Quarters. In Exercises 9–12, refer to the accompanying table of weights (grams) of quarters minted by the U.S. government. This table is available in Data Set 44 “Weights of Minted Quarters” in Appendix B.
Quarters: xbar Chart Treat the 5 measurements from each day as a sample and construct an xbar chart. What does the result suggest?

Verified step by step guidance

1. Step 1: Understand the x̄ (x-bar) chart. An x̄ chart is a type of control chart used to monitor the mean of a process over time. It helps determine if the process is in statistical control by analyzing sample means.
2. Step 2: Calculate the mean (x̄) for each sample. For each day, sum the weights of the 5 quarters and divide by 5. Use the formula: $x=\frac{\sum x}{}$_in, where $n$ is the sample size (5 in this case).
3. Step 3: Calculate the overall mean (grand mean) of all sample means. Add up all the sample means from Step 2 and divide by the number of samples (days). Use the formula: $x_{‾}=\frac{\sum x_i}{k}$, where $k$ is the number of samples.
4. Step 4: Calculate the control limits for the x̄ chart. Use the formulas for the Upper Control Limit (UCL) and Lower Control Limit (LCL): $\mathrm{UCL}=x_{‾}+A$₂R and $\mathrm{LCL}=x_{‾}-A$₂R, where $R$ is the average range of the samples, and $A$₂ is a constant based on the sample size (look up the value in a control chart table for n=5).
5. Step 5: Plot the x̄ chart. On the x-axis, represent the days (or samples), and on the y-axis, plot the sample means. Draw the UCL, LCL, and the grand mean as horizontal lines. Analyze the chart to determine if the process is in control: check if all points fall within the control limits and if there are no patterns or trends suggesting non-random variation.

Key Concepts

X-bar Chart

An X-bar chart is a type of control chart used in statistical process control to monitor the mean of a process over time. It displays the average values of samples taken from a process, allowing for the detection of trends or shifts in the process mean. By plotting the sample means against control limits, it helps identify whether the process is in a state of control or if there are variations that need investigation.

Sampling

Sampling is the process of selecting a subset of individuals or measurements from a larger population to estimate characteristics of the whole. In this context, the five measurements taken each day represent a sample from the population of all quarters minted. Proper sampling techniques ensure that the sample is representative, which is crucial for making valid inferences about the population based on the sample data.

Control Limits

Control limits are the boundaries set on a control chart that indicate the expected range of variation in a process. Typically calculated as the mean plus or minus three standard deviations, these limits help determine whether a process is stable or if there are out-of-control signals. If sample means fall outside these limits, it suggests that the process may be affected by special causes that require further investigation.