Suppose That A Researcher Is Interested In Estimating The Me

Suppose That A Researcher Is Interested In Estimating The Mean Systoli

Suppose that a researcher is interested in estimating the mean systolic blood pressure, μ, of executives of major corporations. He plans to use the blood pressures of a random sample of executives of major corporations to estimate μ. Assuming that the standard deviation of the population of systolic blood pressures of executives of major corporations is 24 mm Hg, what is the minimum sample size needed for the researcher to be 99% confident that his estimate is within 6 mm Hg of μ? 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).

Paper For Above instruction

Estimating the mean systolic blood pressure of corporate executives with a specified confidence level and margin of error is a fundamental task in inferential statistics, often employed to inform health policies, clinical practices, and corporate health initiatives. In this scenario, a researcher aims to determine the minimum sample size required to estimate the population mean, μ, with a confidence level of 99% and a margin of error (E) of 6 mm Hg, given that the known population standard deviation (σ) is 24 mm Hg. This exercise involves understanding the relationship between sample size, confidence level, variability, and precision and applying the formula derived from the properties of the normal distribution.

The key to solving this problem lies in the application of the sample size determination formula for a mean with known standard deviation, which is given by:

n = (Z_{α/2} * σ / E)^2

where:

- n is the required sample size,

- Z_{α/2} is the critical value from the standard normal distribution corresponding to the desired confidence level,

- σ is the population standard deviation,

- E is the desired margin of error.

First, identifying the parameters:

- Confidence level = 99%, which corresponds to an α of 1 - 0.99 = 0.01.

- Half of alpha (α/2) = 0.005,

- The critical Z-value (Z_{α/2}) for a 99% confidence level can be found from standard normal distribution tables or statistical software: approximately 2.576.

- Population standard deviation, σ = 24 mm Hg,

- Margin of error, E = 6 mm Hg.

Next, applying the formula:

n = (Z_{α/2} * σ / E)^2

= (2.576 * 24 / 6)^2

= (2.576 * 4)^2

= (10.304)^2

≈ 106.273

Since the sample size must be a whole number and must satisfy the condition, we round up to the next whole number to ensure that the margin of error remains within specified bounds. Therefore, the minimum sample size is 107.

This calculation underscores the importance of understanding the interplay among confidence level, variability, and precision in designing studies. A higher confidence level or greater variability would necessitate a larger sample size, while a more precise estimate (smaller margin of error) also increases the required sample. Properly determining sample size ensures statistical validity and resource efficiency.

In conclusion, the researcher should survey a minimum of 107 executives to achieve the desired level of confidence and accuracy in estimating the average systolic blood pressure of this population.

References

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  • Statistics Solutions. (2018). Calculating sample size for estimating a mean. Retrieved from https://www.statisticssolutions.com/calculating-sample-size-for-estimating-a-mean/

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