Question

At a certain gas station, the amount of gas purchased per vehicle follows a normal distribution with a mean of 20 gallons and a standard deviation of 3.5 gallons. If 39 vehicles sampled at random, then 15% of the time, their mean amount purchased will be less than how many gallons? Round your answer to the nearest gallon.

Answer 1

To determine the 15% cutoff point for the mean amount of gas purchased by 39 vehicles, we use the Z-score for 15% and calculate it using the population mean, standard deviation, and sample size, rounding the result to the nearest gallon.

Step-by-step explanation:

To find the cutoff point where the mean amount of gas purchased by 39 vehicles is in the lowest 15%, we have to use the concept of the sampling distribution of the sample mean. Since we're dealing with a normal distribution, we'll use the Z-score formula to find the cutoff point.

Firstly, the standard error (SE) of the mean is calculated using the formula SE = σ/√n, where σ is the population standard deviation and n is the sample size. Here, SE = 3.5 gallons/√39.

Next, we look up the Z-score corresponding to 15% in the standard normal distribution table, which gives us a Z-score of approximately -1.04.

Finally, we can calculate the cutoff point using the formula:

X = μ + Z(SE), where X is the cutoff point and μ is the population mean.

After calculating, we will round the result to the nearest gallon to provide our answer.