Download chapter 5 the binomial probability distribution

CHAPTER 5 THE BINOMIAL PROBABILITY DISTRIBUTION
AND RELATED TOPICS
BINOMIAL PROBABILITY DISTRIBUTIONS (SECTIONS 5.2, 5.3 OF
UNDERSTANDABLE STATISTICS)
The binomial probability distribution is a discrete probability distribution controlled by the number of
trials, n, and the probability of success on a single trial, p.
SPSS has three main functions for studying probability distributions.
The PDF & Noncentral PDF (probability density function) gives the probability of a specified value
for a discrete distribution, and probability density function value for a specified value from a
continuous distribution.
The CDF & Noncentral CDF (cumulative distribution function) for a value X gives the probability a
random variable with distribution specified in a sub-function is less than or equal to X.
The Inverse DF gives the inverse of the CDF for continuous distributions. In other words, for a
probability P, Inverse DF returns the value X such that P ≈ CDF(X). Since binomial distribution is a
discrete distribution, it is not covered by this function.
The three functions PDF, CDF, and Inverse DF apply to many probability distributions. To apply PDF and
CDF to a binomial distribution, we need to use the menu selections TransformhCompute followed by
function selections.
TransformhCompute
Dialog Box Responses
Enter name of the Target Variable
Enter the function formula into the Numeric Expression box:
Under the Function group, select PDF & Noncentral PDF for probability; CDF &
Noncentral CDF for cumulative probability for CDF; Inverse DF for inverse cumulative
probability.
Then under Functions and Special Variables, select the sub-function for the specified
distribution.
For example, the PDF function for a binomial distribution is PDF.BINOM(quant, n, prob) and the CDF
function for a binomial distribution is CDF.BINOM(quant, n, prob). Here n is the number of trials, that
is, the value of n in a binomial experiment. prob is the probability of success, that is, the value of p, the
probability of success on a single trial. The input quant is the values of r, the number of successes in a
binomial experiment. You may enter a value for quant, or you may store the values for quant in a variable
(column) and enter the variable name for quant.
Example
A surgeon performs a difficult spinal column operation. The probability of success of the operation is
p = 0.73. Ten such operations are scheduled. Find the probability of success for 0 through 10 successes out of
these ten operations.
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Part IV: SPSS Guide
First enter the possible values of r, 0 through 10, in the first column and name this variable r (Choose 0
under Decimals in the Variable View to have integers for r). We will put the probabilities in the second
column, so name the column prob (Choose 6 under Decimals in the Variable View to give enough
precision.) Fill in the dialog box as shown below.
Click on OK. The results follow.
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Technology Guide Understandable Statistics, 8th Edition
Next use the CDF.BINOM function to find the probability of r or fewer successes. Let us put the
probabilities in the third column and name it cprob (Choose 6 under Decimals in the Variable View to
give enough precision.) Fill in the dialog box as shown below.
Click on OK. The results are shown below. From this screen we see, for example, P(r ≤ 5)= 0.103683.
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Part IV: SPSS Guide
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LAB ACTIVITIES FOR BINOMIAL PROBABILITY DISTRIBUTIONS
1. You toss a coin 8 times. Call heads success. If the coin is fair, the probability of success P is 0.5. What is
the probability of getting exactly 5 heads out of 8 tosses? of exactly 20 heads out of 100 tosses?
2. A bank examiner’s record shows that the probability of an error in a statement for a checking account at
Trust Us Bank is 0.03. The bank statements are sent monthly. What is the probability that exactly two of
the next 12 monthly statements for our account will be in error? Now use the CDF function to find the
probability that at least two of the next 12 statements contain errors. Use this result with subtraction to find
the probability that more than two of the next 12 statements contain errors.
3. Some tables for the binomial distribution give values only up to 0.5 for the probability of success p. There
is symmetry to the values for p greater than 0.5 with those values of p less than 0.5.
(a) Consider the binomial distribution with n = 10 and p = .75. Since there are 0–10 successes possible,
put 0 – 10 in the first column. Use PDF function with this column and store the distribution
probabilities in the second column. Name the second column as P.75. We will use the results in part
(c).
(b) Now consider the binomial distribution with n = 10 and p = .25. Use PDF function with the first
column as quant and store the distribution probabilities in the third column. Name the third column as
P.25.
(c) Now compare the second and third column and see if you can discover the symmetries of P.75 with
P.25. How does P(K = 4 successes with p = .75) compare to P(K = 6 successes with p = .25)?
(d) Now consider a binomial distribution with n = 20 and p = .35. Use PDF on the number 5 to get
P(K = 5 successes out of 20 trials with p = .35). Predict how this result will compare to the probability
P(K = 15 successes out of 20 trials with p = .65). Check your prediction by using the PDF on 15 with
the binomial distribution n = 20, p = .65.
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