Question

The annual per capita consumption of bottled water was 34.4 gallons. Assume that the per capita consumption of bottled water is approximately normally distributed with a mean of 34.4 and a standard deviation of 12 gallons. a. What is the probability that someone consumed more than 34 gallons of bottled water? b. What is the probability that someone consumed between 25 and 35 gallons of bottled water? c. What is the probability that someone consumed less than 25 gallons of bottled water? d. 95% of people consumed less than how many gallons of bottled water? a. The probability that someone consumed more than 34 gallons of bottled water is (Round to four decimal places as needed.) b. The probability that someone consumed between 25 and 35 gallons of bottled water is (Round to four decimal places as needed.) c. The probability that someone consumed less than 25 gallons of bottled water is (Round to four decimal places as needed.) d. 95% of people consumed less than gallons of bottled water. (Round to two decimal places as needed.)

Answer 1

Using standard normal distribution methods, we calculate Z-scores to find the probabilities of consuming more than, between, or less than certain amounts of bottled water and solve for the number of gallons that 95% of people consumed less than.

Step-by-step explanation:

To solve these problems involving the standard normal distribution, we will use the Z-score formula, which is Z = (X - μ) / σ, where X is the value in question, μ is the mean, and σ is the standard deviation. For the calculations, we can use the Z-table, a calculator, or statistical software to find the probabilities.

a. To find the probability that someone consumed more than 34 gallons, we calculate the Z-score for 34 gallons and look up the corresponding area to the right of this Z-score in the standard normal distribution table. Z = (34 - 34.4) / 12 = -0.0333. The probability of consuming more than 34 gallons corresponds to 1 - P(Z ≤ -0.0333), which can be found in the table.

b. To find the probability that someone consumed between 25 and 35 gallons, we calculate the Z-scores for both values and find the area between these Z-scores. Z1 = (25 - 34.4) / 12, Z2 = (35 - 34.4) / 12. The probability is P(25 ≤ X ≤ 35) = P(Z1 ≤ Z ≤ Z2).

c. To find the probability that someone consumed less than 25 gallons, we calculate the Z-score for 25 gallons and look up the corresponding area to the left of this Z-score. Z = (25 - 34.4) / 12. The probability is P(X < 25) = P(Z ≤ Z-score for 25 gallons).

d. To find the number of gallons that 95% of people consumed less than, we refer to the standard normal distribution table to find the Z-score that corresponds to a cumulative area of 0.95. Then we solve for X using the Z-score formula in reverse: X = Zσ + μ.