Final answer:
To estimate the mean weight of all quarters, we use the sample mean of 5.664 grams. The 95% confidence interval is found by calculating the margin of error using the formula E = 1.96 * (0.056 / sqrt(50)) and applying it to the sample mean.
Step-by-step explanation:
To estimate the mean weight of all quarters in circulation, we use the sample mean weight which is 5.664 grams. To find the 95% confidence interval for the average weight of all quarters, we need to use the sample standard deviation and the size of the sample to calculate the margin of error, and then apply that to the sample mean.
The formula for the margin of error (E) in a sample mean is E = z * (s / sqrt(n)), where 'z' is the z-score corresponding to the desired confidence level (1.96 for 95% confidence), 's' is the sample standard deviation, and 'n' is the sample size.
In this case, the formula becomes E = 1.96 * (0.056 / sqrt(50)), which gives us a margin of error to be added and subtracted from the sample mean to find the confidence interval.
Once we compute E, we find the lower limit of the confidence interval by subtracting E from the sample mean and the upper limit by adding E to the sample mean. These two values give us the range in which we are 95% confident the true population mean lies.