Question 39·Hard·Inference from Sample Statistics and Margin of Error
A company surveyed 350 of its 4,700 employees about whether they would prefer a four-day workweek. The survey found that 231 of the employees surveyed would prefer the change. The 95% margin of error for the sample proportion is approximately ±5 percentage points.
Based on these results, which statement is supported?
For margin-of-error questions, first turn the raw counts into a sample proportion or percentage. Then immediately apply the given margin of error to form a confidence interval (sample percent ± margin of error). Finally, test each answer choice against this interval: a correct statement must hold for every value in the interval, while any choice that requires a value outside the interval (for example, insisting on at least 65% when the lower bound is 61%) should be eliminated. This interval-checking approach is quick, systematic, and prevents over- or under-interpreting the data.
Hints
Turn the survey result into a percentage
First, compute the fraction of surveyed employees who prefer a four-day workweek by calculating . Then convert this to a percentage.
Use the margin of error
You are told the margin of error is ±5 percentage points. Build a low and high estimate by subtracting and adding 5 percentage points to your sample percentage.
Compare the interval to the claims
Look at the lower end of your interval. Is it above 50%? above 60%? above 65%? Use these comparisons to rule out any statements that guarantee too high or too low a percentage.
Desmos Guide
Compute the sample proportion
In Desmos, type 231/350 and press Enter. Note the decimal value; this is the sample proportion of employees who prefer the four-day workweek.
Find the confidence interval bounds
On a new line, enter (231/350) - 0.05 to get the lower bound and (231/350) + 0.05 to get the upper bound, since a 5 percentage point margin of error is 0.05 in decimal form. Observe both results to see the approximate interval for the true proportion.
Relate the interval to the answer choices
Compare the lower bound from Desmos to 0.5, 0.6, and 0.65. Decide which answer choice makes a claim that is true for every value in your interval and does not demand a higher minimum percentage than your lower bound supports.
Step-by-step Explanation
Find the sample proportion
The survey found that 231 out of 350 employees prefer a four-day workweek.
The sample proportion is
This means 66% of the sampled employees preferred a four-day workweek.
Apply the margin of error
The 95% margin of error is ±5 percentage points, or ±0.05 in decimal form.
So the 95% confidence interval for the true population proportion is approximately:
- Lower bound: (or )
- Upper bound: (or )
So the company can be 95% confident that the true percentage of all employees who prefer a four-day workweek is between 61% and 71%.
Interpret what the interval tells you
The interval for the true percentage is from 61% to 71%.
Key observations:
- Every possible value in this interval is greater than 50%.
- Every possible value in this interval is also greater than 60%.
- However, some possible values (like 61%, 62%, 63%, 64%) are less than 65%.
So you can be confident the true proportion is above 50% and above 60%, but you cannot be sure it is at least 65%.
Match the interval to the answer choices
Now check each statement against what the interval [61%, 71%] allows:
- “At least 65%” is too strong, because the true percentage could be as low as 61%, which is below 65%.
- Saying you cannot conclude that more than half prefer the change is wrong, because all values in the interval are above 50%.
- Saying you must survey everyone to make a 95% confident conclusion is wrong, because this 95% confidence interval is already based on a sample.
The only statement that correctly reflects the interval is that the company can be 95% confident that more than 60% of all employees prefer a four-day workweek.