Problem 15 - Basic Statistics for Business and Economics

Question

The frequency distribution representing the number of frequent flier miles accumulated by employees at Brumley Statistical Consulting Company is repeated from Exercise \(11 .\) $$ \begin{array}{|cc|} \hline \begin{array}{c} \text { Frequent Flier Miles } \\ (000) \end{array} & \text { Frequency } \\ \hline 0 \text { up to } 3 & 5 \\ 3 \text { up to } 6 & 12 \\ 6 \text { up to } 9 & 23 \\ 9 \text { up to } 12 & 8 \\ 12 \text { up to } 15 & 2 \\ \text { Total } & \frac{2}{50} \\ \hline \end{array} $$a. How many employees accumulated less than 3,000 miles? b. Convert the frequency distribution to a cumulative frequency distribution. c. Portray the cumulative distribution in the form of a cumulative frequency polygon. d. Based on the cumulative frequency polygon, about 75 percent of the employees accumulated how many miles or less?

Step-by-Step Solution

Answer

a. 5 employees. b. Cumulative frequencies: 5, 17, 40, 48, 50. c. Draw using class boundaries and cumulative frequencies. d. About 9,000 miles or less.

Step 1: Find Employees with Less than 3,000 Miles

To determine how many employees accumulated less than 3,000 miles, we look at the first class interval in the frequency distribution, which is "0 up to 3". The frequency for this interval is 5. Therefore, 5 employees accumulated less than 3,000 miles.

Step 2: Convert to Cumulative Frequency Distribution

To create a cumulative frequency distribution, we sum the frequencies progressively. - For "0 up to 3": Cumulative frequency = 5. - For "3 up to 6": Cumulative frequency = 5 + 12 = 17. - For "6 up to 9": Cumulative frequency = 17 + 23 = 40. - For "9 up to 12": Cumulative frequency = 40 + 8 = 48. - For "12 up to 15": Cumulative frequency = 48 + 2 = 50. The cumulative frequencies are now: 5, 17, 40, 48, and 50.

Step 3: Draw a Cumulative Frequency Polygon

To portray the cumulative distribution as a polygon, plot the cumulative frequency against the upper class boundary of each interval on a graph: - Plot (3, 5) - Plot (6, 17) - Plot (9, 40) - Plot (12, 48) - Plot (15, 50) Join these points using straight lines to form the cumulative frequency polygon.

Step 4: Determine 75% of Employees from the Polygon

The total number of employees is 50. To find the number of miles that 75% (or 0.75) of employees accumulated or less, calculate 75% of 50. This gives 0.75 × 50 = 37.5. Since employees must be counted in whole numbers, check the cumulative frequency to find the number of miles: - The cumulative frequency of 40 corresponds to "6 up to 9". Thus, about 75% of employees accumulated 9,000 miles or less.

Key Concepts

Frequency Distribution

Frequency distribution is a way of organizing data to show how frequently specific data points, or ranges of data, occur in a dataset. This concept is particularly useful as it allows us to easily see patterns in larger datasets. In this exercise, frequent flier miles accumulated by employees is the data being organized.
- The frequent flier miles are grouped into intervals, such as "0 up to 3", "3 up to 6", and so on, representing thousands of miles.
- The term "frequency" refers to the number of data points that fall within each range, as per the given example where 5 employees accumulated 0 to 3,000 miles.
This summarized representation helps visualize the data, making it easier to understand trends and frequencies across different ranges.

Cumulative Frequency Polygon

A cumulative frequency polygon is a graphical representation of the cumulative frequency distribution. It helps in understanding data by showing the total number of observations that fall below the upper boundary of each interval. Constructing this type of chart involves plotting cumulative frequencies against the upper bounds of their respective class intervals.
- For example, you would plot points like (3, 5) where 3 is the upper boundary of the first interval and 5 is the cumulative frequency.
- Subsequent points continue this pattern, like (6, 17), until the full graph is plotted.
The resulting pattern, when connected with straight lines, forms a polygon. This visual tool serves to help easily estimate values such as median, quartiles, or any percentile by interpreting where these fall on the graph.

Class Interval

Class intervals are the foundational structures in a frequency distribution, facilitating the grouping of data into smaller, manageable portions. Each interval has a lower and an upper limit, effectively segmenting the data range.

• The interval "0 up to 3" means that data in the range from 0 (inclusive) to just before 3,000 miles (exclusive) fit within this class.
• Each class interval usually maintains a consistent size, which in the provided dataset is 3,000 miles.

Organizing data into class intervals allows for clearer analysis and helps in the simplification of massive datasets, enhancing the ease of plotting graphs such as histograms and cumulative frequency polygons.

Cumulative Frequency

Cumulative frequency is a running total of frequencies in a frequency distribution. It shows how many observations fall below a particular value or class interval in the data set.
- For example, the cumulative frequency for "0 up to 3" begins as 5. Then, as you move to the next interval, "3 up to 6", you add 12 (the frequency for this interval) to 5, reaching a cumulative frequency of 17.
This concept is beneficial for identifying percentiles. For instance, to find how many miles 75% of employees have accumulated or less, we search for the interval at which the cumulative frequency reaches or exceeds 37.5 (75% of 50 employees), which in this case is the interval "6 up to 9". Cumulative frequency thus provides insight into the spread and concentration of data points across the intervals.