Question

A set of \(n=10\) measurements consists of the values \(5.2,3,6,1,2,4,5,1,3 .\) a. Use the range approximation to estimate the value of \(s\) for this set. (HINT: Use the table at the end of Section \(2.5 .\) b. Use your calculator to find the actual value of \(s\). Is the actual value close to your estimate in part a? c. Draw a dotplot of this data set. Are the data mound shaped? d. Can you use Tchebysheff's Theorem to describe this data set? Why or why not? e. Can you use the Empirical Rule to describe this data set? Why or why not?

Short answer

Explain your reasoning. Answer: Yes, Tchebysheff's theorem can be applied to this dataset because it can be applied to any dataset with a defined mean and standard deviation, regardless of the shape of the data distribution. The dataset in this exercise has a mean and standard deviation, making it suitable for the application of Tchebysheff's theorem.

Step-by-step solution

Estimate the standard deviation using the range approximation method

To estimate the value of \(s\) (standard deviation) using the range approximation method, first find the range of the dataset by subtracting the smallest value from the largest value. Then, using the table from Section 2.5, divide the range by a value corresponding to the number of measurements (n=10) in the dataset. Range = Max - Min = 6 - 1 = 5 For n = 10, the corresponding value from the table is 3.5. Then: Estimated standard deviation (s) = Range / 3.5 = 5 / 3.5 = 1.43

Calculate the actual standard deviation using a calculator

Using a calculator, input the data values and find the standard deviation. The calculator returns a value of: Actual standard deviation (s) ≈ 1.51

Compare the estimated and actual standard deviation values

Now compare the estimated value to the actual value. Estimated standard deviation (s) = 1.43 Actual standard deviation (s) ≈ 1.51 The actual value is close to the estimated value, confirming that the range approximation method provides a reasonable estimate of the standard deviation.

Draw a dotplot and determine if the data is mound shaped

To draw a dotplot, plot each data value on the horizontal axis and draw a dot at each value's position. The dotplot will look like this: 1 - x x 2 - x 3 - x x 4 - x 5 - x 6 - x From the dotplot, the data does not appear to be mound shaped.

Evaluate whether Tchebysheff's theorem can be applied to the dataset

Tchebysheff's theorem can be applied to any dataset with a defined mean and standard deviation, regardless of the shape of the data distribution. Since the dataset has a mean and standard deviation, Tchebysheff's theorem can be applied to this dataset.

Evaluate whether the Empirical Rule can be applied to the dataset

The Empirical Rule can only be applied to datasets that are approximately mound shaped and symmetric. Since the data in this exercise does not appear to be mound shaped from the dotplot, the Empirical Rule should not be used to describe this dataset.

Key concepts

Range Approximation

The range approximation is an easy way to estimate the standard deviation of a dataset without performing complex calculations. The range is calculated by subtracting the smallest value from the largest value in the dataset. For example, if we have a dataset with a maximum value of 6 and a minimum value of 1, the range is 6 - 1 = 5. Once we have the range, we use it to find an approximate standard deviation by dividing the range by a value listed in a reference table, which corresponds to the size of the dataset. For a dataset of size 10, we use the number 3.5 from the table. So, the estimated standard deviation (s) is: \[ s \approx \frac{5}{3.5} = 1.43 \]This method provides a quick approximation which can be helpful in getting an initial sense of the dataset's variability before using more precise calculations.

Dotplot

A dotplot is a simple graphical representation of data where each data point is shown as a dot along a number line. It's useful for displaying small datasets and seeing the frequency distribution at a glance. To create a dotplot, plot each data value on a horizontal axis. If a number occurs more than once, stack the dots vertically above that value. For example, if our dataset includes the numbers 1, 2, 3, and 6, with some repetitions, each of these will be represented by a stack of dots. In the context of our exercise, the numbers were plotted as follows: - 1 appears twice, so two dots are placed above 1 - 2 appears once, so one dot is placed above 2 - 3 appears twice, so two dots are placed above 3 - 4, 5, and 6 are plotted similarly Dotplots offer a visual way to determine whether the data is 'mound-shaped', a term often used to suggest a symmetrical, bell-shaped distribution. In this case, the dotplot for our data did not appear mound-shaped.

Tchebysheff's Theorem

Tchebysheff's Theorem offers insight into data distribution without needing the data to follow a specific shape, making it robust for any dataset. It states that for any dataset, no matter the distribution, a certain percentage of data lies within a specific number of standard deviations from the mean. Specifically, at least \[ rac{1 - rac{1}{k^2}}{1} \] of data points will fall within \(k\) standard deviations from the mean, where \(k\) is greater than 1. For example, with \(k = 2\), at least 75% of data will lie within 2 standard deviations of the mean.This theorem confirms applicability as long as a dataset has a defined mean and standard deviation, making it versatile and widely used, regardless of how the data looks when plotted.

Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, is used in statistics to summarize data that follows a normal distribution. This rule states:

• About 68% of data falls within one standard deviation of the mean.
• Approximately 95% lies within two standard deviations.
• And around 99.7% are within three standard deviations.

However, the key requirement for the Empirical Rule to be applicable is that the data must be roughly mound-shaped and symmetric. In the exercise, the data's dotplot revealed it didn't have a mound-shaped distribution. Therefore, the Empirical Rule couldn't be effectively used here. Understanding when this rule holds is crucial for proper statistical interpretation and avoiding misrepresentations where conditions aren't met.