91影视

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

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

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

$n={\left[\frac{z_{\frac{伪}{2}}脳蟽}{E}\right]}^2鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�...\left(1\right)$

Here, E is the margin of error.

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.

$蟽鈮�\frac{\text{range}}{4}鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�...\left(2\right)$

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

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

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

Mathematically,

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

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" $z_{\frac{伪}{2}}$ is 2.33.

a.

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

Therefore, the range of body temperature is as follows.

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

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

$$\begin{gathered} 蟽鈮�\frac{\text{range}}{4} \\ =\frac{3.1}{4} \\ =0.775 \end{gathered}$$

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

$$\begin{gathered} \text{n}={\left[\frac{z_{\frac{伪}{2}}脳蟽}{\text{E}}\right]}^2 \\ ={\left[\frac{2.33脳0.775}{0.1}\right]}^2 \\ =326.07 \\ =327鈥�鈥�鈥�鈥�\left(\text{rounded}鈥�鈥�\text{off}\right) \end{gathered}$$

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

b.

Assume that$蟽=0.62F$ 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$z_{\frac{伪}{2}}$,$s$ , and E in equation (1), as follows.

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

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

c.

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

The sample size required to estimate the mean temperature of all adults $s$ instead of$蟽$ 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 $蟽$ obtained from the range rule of thumb.