Find the sample size required to estimate the population...

Find the sample size required to estimate the population mean. Data Set 1 "Body Data" in Appendix B includes pulse rates of 153 randomly selected adult males, and those pulse rates vary from a low of 40 bpm to a high of 104 bpm. Find the minimum sample size required to estimate the mean pulse rate of adult males. Assume that we want $99 \%$ confidence that the sample mean is within 2 bpm of the population mean. a. Find the sample size using the range rule of thumb to estimate $\sigma .$ b. Assume that $\sigma=11.3$ bpm, based on the value of $s=11.3$ bpm for the sample of 153 male pulse rates. c. Compare the results from parts (a) and (b). Which result is likely to be better?

Find the sample size required to estimate the population mean.
Data Set 1 "Body Data" in Appendix B includes pulse rates
of 153 randomly selected adult males, and those pulse rates vary from a low of 40 bpm to a high of 104 bpm. Find the minimum sample size required to estimate the mean pulse rate of adult males. Assume that we want $99 \%$ confidence that the sample mean is within 2 bpm of the population mean.
a. Find the sample size using the range rule of thumb to estimate $\sigma .$
b. Assume that $\sigma=11.3$ bpm, based on the value of $s=11.3$ bpm for the sample of 153 male pulse rates.
c. Compare the results from parts (a) and (b). Which result is likely to be better?

Key Concepts

Confidence Interval

A confidence interval is a range of values used to estimate a population parameter with a designated level of confidence. It’s constructed around the sample mean, incorporating the margin of error, which accounts for the variability in the data due to sampling.

Margin of Error

The margin of error represents the maximum expected discrepancy between the true population parameter and the sample estimate. It is calculated by multiplying the critical value from the relevant probability distribution by the standard error, setting the precision of the estimate in the context of the desired confidence level.

Standard Deviation and its Estimation

Standard deviation is a measure of how spread out the data are around the mean. When the true population standard deviation is unknown, it can be estimated using sample data. Techniques such as the range rule of thumb provide a quick approximate estimate by using the difference between the maximum and minimum values observed in the sample.

Sample Size Calculation for the Mean

Sample size calculation for estimating the mean aims to determine the number of observations required so that the margin of error falls within a specified range at a given confidence level. The formula n = (Z * sigma / E)^2 is used, where Z is the critical value, sigma is the standard deviation or its estimate, and E is the desired margin of error.

Critical Value

The critical value is a cutoff point from the probability distribution (commonly the standard normal distribution for large samples) that corresponds to the chosen confidence level. It determines how many standard errors away from the mean the population parameter may lie, and is integral to computing the margin of error.

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