Question 45·Hard·Inference from Sample Statistics and Margin of Error
Student Employment Survey
A random sample of 120 students from Jefferson High School found that 42% of the students have a part-time job. The results have a margin of error of ±9 percentage points at the 95% confidence level. There are 1,350 students at Jefferson High School. Based on the survey, what is the greatest number of Jefferson High School students who could reasonably be expected to have a part-time job?
For problems involving survey percentages and margin of error, first adjust the reported percentage by the margin of error to find the relevant bound (upper bound if the question asks for the greatest possible value, lower bound if it asks for the least). Convert that percentage to a decimal and multiply by the total population size to turn the proportion into a count. Finally, make sure your answer is a whole number and that you round in the direction (up or down) that matches what the question is asking for (greatest vs. least).
Hints
Think about what margin of error does to the percentage
The survey found that 42% of students have a part-time job, but the margin of error is ±9 percentage points. What are the lowest and highest percentages that are still consistent with this survey?
Use the bound that gives the greatest number
The question asks for the greatest number of students with part-time jobs that would still be reasonable. Should you use the lower percentage, the middle percentage, or the higher percentage from the margin-of-error range?
Convert percentage to a count of students
Once you pick the correct percentage, turn it into a decimal and multiply by the total number of students, 1,350. How should you handle the result if it is not a whole number?
Desmos Guide
Calculate the maximum expected number directly
In Desmos, type 1350*(0.42+0.09) and look at the output. This gives the number of students corresponding to the highest reasonable percentage from the margin-of-error range; if the result is not a whole number, think about how to adjust it to represent a count of students.
Step-by-step Explanation
Use the margin of error to find the highest reasonable percentage
The survey result is 42% with a margin of error of ±9 percentage points at the 95% confidence level.
That means the true percentage of students with a part-time job could reasonably be anywhere from
- lower bound:
- upper bound:
To get the greatest number of students, use the upper bound .
Convert the upper percentage to a decimal and set up the calculation
A percentage must be written as a decimal to multiply by the total number of students.
- as a decimal is .
Now set up the calculation using the total of 1,350 students:
Multiply and round to a whole number of students
Compute the product:
You cannot have half a student, and the question asks for the greatest number that could reasonably be expected.
So round up to the next whole number.
The greatest number of Jefferson High School students who could reasonably be expected to have a part-time job is 689.