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

Jessica Felix · Accepted Answer

Final Answer : C, Explanation :  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.

Definitions

Learning Objectives

Learning Objectives

Related questions