Question 50·Medium·Inference from Sample Statistics and Margin of Error
A city planner wants to estimate the average daily household water usage in the city. From a simple random sample of 500 households, the planner calculates a sample mean of 148 gallons and reports an associated margin of error of 7 gallons.
Which of the following is the best interpretation of this estimate?
For margin-of-error questions, first recognize what is being estimated: usually the population mean or proportion, not individual values. Quickly form the interval by doing “estimate ± margin of error,” then interpret that interval in words as a range of likely values for the population parameter. Finally, eliminate answer choices that (1) talk about every single individual, (2) describe a percent of individuals inside the interval, or (3) mistakenly put plausible values outside the margin-of-error range.
Hints
Connect the margin of error to the sample mean
You are told the sample mean is 148 gallons and the margin of error is 7 gallons. How can you use these two numbers to form a range around 148?
Decide what quantity is being estimated
The planner wants to estimate the average daily household water usage in the city. Is that about all individual households separately, or about a single overall value that summarizes them?
Be careful about interpreting the range
Does a margin of error usually describe (1) a range for the true average, (2) a range that contains every single individual, or (3) a range that contains a certain percent of individuals? Think about what you learned about confidence intervals.
Desmos Guide
Compute the bounds of the interval
In Desmos’s expression lines, type 148-7 on one line and 148+7 on another. Look at the numerical outputs; they give you the lower and upper bounds of the plausible range for the average daily household water usage.
Step-by-step Explanation
Understand what the margin of error applies to
The planner took a simple random sample and computed a sample mean of 148 gallons. The margin of error of 7 gallons tells us how far the true population mean (the average for all households in the city) is likely to be from this sample mean. It does not describe individual households or what percent of households fall in a certain range.
Compute the interval given by the margin of error
Use the sample mean and margin of error to find the lower and upper bounds:
- Lower bound:
- Upper bound:
So the estimate says the true average daily household water usage for the entire city is likely to be somewhere between 141 and 155 gallons.
Match the correct statistical interpretation to the answer choices
The interval 141 to 155 gallons reflects plausible values for the population mean (the average for all households), not for each individual household and not for 95% of households. The only choice that correctly states this is: “Plausible values for the average daily household water usage for all households in the city are between 141 gallons and 155 gallons.”