In a survey of 2035 workers, $73\%$ reported working out 3 or more days a week. What is the margin of error? What is the interval that is likely to contain the exact percent of all people who work out 3 or more days a week? Show all work.
Use this formula:
$\pm \frac{1}{\sqrt{n}}$
Answer (1)
Calculate the margin of error: 2035 1 ≈ 0.0222 .
Convert the margin of error to percentage: 0.0222 ≈ 2.22% .
Calculate the lower bound of the interval: 73% − 2.22% = 70.78% .
Calculate the upper bound of the interval: 73% + 2.22% = 75.22% . The interval is ( 70.78% , 75.22% ) .
Explanation
Understand the problem and provided data We are given a survey of 2035 workers, where 73% reported working out 3 or more days a week. We need to find the margin of error and the interval that likely contains the exact percentage of all people who work out 3 or more days a week. The formula for the margin of error is given as ± n 1 , where n is the sample size.
Calculate the margin of error First, we calculate the margin of error using the given formula and the sample size n = 2035 .
M a r g in o f E rror = ± 2035 1 M a r g in o f E rror = ± 45.110974 1 M a r g in o f E rror ≈ ± 0.02216755 Converting this to percentage, we get approximately ± 2.22% .
Calculate the interval Next, we calculate the interval that is likely to contain the exact percent of all people who work out 3 or more days a week. We do this by subtracting and adding the margin of error to the reported percentage (73%).
Lower bound = 73% - Margin of Error Upper bound = 73% + Margin of Error
Lower bound = 73% - 2.22% Upper bound = 73% + 2.22%
Lower bound = 70.78% Upper bound = 75.22%
State the final answer Therefore, the interval that is likely to contain the exact percent of all people who work out 3 or more days a week is approximately ( 70.78% , 75.22% ) .
Examples
In political polling, the margin of error helps to understand the accuracy of survey results. For example, if a poll finds that 52% of voters support a candidate with a margin of error of ± 3% , it means the true percentage of voters who support the candidate likely falls between 49% and 55%. This helps to interpret the poll's results more cautiously and understand the range of possible outcomes. Similarly, in market research, companies use margins of error to assess the reliability of consumer surveys and make informed decisions about product development and marketing strategies.
