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 ?

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To determine the minimum sample size needed for the researcher to be confident that his estimate is within a certain margin of error, we can use the formula for sample size calculation for estimating a population mean.

The formula for sample size calculation to estimate the population mean is given by:

n = [((Z * σ) / E) ^ 2]

Where:
n = sample size
Z = Z-score (corresponding to the desired confidence level)
σ = standard deviation of the population
E = margin of error (the maximum difference between the sample mean and the population mean)

In this case, the researcher wants to estimate the mean systolic blood pressure, μ. The standard deviation of the population is given as mm Hg. The researcher wants the estimate to be within mm Hg of the true mean.

To determine the value of Z (Z-score) corresponding to the desired confidence level, we need to specify the confidence level. Let's assume a 95% confidence level, which corresponds to a Z-score of approximately 1.96.

Plugging in the given values into the formula:

n = [((1.96 * mm Hg) / mm Hg) ^ 2]

Simplifying this equation:

n = [(1.96 ^ 2 * σ ^ 2) / E ^ 2]

n = [(3.8416 * σ ^ 2) / E ^ 2]

Now we can substitute the values into the equation:

n = [(3.8416 * σ ^ 2) / E ^ 2]
n = [(3.8416 * ( mm Hg) ^ 2) / ( mm Hg) ^ 2]

Simplifying further:

n = 3.8416

Therefore, the minimum sample size needed for the researcher to be confident that his estimate is within mm Hg of the mean is 3.8416.