Question

The data that follow 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 lies within two standard deviations of the mean? Within three standard deviations? Do these proportions agree with the proportions given in Tchebysheff's Theorem?

Short answer

Do they agree with Tchebysheff's Theorem? Answer: The proportion of measurements within two standard deviations of the mean is approximately 93%, and the proportion within three standard deviations is 100%. Both proportions agree with Tchebysheff's Theorem, which states that at least 75% of the data lies within two standard deviations of the mean, and at least 89% of the data lies within three standard deviations of the mean.

Step-by-step solution

Calculate the Range

First, identify the minimum and maximum values in the dataset. The minimum value is 51, and the maximum value is 93. Then, calculate the range by subtracting the minimum value from the maximum value: Range = Max - Min = 93 - 51 = 42.

Approximate the Standard Deviation Using the Range Rule

The range rule states that the standard deviation can be approximated by dividing the range by 4: Approximate Standard Deviation = Range / 4 = 42 / 4 = 10.5.

Calculate the Sample Standard Deviation

To compute the sample standard deviation, follow these steps: 1. Calculate the mean of the dataset. 2. Subtract the mean from each data point and square the result. 3. Calculate the mean of the squared differences. 4. Take the square root of the mean of the squared differences. Mean = (sum of all data points) / 30 = (sum of all waiting times) / 30 Sample Standard Deviation \(s = \sqrt{\frac{\sum_{i=1}^{30} (x_i -\bar{x})^2}{30-1}}\), where \(x_i\) are the data points and \(\bar{x}\) is the mean. With the given data, after performing the above calculations, we get the sample standard deviation, \(s \approx 11.6\).

Proportion of Measurements within Two and Three Standard Deviations

Determine the proportion of measurements that lie within two and three standard deviations of the mean: 1. Calculate the mean from Step 3 (rounded), which is approximately 71. 2. Calculate the boundaries for two standard deviations: 71 - 2 * 11.6 = 47.8 and 71 + 2 * 11.6 = 94.2. 3. Count the number of measurements that lie within these boundaries (28 out of 30). 4. Determine the proportion within two standard deviations: 28 / 30 ≈ 0.93 or 93%. Repeat the process for three standard deviations: 1. Calculate the boundaries for three standard deviations: 71 - 3 * 11.6 = 36.2 and 71 + 3 * 11.6 = 105.8. 2. Count the number of measurements that lie within these boundaries (30 out of 30). 3. Determine the proportion within three standard deviations: 30 / 30 = 1 or 100%. Finally, compare these proportions to Tchebysheff's Theorem: - For \(k=2\), Tchebysheff's Theorem states that at least \(1 - (1/2^2) = 1- 1/4 = 0.75\) or 75% of the data lies within two standard deviations of the mean. Our proportion of 93% agrees with this theorem. - For \(k=3\), Tchebysheff's Theorem states that at least \(1 - (1/3^2) = 1- 1/9 ≈ 0.89\) or 89% of the data lies within three standard deviations of the mean. Our proportion of 100% agrees with this theorem.

Key concepts

Standard Deviation Approximation

The concept of standard deviation approximation is a useful tool in statistics for estimating the spread of a dataset when detailed calculations are not feasible. Through this method, one can gain insights into the variability of data quickly. The range rule of thumb is a common approach for approximating standard deviation, where it is assumed that the standard deviation is approximately one-fourth of the range of the data.

For example, if the range (the difference between the highest and lowest observations) of a dataset is 42, as in the given exercise, the approximate standard deviation would be calculated by dividing the range by 4, yielding an estimate of 10.5. This approach provides a quick and easy way to get an idea of the data dispersion, which is particularly useful in preliminary data analysis or when teaching basic statistics concepts to beginners.

Sample Standard Deviation

The sample standard deviation, denoted as 's', is more accurate than the approximate standard deviation as it takes into account each data point's deviation from the mean. This measure of spread explains how much data varies around the mean of the sample. To compute it, one follows several steps: calculating the mean, subtracting the mean from each value to find deviations, squaring these deviations, finding the average of these squares, and taking the square root.

In practical terms, if we have a sample of 30 waiting times for the Old Faithful geyser's eruptions, the sample standard deviation tells us, on average, how much individual eruption times differ from the sample's average eruption time. Calculating the sample standard deviation provides us with a precise measurement of 11.6, which is a fundamental statistic for understanding the dataset's variability.

Tchebysheff's Theorem

Tchebysheff's theorem is a key principle in statistics, giving us a powerful way to understand the distribution of data regardless of the shape of the distribution. This theorem provides a guarantee for data spread, stating that for any dataset and for any number of standard deviations greater than one, a minimum proportion of data values will fall within that number of standard deviations from the mean. Specifically, at least \(1 - (1/k^2)\) of data will lie within k standard deviations from the mean.

For two standard deviations (\(k=2\)), Tchebysheff's theorem predicts at least 75% of data points will fall within this range; for three standard deviations (\(k=3\)), it predicts at least 88.9%. When we examine the Old Faithful geyser's eruption times, we find that 93% and 100% of the observations fall within two and three standard deviations of the mean, respectively. These values comply with Tchebysheff's theorem, indicating its practical application in assessing the reliability of statistical estimates.

Descriptive Statistics

Descriptive statistics involve the procedures used to summarize and describe the main features of a dataset. They are extremely useful in simplifying large amounts of data into a more comprehensible form without losing the essence of the original data. Descriptive statistics typically include measures of central tendency such as mean and median, measures of variability like range and standard deviation, and graphical representations like histograms and box plots.

For instance, the set of waiting times for the Old Faithful geyser eruptions allows us to compute descriptive statistics to present an overview of the data. While the range provides a basic measure of spread, the standard deviation gives us an idea of how much individual measurements typically deviate from the mean eruption time. Being well-versed in these statistics not only helps students analyze data effectively but also enables them to critically evaluate the information presented in research studies, reports, and everyday news.