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Chapter 5: Problem 97

On average, 20 households in 50 own answering machines. a. Using the Poisson formula, find the probability that in a random sample of 50 households, exactly 25 will own answering machines. b. Using the Poisson probabilities table, find the probability that the number of households in 50 who own answering machines is i. at most 12 ii. 13 to 17 iii. at least 30

Short Answer

Expert verified
a. The exact calculation depends on the ability to compute factorial and exponential functions, but it can be done using a calculator. Use the Poisson formula to substitute the given values into the formula and compute the probability. b. i. Sum the probabilities for owning a machine from 0 to 12, ii. Sum the probabilities from 13 to 17, iii. Sum the probabilities from 30 to the end of your table.

Step by step solution

01

Calculate the Poisson parameter

First, calculate the Poisson parameter, \( \lambda \), which is the average number of successes - in this case, the average number of households owning an answering machine. Since 20 households out of 50 own answering machines on average, we have \( \lambda = 20 \). We will use this value for the calculations in the following steps.
02

Direct application of the Poisson formula

Using the Poisson formula: \( P(X=k) = \frac{\lambda^ke^{-\lambda}}{k!} \), where \( k = 25 \), \( p = \frac{20^{25}*e^{-20}}{25!} \). This is the answer for part (a).
03

Use Poisson Probabilities Table for part (b)

For part (b), refer to the Poisson probabilities table; each sub-problem refers to different range of values for the number of households owning an answering machine. Use the table to find and add the needed probabilities: i. 'At most 12' would mean summing the probabilities from 0 to 12. ii. '13 to 17' would mean summing the probabilities from 13 to 17. iii. 'At least 30' would mean summing the probabilities from 30 to the end of your table (or to the highest possible number of households if your table does not go to infinity).

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Most popular questions from this chapter

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