Binomial Distribution
We’ve been using simulation techniques to create a picture or what outcomes seem reasonable based
on the “no preference” or “no discrimination” scenario. However, instead of repeatedly “flipping” coins
and ____________________ probabilities, we will focus on the situation in which we simulate this
experiment an ____________________ number of times. This will provide us with the
___________________ probabilities of interest. Also, this will fix the “problem” of different people
getting different answers to the research question using the same data.
Consider the Helper vs. Hinderer study once again. The following graphic shows what the distribution of
“no preference” would look like if we kept flipping 16 coins over and over again (i.e. an infinite number
of times), each time counting and plotting the number of heads obtained. This is known as the
____________________ distribution.
Using the Binomial Distribution to Make Decisions
To answer research questions which involve a single categorical variable, statisticians do not necessarily
always turn to simulations involving taking random samples or flipping coins over and over again.
Instead, we use the Binomial distribution which shows us how often each of our possible outcomes
would occur if we repeated the previous simulations an infinite number of times.
Binomial Distribution – when can we use it?
The Binomial distribution can be used whenever the following conditions are met:
1. There are a fixed number of trials, _____.
2. There are only _____ possible outcomes for each trial – a “success” and a “failure”
3. The probability of ____________________ (p) remains the same for __________ trial.
4. The trials are ____________________.
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Example: Check the conditions in context for the Helper vs. Hinderer study.

There are a fixed number of trials, n.

There are only 2 possible outcomes for each trial – a “success” and a “failure”.

The probability of success (p) remains the same for each trial.

The trials are independent.
Using JMP to Calculate Binomial Probabilities
There is a file on the course website called Binomial_Probabilities.jmp which we will use to calculate
binomial probabilities.
We will need the following pieces of information to model the Helper vs. Hinderer situation:

p = probability infants choose the helper toy (assuming no toy preference) = __________

n = sample size (number of infants in the study) = __________
To enter these values into JMP, right-click on the n column and select Formula:
Then, change the value of n to 16 (since there were 16 infants in the study) as follows:
Click Apply and OK.
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Similarly, right-click on the p column and select Formula. Change the value to p = 0.50 (since there is a
50% chance of choosing the helper toy, assuming infants really have no preference.)
Click Apply and OK. JMP should return the following output:
The graph below shows the individual binomial probabilities for this scenario.
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Questions:
1. What do the “Individual Binomial Probabilities” represent?
2. What do the “Cumulative Binomial Probabilities” represent?
3. What is the probability of seeing exactly 8 infants choose the helper toy, assuming they have no
preference?
4. What is the probability of seeing exactly 14 infants choose the helper toy, assuming they have
no preference?
5. What is the probability of seeing 14 or more infants choose the helper toy, assuming they have
no preference?
6. Does the p-value from Question 5 provide evidence that infants prefer the helper toy? Explain.
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Example Revisited: Are Females Passed Over for Managerial Training?
Recall, this example involves the possibly discrimination against female employees. Suppose a large
supermarket chain occasionally selects employees to receive management training. A group of women
employees have claimed that female employees are less likely than male employees of similar
qualifications to be chosen for training.
The large employee pool that can be tapped for management training is 60% female and 40% male;
however, since the management program began, 9 of the 20 (40%) employees chosen for management
training were female.
Research Question – Is there evidence of gender discrimination against females?
We have already carried out a simulation study by repeatedly taking samples of size 20 from a
population which was 60a5 female.
Question:
7. Based on the simulated results above, what is the estimated probability of observing 9 or fewer
women selected out of 20 if there is no discrimination (i.e. find the estimated p-value)?
However, we can find the exact probability of observing 9 or fewer women selected out of 20 using the
Binomial distribution. First, we’ll need to make sure the conditions for the Binomial distribution have
been satisfied.

There are a fixed number of trials, n.

There are only 2 possible outcomes for each trial – a “success” and a “failure”.
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
The probability of success (p) remains the same for each trial.

The trials are independent.
Now, we’ll need the following information to set up the JMP file for this scenario:

p = proportion of females in the employee pool (no discrimination) = __________

n = sample size (number of employees selected) = __________
To enter these values into JMP, right-click on the n column and select Formula. Then change the value
of n to 20 as follows:
Click Apply and then OK.
Next, right-click on the p column and select Formula. Then change the value of p to 0.60 as follows:
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Click Apply and then OK. JMP should then return the following output:
The graph below shows the individual Binomial probabilities:
Questions:
8. Based on the Binomial probabilities, what is the probability of observing 9 or fewer women
selected if there really is no discrimination? How does this compare to the estimated
probability found from the simulation (Question 7 above)?
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9. If the supermarket chain cannot prove that the males selected for the management program
were more qualified than the women in the pool, is there evidence of gender discrimination?
Explain.
Example: Effectiveness of an Experimental Drug
Suppose a commonly prescribed drug for relieving nervous tension is believed to be only 70% effective.
A new drug was formulated and administered to a random sample of 20 adults who were suffering from
nervous tension. Of those 20 adults, 18 experienced relief when taking the new drug.
Research Question – Is the new experimental drug more effective than the old drug?
Questions:
10. Define the population of interest.
11. Define the variable of interest.
12. Can this scenario be modeled using the Binomial distribution? Describe in context how it meets
the four conditions of a Binomial.
13. Assuming that the new drug is no better than the previous drug, how many of the 20 adults
would you expect to experience relief?
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14. Recall that in the actual study, 18 out of the 20 adults experienced relief. Researchers were
hoping to show that the new drug was more effective than the previous drug (i.e. better than
70% effective). What possible outcomes for the number of subjects who experience relief are at
least as extreme as the observed results? Explain.
15. Assuming the new drug is no better than the previous drug, what is the probability of observing
results at least as extreme as observed? You’ll need to use JMP to find this probability. (Hint:
You’ll use your answer from Question 14 to help you find the appropriate probability.)
16. Does this study provide evidence that the new drug is superior to the one previously prescribed?
Explain. (Hint: You’ll use your answer from Question 15 in your explanation.)
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