Question

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

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.

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

b. Assume that $\sigma =12.5 bpm$based on the value of $s=12.5 bpm$for the sample of 147 female pulse rates.

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 pulse rate of adult females using an estimate of $\sigma$is 480.

b. The sample size required to estimate the mean pulse rate of adult females instead of $\sigma$is 260.

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

Step-by-step solution

Given information

From the “body data,” 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.

Describe the determination of 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

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

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

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

Mathematically,

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

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 $z_{\frac{\alpha }{2}}$is 2.575.

Find the estimate of using the range rule of thumb

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:

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

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

$$\begin{gathered} \sigma \approx \frac{\text{range}}{4} \\ =\frac{68}{4} \\ =17 \end{gathered}$$

Thus, let$\sigma =17\text{bpm}$.

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).

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

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

Find the required sample size using instead s of   σ

b.

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

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

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

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

Compare the results of (a) and (b)

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.