Question

A gasoline tank for a certain car is designed to hold 15 gallons of gas. Suppose that the random variable \(x=\) actual capacity of a randomly selected tank has a distribution that is well approximated by a normal curve with mean 15.0 gallons and standard deviation 0.1 gallon. a. What is the probability that a randomly selected tank will hold at most 14.8 gallons? b. What is the probability that a randomly selected tank will hold between 14.7 and 15.1 gallons? c. If two such tanks are independently selected, what is the probability that both hold at most 15 gallons?

Short answer

a) The probability that a randomly selected tank will hold at most 14.8 gallons is 0.02275. b) The probability that a randomly selected tank will hold between 14.7 and 15.1 gallons is 0.83999. c) The probability that both independently selected tanks hold at most 15 gallons is 0.25.

Step-by-step solution

Calculating the Z-score for part a

First, let's calculate the Z-score, which is the number of standard deviations from the mean, for a tank capacity of 14.8 gallons using the given mean μ and standard deviation σ. The Z-score is calculated through the formula \(Z = (X - μ) / σ\). In this case, \(Z = (14.8 - 15.0) / 0.1 = -2\).

Finding the Probability for part a

Using the standard normal distribution table or a z-table, find the probability that corresponds to a Z-score of -2. This probability represents the chance of a randomly selected tank holding at most 14.8 gallons. The probability associated with a Z-score of -2.00 is 0.02275.

Calculating the Z-scores for part b

Here, we need to calculate the Z-score for both 14.7 gallons and 15.1 gallons. Using the Z-score formula again, \(Z1 = (14.7 - 15) / 0.1 = -3\) for the lower limit and \(Z2 = (15.1 - 15) / 0.1 = 1\) for the upper limit.

Finding the Probability for part b

We look up both Z-scores in the Z-table. The probability associated with a Z-score of -3.00 is 0.00135 and with a Z-score of 1.00 is 0.84134. We subtract the probability of Z1 from the probability of Z2 to find the probability that a randomly selected tank will hold between 14.7 and 15.1 gallons: \(P = 0.84134 - 0.00135 = 0.83999\).

Finding the Probability for part c

First, we find the probability for one such tank holding at most 15 gallons. In the z-score context, 15 is the mean, which corresponds to a z-score of 0 with a probability of 0.5. As the tanks are filled independently, the total probability is the product of the individual probabilities. Hence, the probability that both tanks hold at most 15 gallons is \(0.5 * 0.5 = 0.25\).