Question

Sample Size. In Exercises 29–36, find the sample size required to estimate the population mean.

Mean Body Temperature Data Set 3 “Body Temperatures” in Appendix B includes 106 body temperatures of adults for Day 2 at 12 am, and they vary from a low of 96.5°F to a high of 99.6°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°F of the population mean.

a. Find the sample size using the range rule of thumb to estimate s.

b. Assume that $\sigma =0.62F$, based on the value of $s=0.6F$for the sample of 106 body temperatures.

c. Compare the results from parts (a) and (b). Which result is likely to be better?

Short answer

a. The sample size required to estimate the mean body temperature of all adults using the estimate of $\sigma$ is 327.

b. The sample size required to estimate the mean temperature of all adults instead of for a sample of 106 body temperatures is 209.

c. On comparing the results (a) and (b), the result obtained in part (b) is likely to be better than that in part (a).

Step-by-step solution

Given information

The temperature varies from a low 96.5°F to a high 99.6°F.

The required confidence level is 98%, and the sample mean is within 0.1°F of the population mean.

Describe the formula for sample size

The sample size n can be determined by using the following formula:

$n={\left[\frac{z_{\frac{\alpha }{2}}\times \sigma }{E}\right]}^2 ...\left(1\right)$

Here, E is the margin of error.

Describe the range rule of thumb 

The range rule of thumb is a simple tool for understanding and interpreting the standard deviation.It is used to estimate the standard deviation roughly from the collection of sample data.

The formula for the range rule of thumb is as follows.

$\sigma \approx \frac{\text{range}}{4} ...\left(2\right)$

Find the critical value  zα2

$z_{\frac{\alpha }{2}}$ is a z score that separates an area of $\frac{\alpha }{2}$ in the right tail of the standard normal distribution.

The confidence level 98% corresponds to $\alpha =0.02 \text{and} \frac{\alpha }{2}=0.01$.

The value $z_{\frac{\alpha }{2}}$, has the cumulative area $1-\frac{\alpha }{2}$to its left .

Mathematically,

$$\begin{gathered} P\left(z<z_{\frac{\alpha }{2}}\right)=1-\frac{\alpha }{2} \\ =0.99 \end{gathered}$$

From the standard normal table, the area of 0.99 is observed as the corresponding to the row value 2.3, column value 0.02, which implies that role="math" localid="1648032893802" $z_{\frac{\alpha }{2}}$ is 2.33.

Find the estimate of  using the range rule of thumb 

a.

Themean body temperature varies from a low 96.5°F to a high 99.6°F.

Therefore, the range of body temperature is as follows.

$$\begin{gathered} \text{Range} =99.6-96.5 \\ =3.1 \end{gathered}$$

The estimate of $\sigma$is obtained by substituting the value of range in equation (2). So,

$$\begin{gathered} \sigma \approx \frac{\text{range}}{4} \\ =\frac{3.1}{4} \\ =0.775 \end{gathered}$$

Find the required sample size using the estimate of   σ

The sample size is calculated by substituting the values of $z_{\frac{\alpha }{2}}$, $\sigma$, and E in equation (1), as follows.

$$\begin{gathered} \text{n}={\left[\frac{z_{\frac{\alpha }{2}}\times \sigma }{\text{E}}\right]}^2 \\ ={\left[\frac{2.33\times 0.775}{0.1}\right]}^2 \\ =326.07 \\ =327 \left(\text{rounded} \text{off}\right) \end{gathered}$$

Thus, with 326 samples values, we can be98% confident that the sample mean is within 0.1°F of the population mean.

Find the required sample size using  s instead of   σ

b.

Assume that$\sigma =0.62F$ is based on the value of $s=0.62F$ for the sample of 106 body temperatures.

The sample size is calculated by substituting the values of$z_{\frac{\alpha }{2}}$,$s$ , and E in equation (1), as follows.

$$\begin{gathered} \text{n}={\left[\frac{z_{\frac{\alpha }{2}}\times s}{\text{E}}\right]}^2 \\ ={\left[\frac{2.33\times 0.62}{0.1}\right]}^2 \\ =208.68 \\ =209 \left(\text{rounded} \text{off}\right) \end{gathered}$$

Thus, with 209 sample values, we can be98% confident that the sample mean is within 0.1°F of the population mean.

Compare the results (a) and (b)

c.

The sample size required to estimate the mean body temperature of all adults using the range rule thumb estimate of $\sigma$ is 327.

The sample size required to estimate the mean temperature of all adults $s$ instead of$\sigma$ for the sample of 106 body temperature is 209.

The result obtained in part (a) is larger than the result obtained in part (b).

The result from part (b) is better than the results from part (a) because it uses instead of the estimated $\sigma$ obtained from the range rule of thumb.