If the amount of gasoline purchased per car at a large service station has a population mean of $15 and a population standard deviation of $4 and it is assumed that the amount of gasoline purchased per car is symmetric, there is approximately a 68.26% chance that a random sample of 16 cars will have a sample mean between $14 and $16.

Question

Quizplus · Accepted Answer

The central limit theorem states that for a large enough sample size, the sampling distribution of the sample mean will be approximately normally distributed, even if the population is not normally distributed. In this case, we have a sample size of 16, which is considered large enough for the central limit theorem to apply.

Since we assume that the population is symmetric (which implies it is also unimodal), we can use the 68-95-99.7 rule for normal distributions. This tells us that approximately 68.26% of the observations will fall within one standard deviation of the mean.

The standard error of the mean can be calculated as:

SE = σ / sqrt(n) = 4 / sqrt(16) = 1

where σ is the population standard deviation and n is the sample size.

Using the formula for a z-score, we can compute the z-score for each sample mean:

z = (x - μ) / SE

For x = 14:

z = (14 - 15) / 1 = -1

For x = 16:

z = (16 - 15) / 1 = 1

Using a standard normal table or calculator, we can find the area under the curve between these two z-scores:

P(-1 < Z < 1) = 0.6826

Therefore, there is approximately a 68.26% chance that a random sample of 16 cars will have a sample mean between $14 and $16.

If the Amount of Gasoline Purchased Per Car at a Large