Suppose that a researcher is interested in estimating the mean systolic blood pressure, , of executives of major corporations. He plans to use the blood pressures of a random sample of executives of major corporations to estimate . Assuming that the standard deviation of the population of systolic blood pressures of executives of major corporations is mm Hg, what is the minimum sample size needed for the researcher to be confident that his estimate is within mm Hg of ?
Carry your intermediate computations to at least three decimal places. Write your answer as a whole number (and make sure that it is the minimum whole number that satisfies the requirements).
To calculate the minimum sample size needed, we need to use the formula for sample size estimation in order to ensure that the estimate is within a certain margin of error. In this case, the margin of error is given as mm Hg.
The formula for sample size estimation is:
n = (Z * σ / E)²
Where:
- n is the sample size
- Z is the z-score corresponding to the desired level of confidence
- σ is the standard deviation of the population
- E is the desired margin of error
Since the desired level of confidence is not provided, let's assume a 95% confidence level. The corresponding z-score for a 95% confidence level is approximately 1.96.
Using the given values:
- Z = 1.96
- σ = mm Hg
- E = mm Hg
Plugging these values into the formula:
n = (1.96 * σ / E)²
n = (1.96 * mm Hg / mm Hg)²
n = (1.96)²
n = 3.8416
Rounding up to the nearest whole number, the minimum sample size needed is 4.