Question 56·Medium·Inference from Sample Statistics and Margin of Error
A researcher randomly selects apples from a large orchard and records their masses. From the sample, the estimated mean mass is 0.42 kilogram, with an associated margin of error of 0.03 kilogram.
Which of the following is the best conclusion based on these data?
For questions involving a sample mean and a margin of error, first compute the interval by doing mean ± margin of error. Then remember that this interval applies to the population parameter (usually the mean or proportion), not to every individual in the population. Quickly eliminate any choices that talk about all individuals being in that range, that set absolute maximums or minimums for individuals, or that treat the mean as a value most data points must equal, and select the choice that correctly describes a likely range for the population average or proportion.
Hints
Focus on what the margin of error is about
Ask yourself: Is the margin of error describing individual apples, or is it describing the estimate of a summary value (like the mean) for the whole orchard?
Use the mean and margin of error together
Take the estimated mean of kilogram and think about what values you get if you subtract and add kilogram to it.
Watch out for overly strong statements
Be careful with answer choices that claim something about every apple or that set a strict maximum possible weight. Does the data really guarantee that?
Interpret what a mean tells you
Remember that a mean (average) does not say that most data points are exactly equal to the mean. It only summarizes the center of the data.
Desmos Guide
Compute the margin-of-error interval endpoints
In Desmos, type 0.42 - 0.03 on one line and 0.42 + 0.03 on another. Note the two numerical outputs; these are the lower and upper bounds of the likely interval for the mean mass.
Connect the numbers to the correct type of statement
Look at the two numbers Desmos gave you and scan the answer choices for one that talks about the average mass of all apples being between those two values, rather than about every individual apple or an absolute maximum.
Step-by-step Explanation
Understand what is given
The problem tells you:
- The estimated mean mass from the sample is kilogram.
- The margin of error for this estimate is kilogram.
This information comes from a sample, but the conclusion should be about the entire orchard (the population).
Use the margin of error to find the interval
A margin of error of kilogram means the true population mean is likely within kilogram above or below the sample mean.
Compute the endpoints of that interval:
- Lower bound:
- Upper bound:
So the likely interval for the population mean mass is from kilogram to kilogram.
Interpret what the interval applies to
The margin of error describes uncertainty in the mean of the population, not the weight of each individual apple.
So we can only make a statement like: the average mass of all apples in the orchard is likely somewhere between kg and kg.
We cannot conclude things like:
- Every individual apple must be in that range, or
- No apple can weigh more than the upper bound, or
- Most apples weigh exactly the mean.
Match the correct statistical conclusion to the choices
Now compare each choice to the meaning of margin of error:
- Choices that talk about every apple or set a maximum for individual apples are too strong.
- A choice that says the average mass of all apples is likely between and correctly describes what a mean with margin of error tells us.
Therefore, the best conclusion is: The average mass of all apples in the orchard is likely between 0.39 kilogram and 0.45 kilogram.