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Chapter 6: Problem 73

According to the Centers of Disease Control and Prevention, \(52 \%\) of U.S. households had no landline and only had cell phone service. Suppose a random sample of 40 U.S. households is taken. a. Find the probability that exactly 20 the households sampled only have cell phone service. b. Find the probability that fewer than 20 households only have cell phone service. c. Find the probability that at most 20 households only have cell phone service. d. Find the probability that between 20 and 23 households only have cell phone service.

Short Answer

Expert verified
The answer will be the calculated values from step 2 (Part a), step 4 (Part b), step 6 (Part c) and step 8 (Part d).

Step by step solution

01

Setting up formula for Part a

Here we're asked to find the probability that exactly 20 households only have cell phone service. This corresponds to a binomial probability \(P(20; 40, 0.52)\). Thus we can apply the binomial formula to compute this probability.
02

Applying formula for Part a

Applying the binomial probability formula, we get \(P(20; 40, 0.52) = \binom{40}{20} (0.52)^{20} (1-0.52)^{40-20}\). Calculate this expression to get the value of the probability.
03

Setting up formula for Part b

Here we're asked to find the probability that fewer than 20 households have only cell phone service. This corresponds to the sum of binomial probabilities from x=0 to 19: \(\Sigma_{x=0}^{19} P(x;40,0.52)\).
04

Applying formula for Part b

To get this probability, we have to calculate and sum up 20 binomial probabilities, that is, \(P(x;40,0.52)\) for x from 0 to 19.
05

Setting up formula for Part c

Now we need to find the probability that at most 20 households only have cell phone service. This corresponds to the sum of binomial probabilities from x=0 to 20 which is similar to Part b but includes 20 as well. It can be represented as: \(\Sigma_{x=0}^{20} P(x;40,0.52)\).
06

Applying formula for Part c

Calculate and sum up 21 binomial probabilities, i.e., \(P(x;40,0.52)\) for x from 0 to 20 to get the result.
07

Setting up formula for Part d

We're asked to find the probability that between 20 and 23 households only have cell phone service. This corresponds to the sum of binomial probabilities from x=20 to 23, represented as: \(\Sigma_{x=20}^{23} P(x;40,0.52)\).
08

Applying formula for Part d

Calculate and sum up 4 binomial probabilities, i.e., \(P(x;40,0.52)\) for x from 20 to 23 to get the probability.

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Most popular questions from this chapter

In a game of chance, players draw one cube out of a bag containing 3 red cubes, 2 white cubes, and 1 blue cube. The player wins $$\$ 5$$ if a blue cube is drawn, the player loses $$\$ 2$$ if a white cube is drawn. If a red cube is drawn, the player does not win or lose anything. a. Create a table that shows the probability distribution for the amount of money a player will win or lose when playing this game. b. Draw a graph of the probability distribution you created in part a.

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The Normal model \(N(150,10)\) describes the distribution of scores on the LSAT, a standardized test required by most law schools. Which of the following questions asks for a probability, and which asks for a measurement? Identify the type of problem and then answer the given question. a. A law school applicant scored at the 60 th percentile on the LSAT. What was the applicant's LSAT score? b. A law school applicant scored 164 on the LSAT. This applicant scored higher than what percentage of LSAT test takers?

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Use the table or technology to find the answer to each question. Include an appropriately labeled sketch of the Normal curve for each part. Shade the appropriate region. A section of the Normal table is provided in the previous exercise. a. Find the area to the left of a \(z\) -score of \(0.92\). b. Find the area to the right of a z-score of \(0.92\).
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