Problem 46 - Basic Statistics for Business and Economics with Student

Question

You need to find the "typical" or mean annual dividend per share for large banks. You decide to sample six banks listed on the New York Stock Exchange. These banks and their trading symbols follow. a. After numbering the banks from 01 to \(24,\) which banks would be included in a sample if the random numbers were \(14,08,24,25,05,44,02,\) and \(22 ?\) Go to the following website: \(\quad\) http://bigcharts.marketwatch.com. Enter the trading symbol for each of the sampled banks and record the price earnings ratio (P/E ratio). Determine the mean annual dividend per share for the sample of banks. b. Which banks are selected if you use a systematic sample of every fourth bank starting with the random number \(03 ?\)

Step-by-Step Solution

Answer

Chosen banks for random sampling are those numbered 14, 08, 24, 05, 02, and 22; ignore 25 and 44. For systematic sampling, select banks numbered 03, 07, 11, 15, 19, and 23.

Step 1: Understand the Random Sampling

The task is to randomly sample banks based on a set of given random numbers after numbering them from 01 to 24. We need to identify which banks correspond to these random numbers. Since 25 and 44 exceed the available numbered banks (01-24), they should be ignored or corrected if possible; in this context, they are ignored.

Step 2: List the Sampled Banks from Random Numbers

From the numbers provided (14, 08, 24, 05, 02, and 22), the banks with these corresponding numbers are selected for the sample. To proceed, select bank numbers 14, 08, 24, 05, 02, and 22.

Step 3: Find P/E Ratios for Selected Banks

Visit [BigCharts](http://bigcharts.marketwatch.com) and enter the trading symbol for each bank selected (numbers 14, 08, 24, 05, 02, and 22). Record the P/E ratio for each of these banks. These will be used to further investigate dividend data if needed.

Step 4: Calculate the Mean Annual Dividend per Share

To determine the mean annual dividend per share, you would ideally find the dividend per share data for each of the banks and compute the average. However, since the exercise does not provide dividend data directly, it emphasizes understanding the process of a statistical sample.

Step 5: Understand Systematic Sampling

For systematic sampling, select every fourth bank from a random starting point. Since the random number is 03, you start with bank number 03 and then add 4 repeatedly to select the subsequent banks: 03, 07, 11, 15, 19, and 23.

Key Concepts

Mean Annual Dividend

The mean annual dividend is an average of cash payouts a shareholder receives yearly from a company's profits, based on the stocks they own. Calculating this average provides investors with an idea of what their returns from dividends might be over a year.
The calculation of the mean annual dividend involves adding up all the dividends paid over the year by the selected companies and then dividing by the number of companies in the sample. In the context of the given exercise, you would find the dividend per share for each bank in your sample. Once you have this information, you sum up all the dividends and divide by six, the number of banks in the sample, to find the mean dividend per share.

It's important to note that in real-world scenarios, you'll need the actual dividend data for each bank to compute this mean. Gathering such data requires accessing reliable financial resources or databases.

Random Sampling

Random sampling is a technique where each member of a population has an equal chance of being selected. This is crucial for ensuring that the sample accurately represents the whole population.
In the exercise, a set of random numbers were used to select banks from a possible list of 24. Random numbers meant any bank had an equal chance of being part of the sample. This helps in mitigating biases, providing a more reliable sample to draw conclusions from, especially when calculating metrics like the mean annual dividend.

To perform this, you simply align your numbered list of banks with the given random numbers. Since numbers 25 and 44 exceed the list (01-24), they are discarded in this context, demonstrating the necessity to handle out-of-range numbers appropriately in practical situations.

Systematic Sampling

Systematic sampling is another statistical method used to select a sample from a larger population. Unlike random sampling, this involves selecting samples based on a fixed periodic interval, which is called the sampling interval.
In the exercise, systematic sampling was demonstrated by choosing every fourth bank, beginning with bank number 03. This pattern leads to selecting banks 03, 07, 11, 15, 19, and 23. This method is straightforward and usually easier to perform than random sampling, especially with a large list of members.

• First, determine your starting point.
• Then, consistently apply the interval (here every fourth bank) to select the sample.

This sampling strategy helps maintain a spread across the list, reducing clustering or overrepresentation of certain segments of the population.

Price Earnings Ratio

The Price Earnings Ratio (P/E ratio) is a measure that calculates the current share price relative to its per-share earnings. It's a common metric used by investors to gauge if a stock is overvalued or undervalued compared to its earnings.
To find this ratio, you divide the market price per share by the earnings per share (EPS). For instance, if a bank’s current share price is $50 and its EPS is $5, the P/E ratio will be 10. This implies that investors are willing to pay $10 for every $1 of earnings.

• The P/E ratio is often used to compare companies within the same industry.
• A higher-than-average P/E may indicate investors expect higher earnings growth in the future.
• Conversely, a lower P/E might indicate the stock is undervalued or the company is experiencing issues.

In the exercise, checking the P/E ratios for the sampled banks can help give insight into how these banks are performing relative to each other and the industry.