A researcher is interested in estimating the noise levels in decibels at area urban hospitals. She wants to be 95% confident that her estimate is correct. If the standard deviation is 4.02, how large a sample is needed to get the desired information to be accurate within 0.58 decibels? Show all work.
To determine the sample size needed to estimate the noise levels in decibels accurately, we can use the formula for the sample size for means:
n = (Z * σ / E)^2
Where:
n = sample size
Z = Z-score representing the desired confidence level
σ = standard deviation
E = margin of error
In this case, we want to be 95% confident, so the Z-score corresponding to a 95% confidence level is 1.96. The standard deviation is given as 4.02 decibels, and we want the estimate to be accurate within 0.58 decibels.
Plugging these values into the formula, we have:
n = (1.96 * 4.02 / 0.58)^2
n = (7.8792 / 0.58)^2
n = 13.6^2
n ≈ 184.96
Since we cannot have a fractional sample size, we round up to the nearest whole number. Therefore, a sample size of at least 185 is needed to estimate the noise levels accurately at the urban hospitals with a confidence level of 95% and a margin of error of 0.58 decibels.