Answer to Question #147933 in Statistics and Probability for nawab

Question

he annual per capita consumption of bottled water was 31.9 gallons. Assume that the per capita consumption of bottled water is approximately normally distributed with a mean of 31.9 and a standard deviation of 10 gallons.
a. What is the probability that someone consumed more than 37 gallons of bottled​ water?
b. What is the probability that someone consumed between 20 and 30 gallons of bottled​ water?
c. What is the probability that someone consumed less than 20 gallons of bottled​ water?
d. 99​% of people consumed less than how many gallons of bottled​ water?

Answer 1

$(a) \; P(x > 37) = 1 – P(x < 37) \\ = 1 – P(\frac{x – μ}{σ} < \frac{37 – 31.9}{10}) \\ = 1 – P(z < 0.51) \\ = 0.3050$

$(b)\; P( 20 < x < 30) = P(\frac{20 – 31.9}{10} < \frac{x – μ}{σ} < \frac{30 – 31.9}{10}) \\ = P(-1.19 < z < -0.19) \\ = P(z < -0.19) – P(z < -1.19) \\ = 0.4246 – 0.1170 \\ = 0.3076$

$(c) \; P(x < 20) = P(\frac{x – μ}{σ} < \frac{20 – 31.9}{10}) \\ = P(z < -1.19) \\ = 0.1170 \\ (d) \; P(z < 2.33) = 0.99\\ z = 2.33 \\ x = z \times σ + μ \\ x = 2.33 \times 10 + 31.9 = 55.2$

Answer: 55.2 gallons