Question

In a survey of 975 randomly selected fans of a team, 58% want a coaching change. The margin
of error for the survey is 3.2%.
Which conclusion about the survey is TRUE?
A) Exactly 58% of the fans want a coaching change.
B) More than 60% of the fans want a coaching change.
C) Between 54.8% and 61.2% of the fans want a coaching change.
D) Decreasing the number of respondents would reduce the margin of error.

Answer 1

Given the information from the survey, we know that 58% of the fans want a coaching change, and the margin of error is 3.2%. This means that we can calculate the confidence interval for the proportion of fans wanting a coaching change.

1. **Confidence Interval Calculation**:
- Lower Limit: 58% - 3.2% = 54.8%
- Upper Limit: 58% + 3.2% = 61.2%
- Therefore, the confidence interval is between 54.8% and 61.2%.

Now let's evaluate the possible conclusions:

A) **Exactly 58% of the fans want a coaching change.**
- This is not necessarily true since the survey results indicate a margin of error, which means the actual proportion may vary.

B) **More than 60% of the fans want a coaching change.**
- This is not supported by the data since the upper limit of the confidence interval is 61.2%, not "more than" 60%.

C) **Between 54.8% and 61.2% of the fans want a coaching change.**
- This statement is true as it accurately reflects the calculated confidence interval.

D) **Decreasing the number of respondents would reduce the margin of error.**
- This statement is false. Decreasing the number of respondents would actually increase the margin of error.

Thus, the only true conclusion about the survey is:

**C) Between 54.8% and 61.2% of the fans want a coaching change.**