a.
Find the
a.
Answer to Problem 56CE
The probability that fewer than five employees steal is 0.0262.
Explanation of Solution
In order to qualify as a binomial problem, it must satisfy the following conditions.
- There are only two mutually exclusive outcomes, employees steal from their company and employees do not steal from their company.
- The number of trials is fixed, that is 50 people.
- The probability is constant for each trial, which is 0.20.
- The trials are independent of each other.
Thus, the problem satisfies all the conditions of a binomial distribution.
The mean can be obtained as follows:
The expected number of employees who steal is 10.
The standard deviation can be obtained as follows:
The standard deviation of number of employees who steal is 2.83.
The conditions for normal approximation to the binomial distribution are checked below:
The number of employees
Condition 1:
The condition 1 is satisfied.
Condition 2:
The condition 2 is satisfied.
The conditions 1 and 2 for normal approximation to the binomial distribution are satisfied.
Let the random variable X be the number of employees who steal from their company follows normal distribution with population mean
The probability that fewer than five employees steal can be obtained as follows:
Step-by-step procedure to obtain probability of Z less than –1.94 using Excel:
- Click on the Formulas tab in the top menu.
- Select Insert function. Then from category box, select Statistical and below that NORM.S.DIST.
- Click OK.
- In the dialog box, Enter Z value as –1.94.
- Enter Cumulative as TRUE.
- Click OK, the answer appears in Spreadsheet.
The output obtained using Excel is represented as follows:
From the above output, the probability of Z less than –1.94 is 0.0262.
Now consider the following:
Therefore, the probability that fewer than five employees steal is 0.0262.
b.
Find the probability that more than five employees steal.
b.
Answer to Problem 56CE
The probability that more than five employees steal is 0.9441.
Explanation of Solution
The population mean
The probability that more than five employees steal can be obtained as follows:
Step-by-step procedure to obtain probability of Z less than –1.59 using Excel:
- Click on the Formulas tab in the top menu.
- Select Insert
function . Then from category box, select Statistical and below that NORM.S.DIST. - Click OK.
- In the dialog box, Enter Z value as –1.59.
- Enter Cumulative as TRUE.
- Click OK, the answer appears in Spreadsheet.
The output obtained using Excel is represented as follows:
From the above output, the probability of Z less than –1.59 is 0.0559.
Now consider the following:
Therefore, the probability that more than five employees steal is 0.9441.
c.
Find the probability that exactly five employees steal.
c.
Answer to Problem 56CE
The probability that exactly five employees steal is 0.0297.
Explanation of Solution
The probability that exactly five employees steal can be obtained as follows:
From the part (a), the probability of Z less than –1.94 is 0.0262.
From the part (b), the probability of Z less than –1.59 is 0.0559.
Now consider,
Therefore, the probability that exactly five employees steal is 0.0297.
d.
Find the probability that more than 5 but fewer than 15 employees steal.
d.
Answer to Problem 56CE
The probability that more than 5 but fewer than 15 employees steal is 0.8882.
Explanation of Solution
The probability that more than 5 but fewer than 15 employees steal can be obtained as follows:
From the part (b), the probability of Z less than –1.59 is 0.0559.
Step-by-step procedure to obtain probability of Z less than 1.59 using Excel:
- Click on the Formulas tab in the top menu.
- Select Insert function. Then from category box, select Statistical and below that NORM.S.DIST.
- Click OK.
- In the dialog box, Enter Z value as 1.59.
- Enter Cumulative as TRUE.
- Click OK, the answer appears in Spreadsheet.
The output obtained using Excel is represented as follows:
From the above output, the probability of Z less than 1.59 is 0.9441.
Now consider,
Therefore, the probability that more than 5 but fewer than 15 employees steal is 0.8882.
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Chapter 7 Solutions
EBK STATISTICAL TECHNIQUES IN BUSINESS
- 6. Show that 1{AU B} = max{1{A}, I{B}} = I{A} + I{B} - I{A} I{B}; I{AB} = min{I{A}, I{B}} = I{A} I{B}; I{A A B} = I{A} + I{B}-21{A} I {B} = (I{A} - I{B})².arrow_forwardTheorem 3.5 Suppose that P and Q are probability measures defined on the same probability space (2, F), and that F is generated by a л-system A. If P(A) = Q(A) for all A = A, then P = Q, i.e., P(A) = Q(A) for all A = F.arrow_forward6. Show that, for any random variable, X, and a > 0, Lo P(x -00 P(x < xarrow_forward5. Suppose that X is an integer valued random variable, and let mЄ N. Show that 8 11118 P(narrow_forward食食假 6. Show that I(AUB) = max{1{A}, I{B}} = I{A} + I{B} - I{A} I{B}; I(AB)= min{I{A}, I{B}} = I{A} I{B}; I{A A B} = I{A} + I{B}-21{A} I{B} = (I{A} - I{B})². -arrow_forward11. Suppose that the events (An, n ≥ 1) are independent. Show that the inclusion- exclusion formula reduces to P(UAL)-1-(1-P(Ak)). k=1 k=1arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
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