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The Practice of StatisticsThird Edition
Chapter 8:The Binomial and
Geometric Distributions
Copyright © 2008 by W. H. Freeman & Company
Finding Binomial Probabilities
• We will rarely use the formula for finding
the probability of a binomial distribution.
• We will let the graphing calculator do it for
us.
• Look at Example 8.7 on Page 520.
– binompdf
– pdf stands for probability distribution function
binompdf
• Back to the bad switches example
• To find P(X ≤ 1), n = 10, p = .1
• 2nd|(VARS)/0:binompdf (10,.1,0) = .3487
• 2nd|(VARS)/0:binompdf (10,.1,1) = .3874
• So P(X ≤ 1) = .3487 + .3874 = .7361
• 73.61% of all samples will contain no more
than 1 defective switch.
Probability Distribution for
binomial distribution with n = 10
and p = .1 or B(10, .1)
About 74% of all samples will contain no more than 1 bad switch.
A sample size of 10 cannot be trusted to alert the engineer to the
presence of unacceptable items in the shipment.
Doesn’t really matter what the x axis label is… B(10, .1) is always going to
look like this. Every Binomial Distribution has its own look.
http://homepage.divms.uiowa.edu/~mbognar/applets/bin.html
Example 8.8 on page 521.
What is the probability that Corinne will only make at most 7
out of 12 Free Throws, if she is a 75% shooter?
Adding This All up is to Much Work
• P(X ≤ 7) = P(X = 0) + P(X = 1) + P(X + 2)
+…+ P(X = 7) = .1576
• Have to do binompdf for each of these.
• Is there a shorter way?
• of course …
• binomcdf
2nd|(VARS)/A:binomcdf (n, p, X)
n = 12, p = .75, X = 7
2nd|(VARS)/A:binomcdf (12,.75,7) = .1576
So 15.76% of the time Corrine will make no more than
7 out of 12 foul shots. Not unusual.
Remember for calculator:
Binompdf (n,p) Binomcdf (n, p, X)
“Bananapox”
Binomial Mean and Standard
Deviation
• If count X is binomial with n observations and p
probability, what is the mean μ?
– Corinne shoots 12 free throws with 75%
accuracy, we would expect (12)(.75) = 9 to be μ.
• So in general, the mean of a Binomial Random
Variable is μ = np.
• Standard Deviation σ = √(np(1 – p)).
– Trust me on that one.
• Both on formula sheet.
WARNING: THESE FORMULAS ARE
GOOD ONLY FOR BINMOMIAL
DISTRIBUTIONS. THEY CANNOT BE
USED FOR OTHER DISCRETE RANDOM
VARIABLES!!!
Bad Switches
• Sample size n = 10
• Probability of defective switch p = .1
• Mean µ = np = (10)(.1) = 1
• Standard Deviation α = √(np(1 – p))
• = √((10)(.1)(1 - .1)) = .9487
Normal Approximation of
Binomials
• A useful fact – as the number of trials n gets
larger, the binomial distribution gets close
to a Normal distribution. When n is large,
we can use Normal probability calculations
to approximate hard-to-calculate binomial
probabilities
http://homepage.divms.uiowa.edu/~mbognar/applets/bin.html
Attitudes Towards Shopping
• A survey asked a nation-wide random sample of 2500
adults if they agreed or disagreed that “I like buying new
clothes, but shopping is often frustrating and time-
consuming.” Suppose that in fact 60% of all adult U.S.
residents would say “Agree”. What is the probability that
1520 or more of the sample agree?
Independent?
Attitudes Towards Shopping
• A survey asked a nation-wide random sample of 2500
adults if they agreed or disagreed that “I like buying new
clothes, but shopping is often frustrating and time-
consuming.” Suppose that in fact 60% of all adult U.S.
residents would say “Agree”. What is the probability that
1520 or more of the sample agree?
Find the mean and standard deviation.
Attitudes Towards Shopping
• A survey asked a nation-wide random sample of 2500
adults if they agreed or disagreed that “I like buying new
clothes, but shopping is often frustrating and time-
consuming.” Suppose that in fact 60% of all adult U.S.
residents would say “Agree”. What is the probability that
1520 or more of the sample agree?
http://homepage.divms.uiowa.edu/~mbognar/applets/bin.html
Use the normal calculations with this mean and
standard deviation.
Assignment
• “Binomial pdfs and cdfs” worksheet
• “Normal approximation to the binomial distribution” worksheet
• Calculator (you need to know these calculator functions):https://youtu.be/F6JBimUE43U?list=PLkIselvEzpM7N8zVRRUl7V8aTdoT
sJ919
• Read Technology Toolbox on pages 530 – 532 and do on your
calculator.
• Read pages 540 – 548