Question

To estimate the amount of lumber in a tract of timber, an owner decided to count the number of trees with diameters exceeding 12 inches in randomly selected 50 -by-50foot squares. Seventy 50 -by-50-foot squares were chosen, and the selected trees were counted in each tract. 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 timber trees for all 50 -by-50-foot 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

Based on the provided solution, answer the following question: Question: Calculate the sample standard deviation and construct the following intervals: \(\bar{x} \pm s\), \(\bar{x} \pm 2s\), and \(\bar{x} \pm 3s\). Compare the percentage of squares in these intervals with the Empirical Rule and Tchebysheff's Theorem percentages. Answer: To calculate the sample standard deviation, first find the sample variance using the formula \(s^2 = \frac{1}{69}\sum_{i=1}^{70} (x_i - 7.64)^2\) and then take the square root, obtaining \(s\). Construct the intervals as the sum of \(\bar{x}\) and multiples of \(s\), and then compare the percentages of squares in these intervals with the percentages given by the Empirical Rule (68%, 95%, and 99.7%) and Tchebysheff's Theorem (lower bounds 1 - (1/s^2), 1 - (1/4s^2), 1 - (1/9s^2)).

Step-by-step solution

Find Relative Frequencies and Construct Histogram

First, let's find the range of the dataset by calculating the difference between the maximum and minimum values: Max = 13; Min = 2; Range = 13 - 2 = 11 Let's use bins of width 1. Therefore, the number of bins needed is Range/Width = 11/1 = 11. Now, create a frequency table and calculate the relative frequencies for each bin. Then, use these relative frequencies to construct a histogram.

Calculate the Sample Mean

The definition of the sample mean is \(\bar{x} = \frac{1}{n} \sum x_i\), where \(n\) is the sample size and \(x_i\) are the data values. First, find the total number of squares, which is the product of rows and columns: 10 rows × 7 columns = 70 squares. Calculate the sum of all tree counts and divide by the total number of squares: Sample Mean, \(\bar{x} = \frac{1}{70}\sum{x_i} = \frac{1}{70}(7+8+7+10+\cdots+5+9+8) = \frac{535}{70} \approx 7.64\)

Calculate Sample Standard Deviation and Construct Intervals

First, we need to calculate the sample variance \(s^2\) using the formula \(s^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2\). Compute the deviations, square them and then sum them up: \(s^2 = \frac{1}{69}\sum_{i=1}^{70} (x_i - 7.64)^2\) Now, let's calculate the sample standard deviation, \(s = \sqrt{s^2}\). Round your answer to two decimal places. Construct the intervals as the sum of \(\bar{x}\) and multiples of \(s\) as required: \(\bar{x} \pm s\), \(\bar{x} \pm 2s\), and \(\bar{x} \pm 3s\). Next, count the number of squares falling into each interval: 1. \(\bar{x} \pm s\) interval: Count the squares with \(x_i\) in this range and calculate the percentage. 2. \(\bar{x} \pm 2s\) interval: Count the squares with \(x_i\) in this range and calculate the percentage. 3. \(\bar{x} \pm 3s\) interval: Count the squares with \(x_i\) in this range and calculate the percentage. Lastly, compare these percentages with the Empirical Rule and Tchebysheff's Theorem percentages. The Empirical Rule assumes a normal distribution and gives approximate percentages for the intervals \(\bar{x} \pm s\) (68%), \(\bar{x} \pm 2s\) (95%), and \(\bar{x} \pm 3s\) (99.7%). Tchebysheff's Theorem applies to any distribution and gives lower bounds for these intervals: 1 - (1/s^2) for \(\bar{x} \pm s\), 1 - (1/4s^2) for \(\bar{x} \pm 2s\), and 1 - (1/9s^2) for \(\bar{x} \pm 3 s\). Calculate the lower bounds and compare them with the obtained percentages from the data.

Key concepts

Frequency Histogram

A frequency histogram is a tool widely used in statistics to graphically summarize and display the distribution of a set of data. It's a type of bar graph that shows the frequency of items occurring within certain ranges (or bins). The frequency of the data that falls within each bin is presented by the height of the bar.

Creating a frequency histogram involves several steps. It starts with deciding the range of the data and choosing an appropriate bin width. The entire range of data is then split into non-overlapping intervals, or bins. The frequency of data within each bin is counted and plotted as bars.

For our timber tract example, after determining the frequency of tree counts within specified intervals, we can construct a histogram to visualize the distribution of tree counts per 50-by-50-foot square. This graphical representation aids in quickly assessing the concentration of data points and understanding the distribution pattern.

Sample Mean Calculation

The sample mean, denoted as \(\bar{x}\), is a typical measure used to estimate the central tendency of a set of data. It represents the average value and is calculated by adding up all the data values and dividing by the number of data points. In mathematical terms, this is expressed as \(\bar{x} = \frac{1}{n} \sum x_i\), where \(n\) is the sample size, and \(x_i\) are the individual data values.

In the context of the timber square exercise, we count the total number of trees in each 50-by-50-foot tract and divide by the number of tracts surveyed to find the mean number of trees. This calculation gives us an estimate of the typical tree count we might expect in an average tract.

Sample Standard Deviation

The sample standard deviation is a measure of how spread out numbers are in a set of data. It's the square root of the sample variance, which is the average of the squared differences from the sample mean. The formula for the sample variance \(s^2\) is \(s^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2\), where \(n-1\) is the degrees of freedom and \(x_i - \bar{x}\) represents the deviation of each data point from the mean.

For our exercise, we compute the variance and square root to get the standard deviation. We then create intervals around the mean (\(\bar{x} \pm s\), \(\bar{x} \pm 2s\), and \(\bar{x} \pm 3s\)) and determine the proportion of data points within these intervals. This information provides us with semantic insight into the variability of the sample data.

Empirical Rule and Tchebysheff's Theorem

The Empirical Rule and Tchebysheff's Theorem are concepts that help us understand the spread of data in relation to the mean.

The Empirical Rule is applicable only when the data distribution is approximately normal (bell-shaped) and states that roughly 68% of data falls within one standard deviation (\(\bar{x} \- s\) to \(\bar{x} \+ s\)), 95% falls within two standard deviations (\(\bar{x} \- 2s\) to \(\bar{x} \+ 2s\)), and 99.7% falls within three standard deviations (\(\bar{x} \- 3s\) to \(\bar{x} \+ 3s\)).

On the contrary, Tchebysheff's Theorem provides a broader application for any type of data distribution. It guarantees that at least \((1-\frac{1}{k^2}) * 100\%\) of the data values will fall within \(k\) standard deviations of the mean, where \(k\) is any positive number greater than 1.

In our exercise, comparing the percentages of data within the calculated intervals to the predictions made by the Empirical Rule and the lower bounds provided by Tchebysheff's Theorem reinforces the understanding of statistical theories in practical scenarios.