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Chapter 6: Problem 74

According to the Centers of Disease Control and Prevention, \(44 \%\) of U.S. households still had landline phone service. Suppose a random sample of 60 U.S. households is taken. a. Find the probability that exactly 25 of the households sampled still have a landline. b. Find the probability that more than 25 households still have a landline. c. Find the probability that at least 25 households still have a landline. d. Find the probability that between 20 and 25 households still have a landline.

Short Answer

Expert verified
The calculated probabilities are as follows: a) The probability that exactly 25 of the households sampled still have a landline: use Step 1 to calculate; b) The probability that more than 25 households still have a landline: use Step 2 to calculate; c) The probability that at least 25 households still have a landline: use Step 3 to calculate; d) The probability that between 20 and 25 households still have a landline: use Step 4 to calculate.

Step by step solution

01

Calculate the Binomial Probability for 25 Households

For a binomial distribution, the formula for calculating probability is \(P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\), where \(n\) is the number of trials, \(p\) is the probability of success, and \(k\) is the number of successes. Here, \(n=60\), \(p=0.44\) and \(k=25\). We use the binomial coefficient \(C(n, k)\) which is the number of ways to choose \(k\) successes from \(n\) trials, calculated as \(\frac{n!}{k!(n-k)!}\). Factorial (\(n!\)) is the product of all positive integers up to \(n\). Thus, for part (a), we calculate \(P(X=25)\) using these values in the binomial probability formula.
02

Calculate the Probability for More Than 25 Households

For part (b), when the problem asks for more than a certain number (25), we need to calculate the complement of the event. The complement of 'more than 25' is '25 or fewer'. So, we calculate \(P(X<=25)\) by adding up \(P(X=k)\) for \(k=0\) to 25 using the binomial formula. Then, we subtract this from 1 to find \(P(X>25)\).
03

Calculate the Probability for At Least 25 Households

For part (c), when the problem says 'at least', it means that number or more. So 'at least 25' means 25 or more. Thus, this is calculated similarly to part (b), but we include 25 in the calculation. So, we calculate \(P(X>=25)\) by first finding \(P(X<25)\) by adding up \(P(X=k)\) for \(k=0\) to 24, then subtracting this from 1 to find \(P(X>=25)\).
04

Calculate the Probability for Between 20 and 25 Households

For part (d), 'between 20 and 25' means 20, 21, 22, 23, 24, or 25. Thus, we find \(P(20<=X<=25)\) by adding up \(P(X=k)\) for \(k=20\) to 25 using the binomial formula.

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