The EPA city/hwy mpg range for a Saturn Vue FWD automatic 5-speed transmission is 20 to 28 mpg. If you owned this vehicle, how large a sample (e.g. how many tanks of gas) would be required to estimate your mean mpg with an error of ±1 mpg and 90% confidence? Explain your assumption about ó
To estimate the mean mpg with an error of ±1 mpg and 90% confidence, we need to determine the required sample size for our calculation.
First, we need to make an assumption about the population standard deviation (σ), also known as ó. Since it is not provided, we can make an assumption by considering similar vehicles or conducting a preliminary study.
Suppose we assume that the population standard deviation for the Saturn Vue FWD automatic 5-speed transmission is σ=3 mpg, which is a reasonable estimate based on similar vehicles.
To find the sample size required for our estimation, we can use the formula:
n = (Z * σ / E)^2
Where:
n is the required sample size
Z is the z-score corresponding to the desired confidence level (90% in this case)
σ is the assumed population standard deviation
E is the desired margin of error
Using a standard normal distribution table or calculator, the z-score for a 90% confidence level is approximately 1.645.
Plugging these values into the formula, we get:
n = (1.645 * 3 / 1)^2
n ≈ 26.99
Since we cannot have a fraction of a tank of gas, we should round up to the nearest whole number. Therefore, we would need a sample size of at least 27 tanks of gas to estimate the mean mpg with an error of ±1 mpg and 90% confidence, assuming a population standard deviation of 3 mpg.