Berlecia Howe

2022-10-06

1. You want to estimate the population of waiting times for hospital emergency rooms. You want to be 99% confident that the sample standard deviation is within 1% of the population standard deviation. Find the minimum sample size.

Don Sumner

To find the minimum sample size needed to estimate the population of waiting times for hospital emergency rooms with a 99% confidence level and an acceptable error margin of 1% for the sample standard deviation compared to the population standard deviation, we can use the formula for minimum sample size:
$n={\left(\frac{Z·\sigma }{E}\right)}^{2}$
where:
$n$ represents the sample size,
$Z$ is the z-score corresponding to the desired confidence level,
$\sigma$ is the population standard deviation, and
$E$ is the acceptable error margin for the sample standard deviation.
In this case, we want to be 99% confident, which means the z-score corresponding to a 99% confidence level is approximately 2.576 (from the standard normal distribution table).
We can represent the formula and values as follows:
$n={\left(\frac{2.576·\sigma }{0.01·\sigma }\right)}^{2}$
Simplifying the equation, we get:
$n={\left(\frac{2.576}{0.01}\right)}^{2}$
$n=657.69$
Rounding up to the nearest whole number, the minimum sample size needed is $n=658$.
Therefore, a minimum sample size of 658 is required to estimate the population of waiting times for hospital emergency rooms with a 99% confidence level and an acceptable error margin of 1% for the sample standard deviation compared to the population standard deviation.

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