Question

Is the bottled water you are drinking really purified water? In a four-year study of bottled water brands conducted by the Natural Resources Defense Council found that 25% of bottled water is just tap water packed in a bottle. Consider a sample of five brands of bottled water and let X equal the number of these brands that use tap water.

1. Explain why X is (approximately) a binomial random variable.

2. Find that the P (x = 2)

3. Find that the P (x <= 1)

4. Calculate the expected value and standard deviation of the distribution.

Answer 1

1) There is n = 5 identical trials.

Each trials are independent of the others.

Each trial has two outcomes, success and failures

The probability of success P = 0.25 is the same for each outcome.

2) P(X = x) = nCx * p^x * (1 - p)^{n - x}

P(X = 2) = 5C2 * (0.25)^2 * (0.75)^3 = 0.2637

3) P(X < 1) = P(X = 0) + P(X = 1)

                 = 5C0 * (0.25)^0 * (0.75)^5 + 5C1 * (0.25)^1 * (0.75)^4 = 0.6328

4) Expected value = np = 5 * 0.25 = 1.25

Standard deviation = sqrt(np(1 - p))

                           = sqrt(5 * 0.25 * 0.75)

                           = 0.9682