Question 41·Medium·Inference from Sample Statistics and Margin of Error
A researcher surveyed 120 households chosen at random from a town of 12,500 households. Of the households surveyed, 18 reported having a backyard vegetable garden. Based on the survey results, which of the following is the most appropriate conclusion?
For survey and sampling questions, first compute the sample proportion (e.g., number with the trait divided by total surveyed), then multiply that proportion by the total population to get a predicted count. On the SAT, be careful with wording: results from a random sample almost always support a "best estimate" or "approximately" conclusion, not an "exact" statement, and any proposed number must be consistent with the sample proportion.
Hints
Use the information from the sample
First, focus on the 18 out of 120 households in the survey that have gardens. What fraction or proportion of the surveyed households is that?
Extend the sample proportion to the whole population
Once you know what fraction of the 120 surveyed households have gardens, think about how to apply that same fraction to all 12,500 households in the town.
Think about what a sample can and cannot tell you
Does a survey of only 120 households allow you to know the exact number of households with gardens in the town, or does it give you a good prediction? How should that affect the wording of your conclusion?
Desmos Guide
Compute the predicted number of households
In Desmos, type 18/120 * 12500 and look at the numerical output; this is the predicted number of households with gardens based on the sample proportion.
Step-by-step Explanation
Find the sample proportion with gardens
From the survey, 18 out of 120 households have backyard vegetable gardens.
Compute the sample proportion:
So, the survey suggests that about of households in the town have backyard vegetable gardens.
Scale the sample proportion to the whole town
There are 12,500 households in the town. If about of them have gardens, multiply the total number of households by the proportion:
So, using the sample proportion, we would predict about 1,875 households with gardens in the entire town.
Decide between "exact" and "best estimate"
The 1,875 value comes from a random sample, not from checking every single household. Because samples have natural variation, we cannot claim the town has exactly 1,875 households with gardens; we can only say this is our best estimate based on the survey.
Therefore, the most appropriate conclusion is: The best estimate for the number of households with backyard vegetable gardens is 1,875.