91影视

From the 鈥渂ody data,鈥� the pulse rates vary from 36 bpm to 104 bpm.

The given confidence level is 99%; so the sample mean is within 2 bpm of the true mean.

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

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

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:

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

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

The confidence level of 99% corresponds to $伪=0.01鈥�鈥�鈥�鈥�\text{and}鈥�鈥�鈥�鈥�\frac{伪}{2}=0.005$.

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

Mathematically,

$$\begin{gathered} P\left(z<z_{\frac{伪}{2}}\right)=1-\frac{伪}{2} \\ =0.995 \end{gathered}$$

From the standard normal table, the area of 0.995 is observed corresponding intersection of the row value 2.5 and between column value 0.07 and column value 0.08, which implies $z_{\frac{伪}{2}}$is 2.575.

a.

Themean pulse rate of females vary from a low level of 36 bpm to a high level of 104 bpm.

Therefore, the range of pulse rate is given as follows:

$$\begin{gathered} \text{Range}鈥�=104-36 \\ =68 \end{gathered}$$

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

$$\begin{gathered} 蟽鈮�\frac{\text{range}}{4} \\ =\frac{68}{4} \\ =17 \end{gathered}$$

Thus, let$蟽=17\text{bpm}$.

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

$$\begin{gathered} \text{n}={\left[\frac{z_{\frac{伪}{2}}脳s}{\text{E}}\right]}^2 \\ ={\left[\frac{2.575脳17}{2}\right]}^2 \\ =479.06 \\ =480鈥�鈥�鈥�鈥�\left(\text{rounded}鈥�鈥�\text{off}\right) \end{gathered}$$

Thus, with 480 sample values, you can be 99% confident that your sample mean lies within 2 bpm of the true mean.

b.

Assume that $蟽=12.5鈥�鈥�\text{bpm}$based on the value $s=12.5鈥�鈥�\text{bpm}$of for the sample of 147 female pulse rates

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

$$\begin{gathered} \text{n}={\left[\frac{z_{\frac{伪}{2}}脳s}{\text{E}}\right]}^2 \\ ={\left[\frac{2.575脳12.5}{2}\right]}^2 \\ =259.01 \\ =260鈥�鈥�鈥�鈥�\left(\text{rounded}鈥�鈥�\text{off}\right) \end{gathered}$$

Thus, with 260 sample values, you can be 99% confident that your sample mean lies within 2 bpm of the true mean.

c.

The sample size required to estimate the mean pulse rate of females by using the range rule thumb estimate of is 480.

The sample size required to estimate the mean pulse rate of females using instead of is 260.

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

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