Download Chapter 2.2 STANDARD NORMAL DISTRIBUTIONS

Chapter 2.2
STANDARD NORMAL
DISTRIBUTIONS
Normal Distributions
• Last class we looked at a particular type of density
curve called a Normal distribution.
• All Normal distributions are described by two
μ
parameters: ________
and __________
σ
• Because of this, we can abbreviate a Normal
distribution as ____(___,
N μ ___)
σ
• Another important quality of Normal distributions is
that the follow the __________
Empirical rule. This rule states
68 of the data falls within 1 standard deviation
that ____%
of the mean, ____%
falls within 2 standard deviations
95
99.7 falls within 3 standard deviations.
and _____%
The Standard Normal Distribution
• All normal distributions are the same if we measure in units of
size σ about the mean μ as center.
• Changing these units requires that we standardize (like we did
in 2.1)
Z=x-μ
σ
• If the variable we standardize has a normal distribution, then
so does the new variable, z
• The new distribution is called the standard Normal
Distribution
Great…but why is that useful?
• Remember, the area under a density curve is a
proportion of the observations in a
distribution.
1
– The area under the entire density curve is ____.
– The proportion of observations to the left of the
.5
median is_____.
• We can find the proportion of observation
that lie within any range of values simply by
finding the area under the curve.
The standard Normal Table
• Because standardizing Normal distributions
makes them all the same, we can use a single
table to find the areas under a Normal
distribution.
• This table is called the standard Normal table.
– It’s inside the front cover of you textbook!
– You will be given this table on the AP exam
The standard Normal Table
CAREFUL!!!!
• Example: Find the proportion of observations
from the standard Normal distribution that
are less than -2.15.
• For the
value of
z = -2.15,
the area
is 0.0158
Using the standard Normal table…
• Caution: the area that we found was to the LEFT of z = 2.15. In this case, that is what we were looking for.
• HOWEVER if the problem had asked for the area lying to
the right of -2.15. What would that answer be?
Area to the Right
1
• The total area under the curve is _____.
• So if 0.0158 lies to the left of -2.15…
0.9842 lies to the right of -2.15.
1 - 0.0158= _______
• Then _____
How do you avoid making a mistake when asked
to find the area to the RIGHT?
• Always sketch the Normal curve, mark the zvalue, and shade the area of interest (aka the
area you are looking for in the problem)
• THEN, when you get you answer, CHECK TO
SEE IF IT IS REASONABLE!!!
Practice
• Exercise 2.29
Putting it all Together:
Solving Problems Involving Normal Distributions
• Step 1: State the problem in terms of the
observed variable x. Draw a picture of the
distribution and shade the area of interest.
– Hint…use σ and μ
• Step 2: Standardize and draw a picture. We
need to standardize x to restate the problem in
terms of a standard Normal variable z. Draw a
new picture to show the area of interest under
our now standard Normal curve.
Putting it all Together:
Solving Problems Involving Normal Distributions
• Step 3: Use the table. Find the are under the
standard Normal curve using Table A. (careful
if the problem asks for the area to the right)
• Step 4: Conclusion. Write your conclusion in
the context of the problem.
– Just saying “the area under the curve that is less
that 2.1” means nothing! Your results should tell
you something about the data.
Example: Cholesterol and Young Boys
•
•
For 14-year-old boys, the mean is μ = 170 milligrams of cholesterol per deciliter of blood
(mg/dl) and the standard deviation σ = 30 mg/dl.
Levels above 240 mg/dl may require medical attention. What percent of 14-year-old boys
have more than 240 mg/dl of cholesterol?
Finding a Value when Given a Proportion
• What if you wanted to know what score you would have to get
in order to place among the top 10% of your class on a test?
• Sometimes, we may be asked to find the observed value with a
given proportion of the observations above or below it.
• To do this, we just read Table A going backwards. In other
words, find the proportion you are looking for in the body of
the table, figure out the corresponding z-score, and then
“unstandardize” to get the observed value.
Inverse Normal Calculation Example
• Scores on the SAT Verbal test in recent years
follow approximately the N(505, 110)
distribution. How high must a student score in
order to place in the top 10% of all students
taking the SAT?
Inverse Normal Calculation Example
• Scores on the SAT Verbal test in recent years follow approximately the
N(505, 110) distribution. How high must a student score in order to place
in the top 10% of all students taking the SAT?
Practice!
• 2.31a, b
• 2.32
Assessing Normality
• The normal distribution provides a good
model for some distributions of real data.
• However, not all distributions are Normal.
• It is important to assess the Normality of
distributions before we assume that they are
normal.
• This will be very important when we learn
about statistical inference procedures (much
later)
Assessing Normality Method 1
• One method for assessing normality is to
construct a histogram or a stemplot and then
see if the graph is approximately bell-shaped
and symmetric about the mean.
• Histograms and stemplots can reveal
important “non-Normal” features of a
distributions such as skewness, outliers, or
gaps and clusters.
Method 1 Continued
• For example, this distribution of vocabulary scores
appears Normal.
– The distribution is bell-shaped, it is roughly symmetric,
there are no gaps or clusters, and there do not appear
to be any outliers.
Method 2
We can improve the effectiveness of
our plots by marking x, x ± s, x ± 2s on
the horizontal axis. Then compare the
counts of observations in each interval
using the empirical rule.
• MEAN = 6.8585
• STDEV = 1.5952
1
21
2.07
x - 3s
129
3.67
x - 2s
5.26
x–s
331
6.86
x
318
8.45
x+s
125
21
10.05
x + 2s
1
11.64
x+3s
Method 2 Continued
1
21
2.07
x - 3s
129
3.67
x - 2s
5.26
x–s
331
6.86
x
318
8.45
x+s
125
21
10.05
x + 2s
1
11.64
x+3s
Method 2 Continued…
• Because the actual counts of our distribution
follow the empirical rule very closely, we can
confirm that the Normal distribution with μ =
6.86 and σ = 1.595 fits the data well.
Method #2 for Assessing Normality
• Construct a normal probability plot. This requires
the use of your graphing calculator.
• Basically…without a calculator..
– Arrange the observed data values from smallest to
largest. Record what percentile of the data set each
value occupies. (i.e., the smallest observation in a set
of 20 is the 5% point, the second smallest is the 10%,
etc)
– Use the standard Normal distribution table to find the
z-scores for these percentiles. (i.e., z = -1.645 is the 5%
point of the standard Normal distribution)
– Plot each data point x against the corresponding z.
Method 2 Continued
• Let’s interpret some Normal probability plots!
Normal Probability Plots
• If you draw a
line, it appears
that most of
the data lies
close to a
straight line.
• HOWEVER,
the points
above and
below the line
represent
outliers in our
data.
The only substantial
deviations from the line are
short horizontal runs of
points.
These represent repeated
observations of the same
value. The phenomenon is
called granularity and does
not effect Normality.
Normal Probability Plots
• This is the Normal
probability plot for
guinea pig survival
times.
• Draw a line through the
leftmost points (smallest
observations)
• Notice that the larger
observations fall
systematically ABOVE
the line.
– In other words, the
right-of-center
observations have
larger values than the
Normal distribution
– Therefore, the
distribution is right
skewed
Normal Probability Plots
• The Normal
probability plot
indicates that the
data is left-skewed
because the
smallest
observations fall
below the line.
Interpreting Normal Probability Plot
Graphing Normal Probability Plots on
your Calculator
• Enter the test scores for Mr. Pryor’s statistics class on
page 116 into L1 on your calculator.
• Press 2nd , Y= (STAT PLOT)
• Turn Plot 1 ON
• Select the type on the lower right
• Data List: L1
• Data Axis: x
• Mark (doesn’t matter)
• Press Zoom, 9:ZoomStat
• You should have a probability plot!
Using your calculator: Finding Areas
with normalcdf
•
•
•
You can find the areas under the Normal curve using normalcdf.
For 14-year-old boys, the mean is μ = 170 milligrams of cholesterol per deciliter of
blood (mg/dl) and the standard deviation σ = 30 mg/dl.
Levels above 240 mg/dl may require medical attention. What percent of 14-yearold boys have more than 240 mg/dl of cholesterol?
Using Your Calculator: invNorm
• Finally, we can use our calculators to calculate raw or
standardized values given the area under the Normal curve or
a relative frequency.
• Scores on the SAT Verbal test in recent years follow
approximately the N(505, 110) distribution. How high must a
student score in order to place in the top 10% of all students
taking the SAT?
Next time on Statistics AP
• The Chapter 2 Test will be on TUESDAY!
– It will cover all of Chapter 2
• SO on Friday we will
– Review Chapter 2
• YOUR HOMEWORK:
– Exercises 2.31(c), 2.33, 2.34, 2.36, 2.40, 2.48
– YOU MAY USE YOUR CALCULATOR BUT YOU STILL
NEED THE SKETCH AND Z-SCORE