Question

Estimating Standard Deviation Listed below is a sample of duration times (seconds) of eruptions of the Old Faithful geyser. Use the range rule of thumb to estimate the value of the standard deviation of all duration times, and compare the result to the standard deviation of 33.7 seconds found from a sample of 2634 duration times.

226 228 247 247 253 256 250 254 229 242 250 241 226 240 117

Short answer

The estimated value of the standard deviation of duration times from the given sample is equal to 34.8 seconds.

The estimated value of the standard deviation is quite close to the actual value of the standard deviation of duration times corresponding to the large sample.

Step-by-step solution

Given information

The duration times (in seconds) of eruptions of the Old Faithful geyser are provided. The actual value of the standard deviation of duration times from the given large sample is equal to 33.7 seconds.

Step 2:Range rule of thumb

The range rule of thumb to estimate the value of the standard deviation (denoted by s) is as follows:

$s\approx \frac{\text{Range}}{4}$

The range value for the given dataset is computed as follows:

$$\begin{gathered} \text{Range}=\text{Maximum}\text{value}-\text{Minimum}\text{value} \\ =256-117 \\ =139 \end{gathered}$$

Thus, the estimated value of the standard deviation is equal to:

$$\begin{gathered} s\approx \frac{\text{Range}}{4} \\ \approx \frac{139}{4} \\ \approx 34.8 \end{gathered}$$

Thus, the estimated value of the standard deviation of duration times using the range rule of thumb is equal to 34.8 seconds.

Comparison of actual value and estimated value 

The actual value of the standard deviation of duration times is equal to 33.7 seconds.

The estimated value of the standard deviation of duration times is equal to 34.75 seconds.

As the two values are quite close, it can be said that the range rule of thumb estimates the value of the standard deviation quite accurately.