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 hose limits, the mean weight 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 want 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.

I am just wondering what the formula is because I know that it uses the mean and standard deviation.

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To determine the sample size needed to estimate the population mean within a given confidence interval, you can use the formula for sample size calculation based on a specified confidence level and margin of error.

The formula for calculating the sample size is:

n = (Z * σ / E)^2

where:
- n is the sample size needed
- Z is the Z-score associated with the desired confidence level
- σ is the estimated population standard deviation
- E is the desired margin of error

In this case, the desired confidence level is 95%, which corresponds to a Z-score of 1.96 (derived from the standard normal distribution table). The estimated population standard deviation (σ) is given as 0.068 g, and the desired margin of error (E) is 0.035 g.

Plugging these values into the formula, we get:

n = (1.96 * 0.068 / 0.035)^2

Simplifying the equation:

n ≈ (28.48 / 0.035)^2
n ≈ 816.57^2

Therefore, you would need to randomly select and weigh approximately 667,491 quarters to be 95% confident that the sample mean is within 0.035 g of the true population mean for all quarters.