Problem 13 - Basic Statistics for Business and Economics with Student

Question

Consider all of the coins (pennies, nickels, quarters, etc.) in your pocket or purse as a population. Make a frequency table beginning with the current year and counting backward to record the ages (in years) of the coins. For example, if the current year is \(2007,\) then a coin with 2004 stamped on it is 3 years old. a. Draw a histogram or other graph showing the population distribution. b. Randomly select five coins and record the mean age of the sampled coins. Repeat this sampling process 20 times. Now draw a histogram or other graph showing the distribution of the sample means. c. Compare the shapes of the two histograms.

Step-by-Step Solution

Answer

Create histograms for age distribution and sample means. Compare shapes focusing on spread and concentration.

Step 1: Collect and Record Coin Ages

Gather all the coins from your pocket or purse. Record the year stamped on each coin. Calculate the age of each coin by subtracting the year on the coin from the current year (e.g., if the current year is 2023 and the coin year is 2005, the age is 2023 - 2005 = 18 years). Create a frequency table by listing distinct ages and their respective counts.

Step 2: Create Frequency Histogram

Using the frequency table from Step 1, draw a histogram. On the x-axis, place the ages of the coins, and on the y-axis, place the frequency of each age. Each bar on the histogram corresponds to a different coin age, and the height of the bar represents how many coins are of that particular age.

Step 3: Random Sampling

Randomly select five coins from your collection, without replacement. Record their ages and calculate the mean age of these five coins. Note this mean in a table. Repeat this process 20 times, making sure to record each sample mean in your table.

Step 4: Create Sample Mean Histogram

Using the mean ages recorded from the 20 random samples, draw another histogram. The x-axis represents the mean ages and the y-axis represents the frequency of these means. Each bar represents how many times a specific mean was observed across your samples.

Step 5: Compare Histograms

Inspect both histograms from Steps 2 and 4. Compare their shapes, observing any differences in spread, central tendency (e.g., peakedness), and variation. The population distribution from Step 2 may show a varied distribution of ages, while the sample mean distribution from Step 4 should be more concentrated and normal-shaped due to the Central Limit Theorem.

Key Concepts

Frequency Table

A frequency table is a valuable statistical tool used to organize data. It helps us summarize a data set by showing the number of occurrences (frequency) of different categories or values.
In the context of the coin exercise, the frequency table summarizes how many coins of each age you have. Here's how you can create one:

• First, determine the age of each coin by subtracting the year stamped on it from the current year.
• List each distinct age as a separate category in the table.
• For each age, count how many coins match this age and note the count as the frequency.

By providing a clear view of data distribution, a frequency table sets the stage for creating a histogram, which further visualizes this distribution in a graphical format.

Histogram

A histogram is a graphical representation of a frequency distribution. It uses bars to show the frequency of each category or range of values from a data set.
To create a histogram from a frequency table:

• Place the distinct values (coin ages, in this case) on the x-axis.
• The y-axis shows the frequency of each age group.
• Draw bars for each age group. The height of each bar corresponds to its frequency or count in the table.

This visual representation can quickly convey the shape of the data distribution. In the coin exercise, it shows how many coins you have for each age. Common patterns like peaks, spreads, or gaps can tell you much about the underlying population.

Random Sampling

Random sampling is a statistical method used to select a subset from a population, ensuring each member has an equal chance of being included. It helps create unbiased samples.
In our coin exercise, you are asked to randomly select five coins multiple times:

• Select five coins at random without putting them back into the mix (without replacement).
• Calculate the mean age of these selected coins.
• Record these mean ages, repeating the process 20 times.

By doing so, we gather enough data to analyze the spread and central tendency of the sample means. It demonstrates how samples can provide insights into the entire population.

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in statistics, stating that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the population's original distribution.
In practical terms, when you randomly select five coins, calculate their mean age, and repeat this process 20 times, the distribution of these means should form a bell curve.

• This histogram will likely be more symmetrical and concentrated around the overall mean age.
• Even if the original population histogram (coin ages) had varied shapes, the sample means' histogram looks more normal due to the CLT.

This theorem is powerful because it allows statisticians to make inferences about populations based on sample data, which is essential for hypothesis testing and other statistical applications.