Mint Specs Listed below are weights (grams) from a simple random sample of pennies produced after 1983 (from Data Set 40 “Coin Weights” in Appendix B). Construct a 95% confidence interval estimate of for the population of such pennies. What does the confidence interval suggest about the U.S. Mint specifications that now require a standard deviation of 0.0230 g for weights of pennies?

Verified step by step guidance

1. Step 1: Calculate the sample mean (\( \bar{x} \)) using the formula \( \bar{x} = \frac{\sum x_i}{n} \), where \( x_i \) are the individual weights and \( n \) is the sample size. Add all the weights provided and divide by the total number of weights.
2. Step 2: Calculate the sample standard deviation (\( s \)) using the formula \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \). Subtract the sample mean from each weight, square the result, sum these squared differences, divide by \( n-1 \), and take the square root.
3. Step 3: Determine the critical value (\( t \)) for a 95% confidence interval using a t-distribution table. The degrees of freedom (df) are \( n-1 \), where \( n \) is the sample size.
4. Step 4: Compute the margin of error (ME) using the formula \( ME = t \cdot \frac{s}{\sqrt{n}} \), where \( s \) is the sample standard deviation and \( n \) is the sample size.
5. Step 5: Construct the confidence interval using the formula \( \text{Confidence Interval} = \bar{x} \pm ME \). Interpret the interval in the context of the U.S. Mint specifications, comparing the calculated standard deviation to the required standard deviation of 0.0230 g.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence, typically 95%. It is calculated using the sample mean, the standard deviation, and the sample size, providing a measure of uncertainty around the estimate. For example, if the confidence interval for the mean weight of pennies is (2.490, 2.510), we can be 95% confident that the true mean weight lies within this range.

Standard Deviation

Standard deviation is a statistic that measures the dispersion or variability of a set of values. It indicates how much individual data points differ from the mean of the dataset. In the context of the U.S. Mint specifications, a standard deviation of 0.0230 grams suggests that the weights of the pennies produced are expected to vary by this amount from the average weight, which is crucial for quality control in manufacturing.

Simple Random Sample

A simple random sample is a subset of individuals chosen from a larger population, where each individual has an equal chance of being selected. This method helps ensure that the sample is representative of the population, reducing bias in statistical analysis. In this case, the weights of the pennies were collected from a simple random sample, allowing for valid inferences about the entire population of pennies produced after 1983.