The Tyco Video Game Corporation finds that it is losing income because of slugs used in its video games. The machines must be adjusted to accept coins only if they fall within set limits. In order to set those limits, the mean wight of quarters in circulation must be estimated. A sample of quarters will be weighed in order to determine the mean. How many quarters must we randomly select and weigh if we wan to be 95% confident that the sample mean is within 0.035 g of the true population mean for all quarters? Based on results from a sample of quarters, we can estimate the population standard deviation as 0.068 g.
To determine the sample size required, we need to use the formula for the sample size calculation in estimating the population mean:
n = (Z * σ / E)^2
Where:
n = sample size
Z = Z-score for the desired level of confidence
σ = population standard deviation
E = margin of error
Given:
Desired level of confidence = 95% (which translates to a Z-score of 1.96 for a two-tailed test)
Margin of error (E) = 0.035 g
Population standard deviation (σ) = 0.068 g
Plugging in the values into the formula:
n = (1.96 * 0.068 / 0.035)^2
n ≈ (0.13408 / 0.035)^2
n ≈ 3.83^2
n ≈ 14.68
Since you can't have a fraction of a quarter, you need to round up to the nearest whole number. Therefore, you would need to randomly select and weigh at least 15 quarters in order to be 95% confident that the sample mean is within 0.035 g of the true population mean for all quarters.