Question

The data below are 30 waiting times between eruptions of the Old Faithful geyser in Yellowstone National Park. $$ \begin{array}{lllllllllllllll} 56 & 89 & 51 & 79 & 58 & 82 & 52 & 88 & 52 & 78 & 69 & 75 & 77 & 72 & 71 \\ 55 & 87 & 53 & 85 & 61 & 93 & 54 & 76 & 80 & 81 & 59 & 86 & 78 & 71 & 77 \end{array} $$ a. Calculate the range. b. Use the range approximation to approximate the standard deviation of these 30 measurements. c. Calculate the sample standard deviation \(s\). d. What proportion of the measurements lie within two standard deviations of the mean? Within three standard deviations of the mean? Do these proportions agree with the proportions given in Tchebysheff's Theorem?

Short answer

Answer: The range of the waiting times between eruptions of the Old Faithful geyser is 42, and the approximate standard deviation using the range approximation is 10.5.

Step-by-step solution

Find the Minimum and Maximum Values

The minimum value is the smallest waiting time, and the maximum value is the highest waiting time in the data. In this case: Minimum Value = 51 Maximum Value = 93

Calculate the Range

Range is the difference between the maximum and minimum values. Range = Maximum Value - Minimum Value = 93 - 51 = 42

Approximate the Standard Deviation Using Range Approximation

The range approximation rule states that the approximate standard deviation can be calculated as: Approximate Standard Deviation = Range / 4 = 42 / 4 = 10.5

Calculate the Sample Mean

To calculate the sample mean, we sum all waiting times and divide by the number of measurements, which is 30. Sample Mean = (Σ(waiting times))/30

Calculate the Sample Standard Deviation

The sample standard deviation formula is: \(s = \sqrt{\frac{\sum{(x_i - \bar{x})^2}}{n - 1}}\) Here \(\bar{x}\) is the sample mean, \(x_i\) are the individual measurements, and \(n\) is the number of measurements.

Determine the Proportions Within Two and Three Standard Deviations

Find the number of measurements within two standard deviations (2s) and three standard deviations (3s) of the sample mean, and calculate the corresponding proportions as follows: Proportion within 2s = (Number of Measurements within 2s)/30 Proportion within 3s = (Number of Measurements within 3s)/30

Compare the Proportions with Tchebysheff's Theorem

Finally, compare the calculated proportions (from Step 6) with Tchebysheff's Theorem, which states that at least (1 - 1/k^2) of the data will be within k standard deviations of the mean (\(\bar{x}\)). In this exercise, we will check for k=2 and k=3: Tchebysheff's Proportion for 2s: 1 - 1/2^2 = 0.75 (75%) Tchebysheff's Proportion for 3s: 1 - 1/3^2 = 0.89 (89%) Compare the calculated proportions for 2s and 3s with these Tchebysheff's proportions to check if they agree.

Key concepts

Statistical Range Calculation

Understanding the range of a dataset is crucial for grasping the variability of values. When analyzing something as predictable yet variable as the eruption times of Old Faithful, the range gives us a basic measure of its consistency or lack thereof. To calculate the range, simply subtract the smallest value (minimum) from the largest value (maximum) in the dataset. In the case of the Old Faithful geyser waiting times, the range is 42 minutes.

This is a simple calculation, but its implications are significant for anyone interested in statistics or even planning a visit to Yellowstone National Park. For a geyser watcher, this range informs them that they might wait anywhere from 51 to 93 minutes to see an eruption. For statisticians, it sets the stage for more complex analyses, like standard deviation and Tchebysheff's Theorem.

Standard Deviation Approximation

Standard deviation is a key statistical concept that measures the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. To approximate the standard deviation of the Old Faithful geyser waiting times, one can use the range approximation rule. By dividing the range of 42 minutes by 4, we get an approximate standard deviation of 10.5 minutes.

But why divide by 4? This is based on the assumption that many datasets resemble a normal distribution, where about 95% of the data falls within two standard deviations of the mean—corresponding to roughly 4 'chunks' of this distance. While it's a rough estimate, this approximation helps quickly gauge the variability in waiting times without complex calculations.

Tchebysheff's Theorem

Tchebysheff's Theorem is a statistical rule that applies to all different types of distributions, not just the normal distribution. This versatility makes it incredibly useful for datasets that are non-normal or when the distribution is unknown. According to the theorem, for any dataset, a certain minimum proportion of data points will fall within a specified number of standard deviations from the mean. For example, no matter the shape of the distribution, at least 75% of measurements will fall within two standard deviations (k=2) from the mean, and at least 89% will be within three standard deviations (k=3).

In the Old Faithful geyser problem, we can use this theorem to predict the waiting time distribution. Even without knowing the exact distribution type, by applying Tchebysheff's Theorem, we're guaranteed a certain minimum proportion of the geyser eruptions falling within the calculated ranges of standard deviations. This provides both park visitors and scientists with a statistically sound window into the reliability of the geyser's schedule.