Section 2.2: Binomial Exact Tests
Some Notation:
 Let  = the true percentage engineering majors that are female = 10.9%,
 let ˆ = the observed percentage = 0%, and
 let n = number in sample, 25
Below are the outcomes from 1000 trials done in JMP. This graphic is used to
understand the pattern in the number of females in our engineering class of size 25.
Below is another set of 1000 trials done in JMP. Notice that the empirical distribution
has changed slightly from the one obtained above.
Questions:
1. What is the estimated probability of observing 0 females in a class of 25 students?
Does this value change much when a second empirical distribution is obtained?
Discuss.
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Using the Binomial Model for Inference
Assumptions for a Binomial Probability Model:
The Number of “Successes” follows a binomial probability model
if the following are meet.
1. There are a fixed number of observations under study.
2. There are only two possible outcomes. Traditionally one is
labeled a “Success” and the other a “Failure”.
3. The probability of success remains constant.
4. The observations under study are independent (one person
being selected does not influence another being selected).
Discuss whether or not these assumptions are reasonable for the Engineering Female
students example.
JMP allow us to easily obtain the obtain the binomial probability model.
The first step is to create a column that will be used to represent the Number of
Females selected. This column should include all possible values. If there are a total of
25 people, then the range of possible values for the Number of Females goes from 0 to
25.
Create a second column that will be used to compute the individual probabilities from
the binomial distribution. Right click on this second column and select Formula.
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In the Functions (grouped) drop-down box, select Probability > Binomial Probability.
This function requires three arguments. The first is  = the true percentage = 10.9%,
the second is n = Number of Observations, and the last argument is the Number of
Females column.
The following shows how a graphical display can be obtained in JMP. Select Graph >
Chart in JMP.
The following is a snipit of the binominal probabilities display along the graphical display
of this distribution that was presented earlier.
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Questions
1. What do these probability values mean?
2. Which value has the highest probability?
3. What is the probability of seeing 0 females in a sample of 25?
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Example 2.1.2 (Investigating Statistical Concepts, Applications, and Methods:
Investigation 1.1.1). In November 2000 researchers began conducting medical
examinations and environmental surveys of workers employed at a microwave popcorn
production plant. As part of this study, current employees at the plant underwent
spirometric testing which measures FVC (forced vital capacity) which is the volume of air
that can be maximally forcefully exhaled. There was a total of 116 employees who were
underwent this testing. On this test, 31 employees had abnormal results, including 21
with airway obstruction.
Questions:
1. What is the proportion of employees with airway obstruction?
2. Do you feel this number is higher than it should be?
3. What additional information is necessary to determine whether or not this
production plant is causing an increased level of airway obstruction?
In an effort to make fair comparisons, additional information from the National Health
and Nutrition Examination Survey was obtained. From this survey it was found that
5.5% is the baseline rate of airway obstruction in the general population.
Questions:
1. What is the expected number of individuals with airway obstruction when
116 are observed?
2. Do you think seeing 21 employees is “too much” deviation from what is
expected?
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The following output from the binominal probability model shows what would
happen if the production plant had no impact on airway obstruction.
3. What is the probability of seeing 21 employees with airway obstruction out
of a sample of 116?
4. What is the probability of seeing 21 or more employees with airway
obstruction out of a sample of 116 employees?
5. Do the results from Problem #3 and #4 support or refute the claim made by
the employees that this production plant is causing elevated levels of airway
obstruction? Discuss.
6. What is a reasonable cutoff for the number of employees we’d expect to see
with airway obstruction out of a sample of 116?
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Example 2.1.3 Smokers and Non-smokers tend to have different FVC measurements as
smoking is known to reduce lung volume. Consider the following breakdown of smokers
and non-smokers from the microwave-popcorn production plant.
Smokers vs
Nonsmokers
Smokers
Non-Smokers
Total
Number
with Airway
Obstruction
8
13
21
Number of
Employees
64
52
116
Rates of Airway
Obstruction in General
Population
8%
2.3%
5.5%
Questions: Answer the following questions for the Smokers group.
1.
What is the expected number of smokers with airway obstruction?
2.
Do you think seeing 8 employees is “too much” deviation from what is
expected?
Using JMP, construct the appropriate binomial probability model and answer the
following questions.
3.
What is the probability of seeing 8 employees with airway obstruction
out of a sample of 64?
4.
What is the probability of seeing 8 or more employees with airway
obstruction out of a sample of 64 employees?
5.
Do the results from Problem #3 and #4 support or refute the claim made
by the employees that this production plant is causing elevated levels of
airway obstruction? Discuss.
6.
What is a reasonable cutoff for the number of smokers we’d expect to
see with airway obstruction out of a sample of 64?
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Questions: Answer the following questions for the Non-Smokers group.
1. What is the expected number of non-smokers with airway obstruction?
2. Do you think seeing 13 employees is “too much” deviation from what is
expected?
Using JMP, construct the appropriate binomial probability model and answer the
following questions.
3. What is the probability of seeing13 employees with airway obstruction out of
a sample of 52?
4. What is the probability of seeing 13 or more employees with airway
obstruction out of a sample of 52 employees?
5. Do the results from Problem #3 and #4 support or refute the claim made by
the employees that this production plant is causing elevated levels of airway
obstruction? Discuss.
6. What is a reasonable cutoff for the number of non-smokers we’d expect to
see with airway obstruction out of a sample of 52?
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