7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

2:14 minutes

Problem 7.2.25b

Textbook Question

Mean Pulse Rate of Males 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.

b. Assume that sigma=11.3 bpm, based on the value of s=11.3 bpm for the sample of 153 male pulse rates.

Verified step by step guidance

Step 1: Identify the formula for determining the minimum sample size required to estimate the population mean. The formula is:

n = \frac{{(z)}^{2}}{σ^{2}}

, where

n

is the sample size,

z

is the z-score corresponding to the confidence level,

σ

is the population standard deviation, and

E

is the margin of error.

Step 2: Determine the values given in the problem. The confidence level is 99%, so the z-score corresponding to this confidence level is approximately 2.576. The population standard deviation

σ

is given as 11.3 bpm, and the margin of error

E

is 2 bpm.

Step 3: Substitute the known values into the formula. Replace

z

with 2.576,

σ

with 11.3, and

E

with 2 in the formula:

n = \frac{{(2.576)}^{2}}{{11.3}^{2}}

Step 4: Simplify the numerator and denominator separately. First, calculate the square of the z-score (

{2.576}^{2}

) and the square of the standard deviation (

{11.3}^{2}

). Then, calculate the square of the margin of error (

2^{2}

Step 5: Divide the simplified numerator by the simplified denominator to find the minimum sample size

n

. If the result is not a whole number, always round up to the nearest whole number, as sample size must be an integer.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sample Size Determination

Sample size determination is a statistical process used to calculate the number of observations needed to achieve a desired level of precision in estimating a population parameter. In this context, it involves using the desired confidence level and margin of error to ensure that the sample mean accurately reflects the population mean within specified limits.

Confidence Interval

A confidence interval is a range of values, derived from a data set, that is likely to contain the population parameter with a specified level of confidence. In this case, a 99% confidence interval means that if we were to take many samples, approximately 99% of the calculated intervals would contain the true mean pulse rate of adult males.

Standard Deviation and Population Variance

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In this scenario, the given standard deviation (sigma = 11.3 bpm) is crucial for calculating the sample size, as it reflects the variability in pulse rates among adult males, influencing the width of the confidence interval and the required sample size.

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Textbook Question

Mean Pulse Rate of Males 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 .

Textbook Question

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