A gasoline tank for a certain car is designed to hold 15 gallons of gas.Suppose that the actual capacity of a randomly selected tank has a distribution that is approximately Normal with a mean of 15.0 gallons and a standard deviation of 0.15 gallons.If four tanks are randomly selected,what is the probability that their average capacity will be between 14.75 and 15.10 gallons?
A)0.2397
B)0.6808
C)0.9084
D)0.9962

Question

Quizplus · Accepted Answer

We know that the distribution of the capacity of a randomly selected tank is approximately Normal with mean $\mu = 15$ gallons and standard deviation $\sigma = 0.15$ gallons.

The sample size, n = 4, is large enough to use the Central Limit Theorem. According to the CLT, the sample mean, $\bar{X}$, will also have an approximately normal distribution with mean $\mu_{\bar{X}} = \mu = 15$ gallons and standard deviation $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{0.15}{\sqrt{4}} = 0.075$ gallons.

Therefore, we need to find P(14.75 < $\bar{X}$ < 15.10) = P($\frac{14.75-15}{0.075}$ < Z < $\frac{15.10-15}{0.075}$), where Z is a standard normal random variable.

Using a standard normal table, we find that P(-2 < Z < 1.33) = 0.9084.

Therefore, the probability that the average capacity of four randomly selected tanks will be between 14.75 and 15.10 gallons is 0.9084. Hence the answer is C.

A Gasoline Tank for a Certain Car Is Designed to Hold