The general manager for Gordon's Self Service Gasoline is interested in estimating the mean number of gallons of gasoline that are purchased by customers at their Philadelphia location. He would like his estimate to be within plus or minus 0.50 gallons, and he would like the estimate to be at a 99% confidence level. Past studies have shown that the standard deviation for purchase amount is 4.0 gallons. Find the required sample size.

Question

The general manager for Gordon's Self Service Gasoline is interested in estimating the mean number of gallons of gasoline that are purchased by customers at their Philadelphia location. He would like his estimate to be within plus or minus 0.50 gallons, and he would like the estimate to be at a 99% confidence level. Past studies have shown that the standard deviation for purchase amount is 4.0 gallons. Find the required sample size.

(please provide complete detailed answer)

Answer 1

To determine the required sample size for estimating the mean number of gallons of gasoline with a specific margin of error and confidence level, we can use the formula:

n = (Z * σ / E)^2

Where:
n = sample size
Z = the Z-score corresponding to the desired confidence level
σ = standard deviation
E = margin of error (desired precision)

Let's break down the formula and calculate the required sample size step by step.

Step 1: Find the Z-score for a 99% confidence level.
Since we want a 99% confidence level, we need to find the Z-score that corresponds to this level. The Z-score can be found using a standard normal distribution table or a calculator.

For a 99% confidence level, we have to consider 1% in the tails of the distribution. Since the distribution is symmetric, each tail will have 0.5% or 0.005. Subtracting this from 1, we get 0.995. We need to find the Z-score that corresponds to this cumulative probability.

Using a standard normal distribution table or calculator, we find that the Z-score for a cumulative probability of 0.995 is approximately 2.576.

Step 2: Plug in the values into the formula.

n = (Z * σ / E)^2

n = (2.576 * 4.0 / 0.50)^2

Step 3: Calculate the required sample size.

n = (10.304 / 0.50)^2

n = 20.608^2

n ≈ 424.80

Since we cannot have a fraction of a person, we round up the sample size to the nearest whole number to ensure we have enough participants, resulting in a required sample size of 425.

Therefore, the general manager needs a sample size of at least 425 to estimate the mean number of gallons of gasoline purchased by customers at the Philadelphia location with a margin of error of plus or minus 0.50 gallons and a 99% confidence level.