You want to estimate the mean amount spent by customers at a local gas station with 98% confidence and a margin of error of no more than $2. Preliminary data suggests that the standard dev. is $5.1 is a reasonable estimate for the standard dev. for all customers. How large a sample do you need?
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To determine the sample size needed to estimate the mean amount spent by customers at a local gas station, we can use the formula for sample size determination with a known population standard deviation. The formula is as follows:
n = (Z * σ / E)²
Where:
n = sample size needed
Z = Z-value for the desired confidence level
σ = population standard deviation
E = margin of error
In this case, we want a 98% confidence level. The Z-value for a 98% confidence level is 2.33 (obtained from a standard normal distribution table or calculator). The margin of error is specified as $2.
Using the given values, we can calculate the needed sample size:
n = (2.33 * 5.1 / 2)²
n = (11.883 / 2)²
n = 5.942²
n = 35.35 (rounded up to the nearest whole number)
Therefore, you would need a sample size of at least 36 to estimate the mean amount spent by customers at the gas station with 98% confidence and a margin of error no more than $2.