142k views
0 votes
A population has a standard deviation σ = 20.5. How large a sample must be drawn so that a 99% confidence interval for μ will have a margin of error equal to 5.4?

1 Answer

4 votes

Final answer:

To calculate the sample size needed for a 99% confidence interval with a margin of error of 5.4 and a population standard deviation of 20.5, we use the margin of error formula, which results in a necessary sample size of at least 98 individuals when rounded up.

Step-by-step explanation:

To determine how large a sample must be drawn so that a 99% confidence interval for the population mean (μ) will have a margin of error equal to 5.4 when the population standard deviation (σ) is 20.5, we can use the formula for the margin of error (EBM):

EBM = Z * (σ / √n)

Where:

  • Z is the Z-value corresponding to the desired confidence level (for a 99% confidence interval, Z is approximately 2.576)
  • σ is the population standard deviation, in this case, 20.5
  • n is the sample size
  • EBM is the desired margin of error, in this case, 5.4

Rearranging the formula to solve for n gives us:

n = (Z × σ / EBM)^2

Plugging in the values we get:

n = (2.576 × 20.5 / 5.4)^2 ≈ 97.456

Since we cannot have a fraction of a person, we would round up to the nearest whole number. Therefore, a sample of at least 98 individuals needs to be drawn for a 99% confidence interval to have a margin of error of 5.4.

User Kevin LE GOFF
by
7.0k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories