You want to estimate the mean amount spent by customers at a local gas station with 95% confidence and a margin of error of no more than $3. Preliminary data suggests that $5.10 is a reasonable estimate for the standard deviation for all customers. How large a sample do you need? Show all work.
To determine the sample size needed to estimate the mean amount spent by customers with a 95% confidence level and a margin of error of no more than $3, we can use the formula:
n = ((Z * σ) / E)²
Where:
n = sample size
Z = Z-value for the desired confidence level
σ = standard deviation
E = margin of error
In this case, we are given the following information:
Z-value for a 95% confidence level is approximately 1.96 (lookup value from a standard normal distribution table)
σ = $5.10 (standard deviation)
E = $3 (margin of error)
Plugging in these values into the formula:
n = ((1.96 * $5.10) / $3)²
n = (9.996 / $3)²
n = (3.332)²
n ≈ 11.11
Since we cannot have a fractional sample size, we round up to the nearest whole number. Hence, we need a sample size of at least 12 to estimate the mean amount spent by customers with a 95% confidence level and a margin of error of no more than $3.