According to a study, $40 \%$ of people have more than $ \$10,000$ in credit card debt. Suppose that in a particular bank there are currently 60 customers waiting in line.
Use a calculator to find the probability that of those 60 customers, between 21 and 23 of them have more than $ \$10,000$ in credit card debt.
Round your answers to three decimal places.
Provide your answer below:
$\sigma_{p^{\wedge}}=\square$
$P(21 \leq X \leq 23)=\square$
Answer (2)
The standard deviation of the sample proportion is approximately 0.063, and the probability that between 21 and 23 customers have more than $10,000 in credit card debt is approximately 0.273.
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Calculate the standard deviation of the sample proportion: σ p ∧ = 60 0.4 ( 1 − 0.4 ) ≈ 0.063 .
Calculate the probability of X = 21: P ( X = 21 ) = ( 21 60 ) ( 0.4 ) 21 ( 0.6 ) 39 ≈ 0.078 .
Calculate the probability of X = 22: P ( X = 22 ) = ( 22 60 ) ( 0.4 ) 22 ( 0.6 ) 38 ≈ 0.092 , and the probability of X = 23: P ( X = 23 ) = ( 23 60 ) ( 0.4 ) 23 ( 0.6 ) 37 ≈ 0.102 .
Calculate the probability that X is between 21 and 23: P ( 21 ≤ X ≤ 23 ) = 0.078 + 0.092 + 0.102 ≈ 0.273 .
Explanation
Problem Analysis We are given that 40% of people have more than $10 , 000 in credit card debt. In a bank with 60 customers, we want to find the probability that between 21 and 23 of them have more than $10 , 000 in credit card debt.
Define the variables and the distribution Let p = 0.4 be the probability that a person has more than $10 , 000 in credit card debt, and let n = 60 be the number of customers. We can model this situation using a binomial distribution, where X is the number of customers who have more than $10 , 000 in credit card debt. We want to find P ( 21 ≤ X ≤ 23 ) , which is the sum of the probabilities P ( X = 21 ) , P ( X = 22 ) , and P ( X = 23 ) .
Calculate the standard deviation First, let's calculate the standard deviation of the sample proportion, which is given by the formula: σ p ∧ = n p ( 1 − p ) = 60 0.4 ( 1 − 0.4 ) = 60 0.4 ( 0.6 ) = 60 0.24 = 0.004 ≈ 0.063
Calculate the probabilities P(X=21), P(X=22), and P(X=23) Now, we need to calculate the probabilities P ( X = 21 ) , P ( X = 22 ) , and P ( X = 23 ) using the binomial probability formula: P ( X = k ) = ( k n ) p k ( 1 − p ) n − k
So we have: P ( X = 21 ) = ( 21 60 ) ( 0.4 ) 21 ( 0.6 ) 39 P ( X = 22 ) = ( 22 60 ) ( 0.4 ) 22 ( 0.6 ) 38 P ( X = 23 ) = ( 23 60 ) ( 0.4 ) 23 ( 0.6 ) 37
Using a calculator, we find: P ( X = 21 ) ≈ 0.078 P ( X = 22 ) ≈ 0.092 P ( X = 23 ) ≈ 0.102
Calculate the final probability and round the answers Finally, we sum these probabilities to find P ( 21 ≤ X ≤ 23 ) :
P ( 21 ≤ X ≤ 23 ) = P ( X = 21 ) + P ( X = 22 ) + P ( X = 23 ) ≈ 0.078 + 0.092 + 0.102 = 0.272
Rounding to three decimal places, we get: σ p ∧ ≈ 0.063 P ( 21 ≤ X ≤ 23 ) ≈ 0.273
State the final answer Therefore, the standard deviation of the sample proportion is approximately 0.063, and the probability that between 21 and 23 customers have more than $10 , 000 in credit card debt is approximately 0.273.
Examples
This type of probability calculation is useful in risk assessment for financial institutions. For example, a bank might use this to estimate the number of customers likely to default on their credit card debt, which helps in planning for potential losses. By understanding the probability of a certain number of customers having high debt, the bank can make informed decisions about lending policies and reserve requirements. This ensures they remain financially stable and can continue to offer services to their customers. The binomial distribution helps quantify these risks and make data-driven decisions.
