of students did he score
better than?
TI-Nspire: NormalCDF
Normalcdf- “Area under the curve between two points”
1.
2.
3.
4.
5.
6.
Select Calculator (on home screen), press center button.
Press menu, press enter.
Select 6: Statistics, press enter.
Select 5: Distributions, press enter.
Select 2: Normal Cdf, press enter.
Enter the following information:
1. Lower: (the lower bound of the region OR 1^-99)
2. Upper: (the upper band of the region OR 1,000,000)
3. µ: (mean)
4. Ơ: (standard deviation)
7. Press enter, number that appears is the p-value
TI-Nspire: NormalPDF
Normalpdf- “Exact percentile/probability of a specific event occurring”
1.
2.
3.
4.
5.
6.
Select Calculator (on home screen), press center button.
Press menu, press enter.
Select 6: Statistics, press enter.
Select 5: Distributions, press enter.
Select 1: Normal Pdf press enter.
Enter the following information:
1. Xvalue (not a percent)
2. µ: (mean)
3. Ơ: (standard deviation)
7. Press enter, number that appears is the p-value
TI-Nspire: InvNorm
invNorm- “Exact x-value at which something occurred”
1.
2.
3.
4.
5.
6.
Select Calculator (on home screen), press center button.
Press menu, press enter.
Select 6: Statistics, press enter.
Select 5: Distributions, press enter.
Select 3: Inverse Norm press enter.
Enter the following information:
1. Area (enter as a decimal)
2. µ: (mean)
3. Ơ: (standard deviation)
7. Press enter, number that appears is the p-value
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?
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?
Suzy bombed her recent AP Stats exam; she
scored at the 25th percentile. The class average
was a 170 with a standard deviation of 30.
Assuming the scores are normally distributed,
what score did Suzy earn of the exam?
Suzy bombed her recent AP Stats exam; she
scored at the 25th percentile. The class average
was a 170 with a standard deviation of 30.
Assuming the scores are normally distributed,
what score did Suzy earn of the exam?
When Can I Use Normal
Calculations?!
• Whenever the distribution is Normal.
• Ways to Assess Normality:
– Plot the data.
• Make a dotplot, stemplot, or histogram and see if the
graph is approximately symmetric and bell-shaped.
– Check whether the data follow the 68-95-99.7
rule.
– Construct a Normal probability plot.
Normal Probability Plot
• These plots are constructed by plotting each observation
in a data set against its corresponding percentile’s zscore.
Interpreting Normal Probability Plot
• If the points on a Normal probability plot lie close to a
straight line, the plot indicates that the data are Normal.
• Systematic deviations from a straight line indicate a nonNormal distribution.
• Outliers appear as points that are far away from the
overall pattern of the plot.
Summary: Normal Distributions
• The Normal Distributions are described by a special family of bellshaped, symmetric density curves called Normal curves. The mean µ and
standard deviation σ completely specify a Normal distribution N(µ,σ). The
mean is the center of the curve, and σ is the distance from µ to the
change-of-curvature points on either side.
• All Normal distributions obey the 68-95-99.7 Rule, which describes what
percent of observations lie within one, two, and three standard
deviations of the mean.
• All Normal distributions are the same when measurements are
standardized. The standard Normal distribution has mean µ=0 and
standard deviation σ=1.
• Table A gives percentiles for the standard Normal curve. By standardizing,
we can use Table A to determine the percentile for a given z-score or the
z-score corresponding to a given percentile in any Normal distribution.
• To assess Normality for a given set of data, we first observe its shape. We
then check how well the data fits the 68-95-99.7 rule.
Additional Help
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.
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.