A researcher with the Department of Energy wants to determine the mean natural gas bill of households throughout the United States. He knows the mean natural gas bill of households for each state, so he adds together these 50 values and divides by 50 to arrive at his estimate. Is this a valid approach? Why or why not?

Verified step by step guidance

1. Understand the research goal: The researcher wants to find the mean natural gas bill of households across the entire United States, which means the average bill per household considering all households nationwide.
2. Recognize what the researcher did: He took the mean natural gas bill for each state (which is an average for that state) and then averaged these 50 state means by adding them together and dividing by 50.
3. Identify the key issue: Each state has a different number of households, so simply averaging the 50 state means gives equal weight to each state regardless of its population size.
4. Explain why this is problematic: To get the true national mean, the researcher should calculate a weighted average of the state means, where each state's mean is weighted by the number of households in that state. This accounts for the fact that states with more households contribute more to the national average.
5. Summarize the correct approach: The valid method is to multiply each state's mean natural gas bill by the number of households in that state, sum these products for all states, and then divide by the total number of households in the United States. Mathematically, this is: \(\text{National Mean} = \frac{\sum_{i=1}^{50} (\text{State Mean}_i \times \text{Households}_i)}{\sum_{i=1}^{50} \text{Households}_i}\).

Key Concepts

Weighted vs. Unweighted Averages

An unweighted average treats all groups equally regardless of size, while a weighted average accounts for the size or importance of each group. In this case, simply averaging state means ignores the number of households per state, leading to a potentially biased estimate.

Population Heterogeneity

Different states have varying numbers of households and consumption patterns. Ignoring these differences by treating each state equally can distort the overall mean, as states with fewer households are given the same influence as larger states.

Sampling and Estimation Principles

Accurate estimation of a population mean requires combining subgroup means weighted by subgroup sizes. This ensures the estimate reflects the true distribution of the population, avoiding misleading conclusions from simple averages of subgroup means.