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5.2
The Standard Normal Distribution
(Read the book, examples, jot down comments, do the problems suggested in the
homework. Bring your questions to class, my office, or the Math Science Center.)
Normal Distribution: a distribution of a continuous random variable with a graph
that is symmetric and bell-shaped.
Uniform Distribution: a probability distribution in which every value of the
random variable is equally likely. The values spread evenly over the range of
possibilities. The graph of a uniform distribution results in a rectangular shape.
*** Read Class Length Example on page 227.
Density Curve (or probability density function) is a graph of a continuous
probability distribution. It must satisfy the following properties:
1. Total area under the curve is 100% or 1 (there is a correspondence
between area and probability).
2. Every point on the curve must have a vertical height that is zero or
greater.
*** Read Class Length Example on page 228.
Normal Density Curves
The shapes of these curves are determined by μ and σ.
*** See figure 5-4 on pg. 229.
Standard Normal Distributions
The standard normal distribution is a normal probability distribution with
mean μ = 0 and standard deviation σ = 1, and the total area under its density
curve is equal to 1.
Finding Probabilities When Given z-Scores
TABLE A-2 gives the area under the standard normal curve to the left of a given
z-score. One page of the table uses negative z-scores and the other page uses
positives. Each value in the body of the table is a cumulative area from the left
up to a vertical line at a specific z-score.
*** Read examples on pages 232 and 233
Note: If Random Variable is Continuous P( z = a) = 0, for any number a.
Note difference from discrete!!
Note: z scores are positive or negative, but probabilities are from 0 to 1
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Let’s practice finding probabilities here:
Recall the Empirical Rule from Chapter 2
If the distribution is approximately bell shaped, then:
About .......... of the scores fall within 1 standard deviation from the mean
About .......... of the scores fall within 2 standard deviation from the mean
About .......... of the scores fall within 3 standard deviation from the mean
Now find the following probabilities and compare your results with the
percentages mentioned in the empirical rule:
(a) P(-1 < z < 1)
(b) P(- 2 < z < 2)
(c) P(-3 < z < 3)
Finding z Scores from Known Areas (Probabilities)
Often probabilities are given as percents.
Let’s practice finding z scores here:
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5.3
Applications of Normal Distributions
These are more realistic cases where μ ≠ 0 and σ≠ 0.
Finding probabilities (areas)
Step 1: Draw the graph shading the area desired. Label the mean
and the specific x-values being considered.
Step 2: Find the z-Score for each x-value involved.
Z-SCORE: z 
x

Step 3: Use table A-2 to find the cumulative left area bounded by z.
Step 4: Answer the problem.
Let’s practice finding probabilities here:
Using the TI-83 to obtain probabilities, percentages, areas
Press
2nd, VARS
Select
2:normalcdf(
Type
left endpoint, right endpoint, , )
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Finding Values from Know Areas (Probabilities)
Working BACKWARDS
Step 1: Draw the graph, shade and label the given area, and identify the
x-value being sought.
Step 2: Find the cumulative left area bounded by x.
Step 3: Use table A-2 to find the z-score.
(Go backwards! From the main body of the table to the z-score)
Step 4 Find the score, x, by using the formula
x    z
Let’s practice finding scores here:
Using the TI-83 to obtain normal scores, percentiles
Press
2nd, VARS
Select
3:invNorm(
Type
total area to the left of the desired value, , )
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5.4 – Which Statistics Make Good Estimators of Parameters?
When using a sample statistic to estimate a population parameter, some
statistics are good in the sense hat they target the population parameter and are
therefore likely to yield good results. Such statistics are called unbiased
estimators. Other statistics are not so good (because they are biased
estimators).
Here is a summary:


Statistics that target population parameters: Mean, Variance,
Proportion.
Statistics that do not target population parameters: Median, Range,
Standard Deviation. Although the sample standard deviation does not
target the population standard deviation, the bias is relatively small in
large samples, so s is often used to estimate σ. Consequently, means,
proportions, variances, and standard deviation will all be considered as
major topics in following chapters, but the median and range will rarely be
used.
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5.5
The Central Limit Theorem
Sampling Distribution of the Mean: The probability distribution of sample
means, for a sample of size n.
Central Limit Theorem
Given:
1. That random variable x has a distribution (which may or may not be
normal) with mean μ and standard deviation σ.
2. A simple random sample of size n is selected from this population
(the samples are selected so that all possible samples of size n
have the same chance of being selected); then:
Conclusions:
1. The distribution of sample means x will, as the sample size
increases, approach a normal distribution.
2. The mean of the sample means is the population mean, μ.
(That is, the normal distribution from Conclusion 1 has mean μ.)
x  
x 
3. The standard deviation of the sample means is

n
(That is, the normal distribution from Conclusion 1
has standard deviation
x
Notation:

)
n
is often called the standard error of the mean.
N(μ ,

n
)
Normal distribution with parameters μ and

n
Practical Rules:
1. For samples of size n larger than 30, the distribution of the sample
means can be approximated reasonably well by a normal
distribution. The approximation gets better as the sample size n
gets larger.
2. If the original population is itself normally distributed, then the sample
means will be normally distributed for any n (not just those
larger than 30).
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Applying the Central Limit Theorem
-
When working with an individual value from a normally distributed
population, use the methods of section 5-3. Use
-
z
x

When working with a mean for some sample (or group) be sure to use the
value

for the standard deviation of the sample means.
n
(x  )
Use z 

( )
n
*** Read Example pages 262-264
Let’s practice using the Central Limit Theorem here:
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Interpreting Results
Recall the Rare Event Rule:
If, under a given assumption, the probability of a particular observed
event is exceptionally small, we conclude that the assumption is probably not
correct.
*** Read Example pages 265-266
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5.6 – Normal Approximation to Binomial Distribution
Let’s recall the conditions required for a Binomial Distribution (from
section 4.3)
1. The procedure must have a fixed number of trials.
2. The trials must be independent.
3. Each trial must have all outcomes classified into two categories.
4. The probabilities must remain constant for each trial.
Methods used in 4.3
Formula
Calculator: built in program binompdf(n,p,x) or binomcdf(n,p,x)
*** Example: 74% of U.S. households have answering machines. If we select
2500 households at random, what is the probability that more than 650 have
answering machines.
P(x > 650) = P(651)+P(652)+P(653)+...+P(2499)+P(2500)
The TI-83 plus may handle this calculation but the regular TI-83 can't.
In this section we are going to use the normal distribution to approximate a
binomial distribution.
Continuity Correction Factor
Notice that we are using a CONTINUOUS model to approximate a DISCRETE
model. We need to make an adjustment for continuity.
*** Read procedure for continuity corrections on pages 275-6.
*** In the following exercises, the given values are discrete. For each problem do
the following 3 steps:
a) Identify the region of the normal curve that corresponds to the indicated
probability.
b) Describe the problem in terms of areas, that is,
Area to the right of …….
Area to the left of ………
Area between ….. and ……
c) Give the continuity correction factor.
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Normal Distribution as Approximation to Binomial Distribution
If np  5 , and nq  5 , then the binomial random variable has a probability
distribution that can be approximated by a normal distribution with the mean an
standard deviation given as
  np ,
  npq
Procedure for Normal Approximation to Binomial
There are 5 Steps to follow:
Back to the example from the first page: 74% of U.S. households have answering
machines. Sample 2500 households. Estimate the probability that more than 650
have answering machines.
1) Test if Normal is Appropriate ( np  5 , and
np = .74 X 2500 = 1850
nq = .26 X 2500 = 650
Both are greater than 5
nq  5 )
2) Find the mean and the standard deviation
  np
  npq
3) Draw a normal distribution centered about μ, and identify the region
representing the probability to be found.
4) Find the continuity correction factor
5) Estimate the probability
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*** Read Example on pages 273-275
*** Read Example on pages 276-278
Using Probabilities to Determine when Results Are Unusual
-
Unusually high: x successes among n trials is an unusually high number
of successes if P(x or more) is very small (such as 0.05 or less)
Unusually low: x successes among n trials is an unusually low number of
successes if P(x or fewer) is very small (such as 0.05 or less)
Interpreting Results
Recall the Rare Event Rule:
If, under a given assumption, the probability of a particular observed
event is exceptionally small, we conclude that the assumption is probably not
correct.
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