Concept explainers
a.
Explain the reason for a negative binomial distribution to be appropriate for a given random variable and define a formula for
a.
Answer to Problem 31P
There are binomial trials with probability of success 0.80 and failure 0.20 with n as a random variable that represents a number of contacts needed to get 12th successful sale.
The formula for probability that Person S makes 12 successful sales is.
Explanation of Solution
Calculation:
There are binomial trials with probability of success
Let n follow a negative binomial distribution that represents the number of contacts needed to get 12th sale.
Probability of success
Probability of failure
Number of successful sales
Negative binomial probability:
The probability that kth success occurs on nth trial is given below:
Here, n is the number of trials in which kth success occurs, k is the number of successes,
The formula for probability that Person S makes 12 successful sales is given below:
Thus, the formula for probability that Person S makes 12 successful sales is
b.
Calculate the given probabilities.
b.
Answer to Problem 31P
The probability that Person S needs 12 contacts to get bonus is 0.0687.
The probability that Person S needs 13 contacts to get bonus is 0.1649.
The probability that Person S needs 14 contacts to get bonus is 0.2144.
Explanation of Solution
Calculation:
The probability that Person S needs 12 contacts to get bonus is given below:
Thus, the probability that Person S needs 12 contacts to get bonus is 0.0687.
The probability that Person S needs 13 contacts to get bonus is given below:
Thus, the probability that Person S needs 13 contacts to get bonus is 0.1649.
The probability that Person S needs 14 contacts to get bonus is given below:
Thus, the probability that Person S needs 14 contacts to get bonus is 0.2144.
c.
Calculate the probability that Person S will require 12 to 14 contacts to get bonus.
c.
Answer to Problem 31P
The probability that Person S will require 12 to 14 contacts to get bonus is 0.4480.
Explanation of Solution
Calculation:
The probability that Person S will require 12 to 14 contacts to get bonus is calculated as follows:
Thus, the probability that Person S will require 12 to 14 contacts to get bonus is 0.4480.
d.
Calculate the probability that Person S will require more than 14 contacts to get bonus.
d.
Answer to Problem 31P
The probability that Person S will require more than 14 contacts to get bonus is 0.5520.
Explanation of Solution
Calculation:
The probability that Person S will require more than 14 contacts to get bonus is calculated follows:
Thus, the probability that Person S will require more than 14 contacts to get bonus is 0.5520.
e.
Calculate the
Calculate the standard deviation of
e.
Answer to Problem 31P
The expected value of
The standard deviation of
Explanation of Solution
Calculation:
The expected value of
Thus, the expected value of
The standard deviation of
Thus, the standard deviation of
Interpretation:
The expected number of contacts that the twelfth sale will occur is 15 with a standard deviation of 1.94.
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