Question

To estimate the amount of lumber in a tract of timber, an owner randomly selected seventy 15 -by-15-meter squares, and counted the number of trees with diameters exceeding 1 meter in each square. The data are listed here: $$ \begin{array}{rrrrrrrrrr} 7 & 8 & 7 & 10 & 4 & 8 & 6 & 8 & 9 & 10 \\ 9 & 6 & 4 & 9 & 10 & 9 & 8 & 8 & 7 & 9 \\ 3 & 9 & 5 & 9 & 9 & 8 & 7 & 5 & 8 & 8 \\ 10 & 2 & 7 & 4 & 8 & 5 & 10 & 7 & 7 & 7 \\ 9 & 6 & 8 & 8 & 8 & 7 & 8 & 9 & 6 & 8 \\ 6 & 11 & 9 & 11 & 7 & 7 & 11 & 7 & 9 & 13 \\ 10 & 8 & 8 & 5 & 9 & 9 & 8 & 5 & 9 & 8 \end{array} $$ a. Construct a relative frequency histogram to describe the data. b. Calculate the sample mean \(\bar{x}\) as an estimate of \(\mu,\) the mean number of trees for all 15 -by-15-meter squares in the tract. c. Calculate \(s\) for the data. Construct the intervals \(\bar{x} \pm s, \bar{x} \pm 2 s,\) and \(\bar{x} \pm 3 s .\) Calculate the percentage of squares falling into each of the three intervals, and compare with the corresponding percentages given by the Empirical Rule and Tchebysheff's Theorem.

Short answer

Question: Compare the percentages of squares falling within each of the three intervals with the percentages given by the Empirical Rule and Tchebysheff's Theorem.

Step-by-step solution

Compute the relative frequencies

We need to count the frequency of each value in the given data. Then, divide the frequency of each value by the total number of squares (70) to get the relative frequency.

Calculate the sample mean \(\bar{x}\)

To calculate the sample mean, add up all the tree counts and divide by the total number of squares (70): $$ \bar{x} = \frac{\sum x}{n} $$

Calculate the standard deviation \(s\)

To calculate the standard deviation \(s\), we use the formula: $$ s = \sqrt{\frac{\sum (x - \bar{x} )^2}{n - 1}} $$

Construct the intervals \(\bar{x} \pm s\), \(\bar{x} \pm 2s\), and \(\bar{x} \pm 3s\)

Now we will compute the three intervals: 1. \(\bar{x} \pm s\): \([\bar{x} - s, \bar{x} + s]\) 2. \(\bar{x} \pm 2s\): \([\bar{x} - 2s, \bar{x} + 2s]\) 3. \(\bar{x} \pm 3s\): \([\bar{x} - 3s, \bar{x} + 3s]\)

Calculate the percentage of squares falling into each interval

Count the total number of squares that fall into each interval and divide it by the total number of squares (70) to get the percentage.

Compare with the Empirical Rule and Tchebysheff's Theorem

The Empirical Rule states that for a normal distribution, approximately 68% of the data will fall within the first standard deviation, 95% within the second, and 99.7% within the third. Tchebysheff's Theorem states that for any distribution with mean \(\mu\) and standard deviation \(\sigma\), at least \((1 - \frac{1}{k^2})\) of the data will fall within \(k\) standard deviations from the mean for \(k > 1\). Compare the percentages calculated in Step 5 with the percentages given by the Empirical Rule (68%, 95%, 99.7%) and Tchebysheff's Theorem.

Key concepts

Relative Frequency Histogram

A relative frequency histogram is a graphical representation of the relative frequencies of data within specific intervals. Instead of just showing how many times each value appears, it shows what proportion of the total that count represents. This is particularly useful when comparing datasets of different sizes or when trying to understand the relative importance of certain data points within the context of the entire dataset. For example, if students are given a set of numbers representing tree counts in different land plots as in the textbook exercise, constructing a relative frequency histogram would allow them to visualize how frequently different tree counts occur relative to the whole sample, thereby enabling a better grasp of the distribution of tree counts across the land plots.

Sample Mean

The sample mean, often represented by \(\bar{x}\), is a crucial statistic used to estimate the central tendency, or average, of a population. To calculate the sample mean, we sum all the observed values and then divide by the number of observations. In the context of the textbook exercise, the mean number of trees with diameters exceeding 1 meter in the sampled 15-by-15-meter squares is calculated by adding together all the trees counted in the 70 squares and then dividing by 70, providing an estimate for the average (\(\mu\)) number of trees per square in the entire tract.

Standard Deviation

The standard deviation, symbolized by \(s\) in samples, is a measure of the dispersion or variability within a set of data. Essentially, it tells us how spread out the data points are in relation to the mean. A small standard deviation indicates that the data points tend to be close to the mean, while a large standard deviation suggests that the data points are spread out over a larger range of values. Calculating the standard deviation involves determining the average distance of each data point from the mean, squared, summed up, and then taking the square root of that average. In our exercise, students can see how much the number of trees in each 15-by-15-meter square deviates from the mean count.

Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical rule which states that in a normal distribution:

• About 68% of the data falls within one standard deviation of the mean.
• About 95% falls within two standard deviations.
• About 99.7% falls within three standard deviations.

This is an essential concept for students to understand because it gives them a quick way to assess the distribution of data in a set with a bell-shaped curve. If they learn that the sample from the textbook exercise resembles a normal distribution, the Empirical Rule lets them estimate how many tree counts should fall within each range defined by standard deviations from the mean.

Tchebysheff's Theorem

Tchebysheff's Theorem offers a broader application compared to the Empirical Rule, as it applies to any shaped distribution, not just the normal one. It states that for any dataset with a mean \(\mu\) and standard deviation \(\sigma\), at least \((1 - \frac{1}{k^2})\) of the data must fall within \(k\) standard deviations of the mean, for all \(k > 1\). In simpler terms, Tchebysheff's Theorem provides a guarantee for the minimum proportion of values that lie close to the mean. This concept is powerful for students learning statistics because it helps them develop expectations about their data regardless of its precise distribution. When comparing this theorem to the sample provided in the exercise, they can understand the minimum percentage of 15-by-15-meter squares with a certain range of tree counts.