Question

A​ four-year study of various brands of bottled water found that 25​% of bottled water is just tap water packaged in a bottle. Consider a sample of sevenseven ​bottled-water brands, and let x equal the number of these brands that use tap water. Complete parts a through d.

a. Is x​ (approximately) a binomial random​ variable?
b. Give the probability distribution for x as a formula.
c. Find p(x = 2).
d. Find P(x <= 1).

Answer 1

a. Yes, x is approximately a binomial random variable. b. The formula for the probability distribution of x is P(x) = C(7, x) *
`(0.25)^x * (0.75)^{(7-x)`. c. P(x = 2) can be found by substituting x = 2 into the formula. d. P(x <= 1) can be found by adding the probabilities of x = 0 and x = 1.

Step-by-step explanation:

a. Yes, x is approximately a binomial random variable because it satisfies the conditions of a binomial distribution: there are a fixed number of trials (7), each trial is independent, and there are only two possible outcomes for each trial (tap water or not tap water).

b. The probability distribution for x can be represented by the formula: P(x) = C(7, x) *
`(0.25)^x * (0.75)^{(7-x)` where C(7, x) is the number of combinations of 7 trials taken x at a time.

c. To find P(x = 2), substitute x = 2 into the formula: P(2) = C(7, 2) *

`(0.25)^2 * (0.75)^5`.

d. To find P(x <= 1), add the probabilities of x = 0 and x = 1: P(x <= 1) = P(0) + P(1).

Answer 2

Answers below

Step-by-step explanation:

a. Is x​ (approximately) a binomial random​ variable?

b. Give the probability distribution for x as a formula.

c. Find p(x = 2).

d. Find P(x <= 1).