Evaporation from swimming pools. A new formula for estimating the water evaporation from occupied swimming pools was proposed and analyzed in the journal Heating Piping/Air Conditioning Engineering (April 2013). The key components of the new formula are number of pool occupants, area of pool’s water surface, and the density difference between room air temperature and the air at the pool’s surface. Data were collected from a wide range of pools for which the evaporation level was known. The new formula was applied to each pool in the sample, yielding an estimated evaporation level. The absolute value of the deviation between the actual and estimated evaporation level was then recorded as a percentage. The researchers reported the following summary statistics for absolute deviation percentage: $\overline{\mathbf{x}}\mathbf{=}\mathbf{18}$, $\mathit{s}\mathbf{=}\mathbf{20}$. Assume that the sample contained$\mathit{n}\mathbf{=}\mathbf{15}$ swimming pools

a. Estimate the true mean absolute deviation percentage for the new formula with a 90% confidence interval.

b. The American Society of Heating, Refrigerating, and Air-Conditioning Engineers (ASHRAE) handbook also provides a formula for estimating pool evaporation. Suppose the ASHRAE mean absolute deviation percentage is $\mathit{\mu }\mathbf{=}\mathbf{40}\mathit{\%}$. (This value was reported in the article.) On average, is the new formula “better” than the ASHRAE formula? Explain

Step by step solution

01

Given information

Sample mean $\overline{x}=18$,

Sample standard deviation s=20

Sample size n=5

02

Calculating the true mean absolute deviation percentage for the new formula with 90% confidence interval..

To estimate the true mean absolute deviation percentage for the new formula with 90% confidence interval.

The degree of freedom can be calculated as

$\begin{array}{c}\mathrm{df}=\mathrm{n}-1\\ =15-1\\ =14\end{array}$

Confidence coefficient is given 0.90. hence,

$\begin{array}{c}1-\alpha =0.90\\ \alpha =1-0.90\\ \alpha =0.1\end{array}$

$\frac{\alpha }{2}=0.05$

Therefore, from the “table 3 appendix D” the vale of $t_{0.05,14}=1.761$

90% of confidence interval is obtained as:

$\begin{array}{c}\overline{\mathrm{x}}\pm {\mathrm{t}}_{\frac{\mathrm{\alpha }}{2}}\left(\frac{\mathrm{s}}{\sqrt{\mathrm{n}}}\right)=18\pm 1.761\left(\frac{20}{\sqrt{15}}\right)\\ =18\pm 9.094\\ =\left(-18-9.094,18+9.094\right)\end{array}$

$\overline{\mathrm{x}}\pm {\mathrm{t}}_{\frac{\mathrm{\alpha }}{2}}\left(\frac{\mathrm{s}}{\sqrt{\mathrm{n}}}\right)=\left(8.906,27.094\right)$

Hence, the true mean absolute deviation percentage for the new formula with a 90% confidence interval is$\left(8.906,27.094\right)$

03

Calculating the true mean absolute deviation percentage for the new formula with 90% confidence interval..

Yes the new formula is better than the ASHRAE formula because the ASHRAE mean absolute deviation percentage is$\mu =34\%$ does not lie between the lower and upper limits .that is $\left(8.906,27.094\right)$,

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