7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

1:31 minutes

Problem 7.2.26b

Textbook Question

Mean Pulse Rate of Females Data Set 1 “Body Data” in Appendix B includes pulse rates of 147 randomly selected adult females, and those pulse rates vary from a low of 36 bpm to a high of 104 bpm. Find the minimum sample size required to estimate the mean pulse rate of adult females. Assume that we want 99% confidence that the sample mean is within 2 bpm of the population mean.

b. Assume that sigma=12.5 bpm, based on the value of s=12.5 bpm for the sample of 147 female 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 12.5 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 12.5, and

E

with 2 in the formula:

n = \frac{{(2.576)}^{2}}{{12.5}^{2}}

Step 4: Simplify the numerator by squaring the z-score and the standard deviation. Then, simplify the denominator by squaring the margin of error.

Step 5: Divide the simplified numerator by the simplified denominator to calculate the minimum sample size. 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 is a reliable estimate of the population mean.

Confidence Interval

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

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 = 12.5 bpm) is crucial for calculating the sample size, as it reflects the variability in pulse rates among adult females, influencing the precision of the estimate.

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Textbook Question

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Textbook Question

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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

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

Textbook Question

Mean Body Temperature Data Set 5 “Body Temperatures” in Appendix B includes 106 body temperatures of adults for Day 2 at 12 AM, and they vary from a low of 96.5F to a high of 99.6F. Find the minimum sample size required to estimate the mean body temperature of all adults. Assume that we want 98% confidence that the sample mean is within 0.1F of the population mean.

b. Assume that sigma=0.62F, based on the value of s=0.62F for the sample of 106 body temperatures.

Textbook Question

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Multiple Choice

Gas prices are getting more and more expensive. The average gas price, from a random sample of 100 gas stations, was $3.50. It is assumed that gas prices have a standard deviation of $0.04. Construct an 80% confidence interval for the true mean gas price in the United States.

Multiple Choice

Books get more and more expensive every semester, but the distribution of their prices is always normal. 25 randomly selected students in your school spent, on average $500 with a standard deviation of $50. Construct a 98% confidence interval for the true spending on books.

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