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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.

$1$ . 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.

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Interpret confidence intervals for population means

Full solution

Q.

1

. 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.

Calculate sample proportion: Calculate the sample proportion ( $\hat{p}$ ) by dividing the number of workers who work out $3$ or more days a week by the total number of workers surveyed. $\newline$ $\hat{p} = \frac{73}{100} \times 2035$ $\newline$ $\hat{p} = 1485.55$
Round number of workers: Since we can't have a fraction of a person, round the number of workers to the nearest whole number. $\hat{p} = 1486$ workers
Recalculate sample proportion: Recalculate the sample proportion ( $\hat{p}$ ) using the rounded number of workers. $\newline$ $\hat{p} = \frac{1486}{2035}$ $\newline$ $\hat{p} \approx 0.73$
Calculate standard error: Calculate the standard error (SE) using the formula $SE = \sqrt{\left(\hat{p}(1 - \hat{p})\right) / n}$ , where $n$ is the sample size. $\newline$ $SE = \sqrt{\left(0.73 \times (1 - 0.73)\right) / 2035}$ $\newline$ $SE = \sqrt{\left(0.73 \times 0.27\right) / 2035}$ $\newline$ $SE = \sqrt{0.1971 / 2035}$ $\newline$ $SE = \sqrt{0.0000968}$ $\newline$ $SE \approx 0.0098$
Determine z-score: Determine the z-score for a $95$ % confidence level. Typically, the z-score for a $95$ % confidence level is $1.96$ . $\newline$ $z = 1.96$
Calculate margin of error: Calculate the margin of error (ME) using the formula $ME = z \times SE$ . $ME = 1.96 \times 0.0098$ $ME \approx 0.0192$
Convert margin of error: Convert the margin of error to a percentage by multiplying by $100$ . $\newline$ $\text{ME} \approx 0.0192 \times 100$ $\newline$ $\text{ME} \approx 1.92\%$
Calculate lower bound: Calculate the lower bound of the confidence interval by subtracting the margin of error from the sample proportion. $\newline$ Lower bound = $\hat{p} - ME$ $\newline$ Lower bound = $0.73 - 0.0192$ $\newline$ Lower bound $\approx 0.7108$
Calculate upper bound: Calculate the upper bound of the confidence interval by adding the margin of error to the sample proportion. $\newline$ Upper bound = $\hat{p} + ME$ $\newline$ Upper bound = $0.73 + 0.0192$ $\newline$ Upper bound $\approx 0.7492$
Convert bounds to percentages: Convert the lower and upper bounds of the confidence interval to percentages by multiplying by $100$ . $\newline$ Lower bound $\approx 0.7108 \times 100$ $\newline$ Lower bound $\approx 71.08\%$ $\newline$ Upper bound $\approx 0.7492 \times 100$ $\newline$ Upper bound $\approx 74.92\%$

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