Final answer:
Using the binomial probability formula, we can find the probabilities of various scenarios concerning American households using cell phone service exclusively, with 26% as the given success rate and nine households in the sample.
Step-by-step explanation:
The subject question is related to probability and involves the use of the binomial probability formula. With the given proportion of households using only cell phone service (26%), and the sample size of nine households, we can calculate the probabilities for various scenarios using the binomial distribution. The binomial probability formula is:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
where:
- P(X = k) is the probability of having k successes in n trials
- n is the number of trials (households in this case)
- k is the number of successes (households using cell phone exclusively)
- p is the probability of success on an individual trial (0.26 in this case)
- C(n, k) is the combination of n items taken k at a time
a) Exactly three households use a cell phone exclusively
Using the binomial probability formula:
P(X = 3) = C(9, 3) * (0.26)^3 * (1-0.26)^(9-3) which can be calculated to find the probability.
b) More than six households use a cell phone exclusively
For this, sum the probabilities for 7, 8, and 9 households:
P(X > 6) = P(X = 7) + P(X = 8) + P(X = 9)
c) At least five households use a cell phone exclusively
This is the sum of probabilities starting from 5 to 9 households:
P(X ≥ 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9)
Each of these probabilities needs to be calculated individually and then summed.