91Ó°ÊÓ

Data Set 3 "Body Temperatures" in Appendix B includes 106 body temperatures of adults for Day 2 at \(12 \mathrm{AM}\), and they vary from a low of \(96.5^{\circ} \mathrm{F}\) to a high of \(99.6^{\circ} \mathrm{F}\). 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 \(0.1^{\circ} \mathrm{F}\) of the population mean. a. Find the sample size using the range rule of thumb to estimate \(\sigma\). b. Assume that \(\sigma=0.62^{\circ} \mathrm{F}\), based on the value of \(s=0.62^{\circ} \mathrm{F}\) for the sample of 106 body temperatures. c. Compare the results from parts (a) and (b). Which result is likely to be better?

Step by step solution

01

Determine the required sample size formula

The formula for the minimum sample size required to estimate the population mean is given by \[ n = \frac{(Z \times \frac{\text{standard deviation}}{\text{margin of error}})^2}{MOE^2} \]where Z is the Z-score for the confidence level, \text{standard deviation} is \text{σ}, and \text{margin of error} is \text{E}.

02

Find the Z-score for 98% Confidence Level

For a 98% confidence level, approximately 2% is outside the confidence interval. Since the normal distribution is symmetric, 1% is in each tail. So, we find the Z-score for 0.01 in the standard normal distribution table, which approximately equals 2.33.\[ Z = 2.33 \]

03

Estimate \text{σ} using the range rule of thumb

The range rule of thumb estimates the standard deviation as follows: \[ \text{σ} \text{≈} \frac{\text{Range}}{4} \]Given the range of body temperatures is 99.6°F - 96.5°F = 3.1°F, the standard deviation can be estimated as\[ \text{σ} \text{≈} \frac{3.1}{4} = 0.775°F \]

04

Calculate the sample size using the estimated \text{σ}

Using \text{σ} = 0.775°F and MOE = 0.1°F:\[ n = \frac{(2.33 \times 0.775)^2}{(0.1)^2} = \frac{(1.80775)^2}{0.01} ≈ \frac{3.268}{0.01} = 326.8 \]Rounding up, we get \[ n ≈ 327 \]

05

Calculate the sample size using the given \text{σ}

Given \text{σ} = 0.62°F:\[ n = \frac{(2.33 \times 0.62)^2}{(0.1)^2} = \frac{(1.4446)^2}{0.01} ≈ \frac{2.086}{0.01} = 208.6 \]Rounding up, we get \[ n ≈ 209 \]

06

Compare the results

The sample size estimated using the range rule of thumb (\( n ≈ 327 \)) is higher than the sample size calculated using the given standard deviation (\( n ≈ 209 \)). The result using the given standard deviation is likely to be better as it is based on actual data, which provides a more precise estimate.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Interval

Confidence intervals are ranges that estimate the population parameter, like the mean, with a specified probability—or confidence level. A 98% confidence interval means we are 98% sure the population mean lies within our interval. To find this interval, we use a Z-score, which depends on our desired confidence level. For a 98% confidence level, the Z-score is roughly 2.33. The formula for the interval combines this score with our data’s standard deviation and sample size.
Confidence intervals give us a range instead of a single estimate. This accounts for sample variability and provides more reliable results.

Standard Deviation

Standard deviation (σ) measures the amount of variation in a data set. It shows how much individual data points differ from the mean. In our exercise, the body temperature values range from 96.5°F to 99.6°F, with a given standard deviation of 0.62°F. Another way to estimate standard deviation is using the range rule of thumb, which suggests σ is approximately the range divided by 4. For our temperatures, that calculation gives us 0.775°F.
Lower standard deviations indicate that the data points are close to the mean, whereas higher standard deviations show more spread-out data.

Population Mean

The population mean is the average of a population’s values. Unlike sample means, which are averages of a subset, the population mean encompasses the entire group. To estimate it accurately, we need a good sample size. We increase our confidence in the estimate by using a sufficiently large sample, leading to more precise results. In our exercise, we use the body temperatures data to estimate the mean body temperature of all adults, ensuring a high confidence level by using calculated sample sizes.

Z-score

A Z-score indicates how many standard deviations a value is from the mean. For confidence intervals, Z-scores help define how confident we want to be. At a 98% confidence level, the Z-score is 2.33, meaning we are highly confident that the true population mean lies within our calculated range.
Z-scores are critical because they standardize scores, allowing comparison across different data sets or distributions.

Margin of Error

The margin of error (E) indicates the range within which we expect the true population mean to lie, given our sample data. It’s a critical component of calculating sample size. In our exercise, the margin of error was set to 0.1°F, meaning our sample mean should be no more than 0.1°F away from the true population mean.
Smaller margins of error require larger sample sizes to maintain the same confidence level, leading to more precise estimates.