Question

Question: Evaporation from swimming pools. Refer to the Heating/ Piping/Air Conditioning Engineering (April 2013) study of evaporation from occupied swimming pools, Exercise 6.34 (p. 349). The researchers desired an estimate of the mean absolute value of the deviation between the actual and estimated evaporation level (recorded as a percentage). Using a small sample, the researchers obtained the following summary statistics for absolute deviation percentage x = 18%, s = 20%. How many swimming pools must be sampled to estimate the true mean absolute deviation percentage to within 5% using a 90% confidence interval?

Short answer

The number of swimming pools must be sampled to estimate the true mean absolute deviation percentage to within 5% using a 90% confidence interval is 43.

Step-by-step solution

Given Information

The confidence interval is 90%.

The sample standard deviation is 20%.

The sampling error is 5%.

State the formula used to obtain the sample size

The general formula for the sample size is given below:

$n={\left[\frac{Z_{\frac{\alpha }{2}}\left(\sigma \right)}{SE}\right]}^2$

Where SE represents the sampling error.

The value of $\sigma$is usually unknown. It can be estimated by the standard deviation, s from the prior sample.

 Compute the value

For the confidence level of 90%, the level of significance is 0.90.

$$\begin{gathered} For\left(1-\alpha \right)=0.90 \\ \alpha =0.10 \\ \frac{\alpha }{2}=0.05 \end{gathered}$$

Thecorresponding to$Z_{\frac{\alpha }{2}}$ the standard normal table is,

$$\begin{gathered} Z_{\frac{\alpha }{2}}=Z_{0.05} \\ =1.645 \end{gathered}$$

Compute the sample size

The sample standard deviation is

$$\begin{gathered} s=\frac{20}{100} \\ =0.2 \end{gathered}$$

The sampling error is,

$$\begin{gathered} SE=\frac{5}{100} \\ =0.05 \end{gathered}$$

The sample size is computed as:

$$\begin{gathered} n={\left[\frac{\left(1.645\right)\left(0.2\right)}{0.05}\right]}^2 \\ ={\left[6.58\right]}^2 \end{gathered}$$

Hence, the number of swimming pools must be sampled to estimate the actual mean absolute deviation percentage to within 5% using a 90% confidence interval is 43.