Problem 32 - Basic Statistics for Business and Economics

Question

The Midland National Bank selected a sample of 40 student checking accounts. Below are their end-of-the-month balances. $$ \begin{array}{rrrrrrrrrr} \hline \$ 404 & \$ 74 & \$ 234 & \$ 149 & \$ 279 & \$ 215 & \$ 123 & \$ 55 & \$ 43 & \$ 321 \\ 87 & 234 & 68 & 489 & 57 & 185 & 141 & 758 & 72 & 863 \\ 703 & 125 & 350 & 440 & 37 & 252 & 27 & 521 & 302 & 127 \\ 968 & 712 & 503 & 489 & 327 & 608 & 358 & 425 & 303 & 203 \\ \hline \end{array} $$ a. Tally the data into a frequency distribution using \(\$ 100\) as a class interval and \(\$ 0\) as the starting point. b. Draw a cumulative frequency polygon. c. The bank considers any student with an ending balance of \(\$ 400\) or more a "preferred customer." Estimate the percentage of preferred customers. d. The bank is also considering a service charge to the lowest 10 percent of the ending bal- ances. What would you recommend as the cutoff point between those who have to pay a service charge and those who do not?

Step-by-Step Solution

Answer

a) Frequency distribution created. b) Draw cumulative polygon using frequencies. c) 35% are preferred customers. d) Cutoff is $99 for service charges.

Step 1: Create the Frequency Distribution

First, determine the range of the data by subtracting the smallest value from the largest. The maximum account balance is $968 and the minimum is $27, so the range is $941. Subdivide the range into classes with a class width of $100. The classes are: - $0 to $99 - $100 to $199 - $200 to $299 - $300 to $399 - $400 to $499 - $500 to $599 - $600 to $699 - $700 to $799 - $800 to $899 - $900 to $999. Tally each balance into the appropriate class.

Step 2: Tallying the Frequencies

Now assign each balance to its corresponding class: - $0 to $99: 9 observations - $100 to $199: 7 observations - $200 to $299: 8 observations - $300 to $399: 3 observations - $400 to $499: 5 observations - $500 to $599: 1 observation - $600 to $699: 2 observations - $700 to $799: 4 observations - $800 to $899: 1 observation - $900 to $999: 1 observation.

Step 3: Construct the Cumulative Frequency Distribution

Begin counting cumulative frequencies starting from the first class: - $0 to $99: Cumulative frequency = 9 - $100 to $199: Cumulative frequency = 9 + 7 = 16 - $200 to $299: Cumulative frequency = 16 + 8 = 24 - $300 to $399: Cumulative frequency = 24 + 3 = 27 - $400 to $499: Cumulative frequency = 27 + 5 = 32 - $500 to $599: Cumulative frequency = 32 + 1 = 33 - $600 to $699: Cumulative frequency = 33 + 2 = 35 - $700 to $799: Cumulative frequency = 35 + 4 = 39 - $800 to $899: Cumulative frequency = 39 + 1 = 40 - $900 to $999: Cumulative frequency = 40.

Step 4: Draw the Cumulative Frequency Polygon

Plot the cumulative frequencies against the upper boundary of the class intervals on a graph. Connect the points with line segments. The x-axis is the upper class boundaries, and the y-axis represents the cumulative frequency.

Step 5: Calculate Percentage of Preferred Customers

Preferred customers have balances of \(400 or more. Classes \)400 to $499 and above have 5 + 1 + 2 + 4 + 1 + 1 = 14 customers. Therefore, the percentage of preferred customers is \( \frac{14}{40} \times 100 = 35\% \).

Step 6: Determine Cutoff for Lowest 10%

The lowest 10% of balances correspond to the first 4 (0.10 x 40) accounts among the ordered data. Observing the cumulative frequencies, classes $0 to $99 include the first 9 accounts, indicating all balances in this range are within the lowest 10%. Recommend a cutoff of $99 for the service charge.

Key Concepts

Cumulative Frequency Polygon

A cumulative frequency polygon is a graphical plot that represents cumulative frequencies with respect to class intervals.
To create one, begin with constructing a cumulative frequency table. Here, we calculate the cumulative frequency step by step by adding the frequency of each class interval to the sum of the frequencies of all preceding class intervals.
On a graph, plot cumulative frequencies on the y-axis against the upper boundary of class intervals on the x-axis.
Next, connect these plot points using straight lines. This visualization helps in understanding how data accumulates over class intervals and showcases the increasing trend of the collected data.
Unlike regular frequency polygons, cumulative frequency polygons are not bell-shaped because they do not show individual class frequencies but cumulative information instead.

Preferred Customer Percentage

The preferred customer percentage identifies the proportion of customers meeting a specified condition. In this situation, Midland National Bank considers students with balances of \(400 or more as preferred customers.
To estimate this percentage, identify those class intervals that meet the criteria—that is, balances from \)400 and up.

• First, count the total number of observations within these intervals.
• Then, compute the percentage through the formula: \( \frac{\text{Number of preferred customers}}{\text{Total number of observations}} \times 100 \).

In the exercise, 14 of 40 accounts meet the preferred balance requirement, equating to \( \frac{14}{40} \times 100 = 35\% \). This percentage insight can help the bank tailor marketing or services for such preferred customers.

Service Charge Cutoff

The service charge cutoff is a threshold to determine which accounts are subject to a fee. In this context, Midland National Bank seeks to apply this fee to the lowest 10% of balances.
The strategy is to determine the cutoff balance value by identifying the class interval that encompasses this lowest percentile of accounts.
Beginning with ordered cumulative data, locate the class interval containing the N-th observation, where \(N = 0.10 \times 40 = 4 \).
Since the cumulative frequency for the \(0 to \)99 class is 9, this range covers the lowest 10%, meaning that balances below \(100 fall within this group.
Therefore, a cutoff point of \)99 is recommended to determine which customers should incur the service charge.

Class Intervals

Class intervals divide the data range into segments or groups. They form the backbone of frequency distributions and ease the analysis of data patterns.
To construct class intervals, start by determining the data range, then divide it into equal parts. Each segment is a class. For example, in this exercise, the range is $941, split into intervals of width $100.

• Interval examples: $0-$99, $100-$199, $200-$299, etc.
• Every observation falls into one specific class based on its value.

Class intervals allow complex datasets to be summarized efficiently, permitting further calculations like frequencies or cumulative frequencies.
They also help identify the distribution trend across the dataset, revealing how data is spread out over different value ranges.