Problem 45 - Basic Statistics for Business and Economics with Student

Question

One of the most popular candies in the United States is M\&M's, which are produced by the Mars Company. In the beginning M\&M's were all brown; more recently they were produced in red, green, blue, orange, brown, and yellow. You can read about the history of the product, find ideas for baking, purchase the candies in the colors of your school or favorite team, and learn the percent of each color in the standard bags at http://global.mms.com/us/about/products/milkchocolate/ Recently the purchase of a 14 -ounce bag of M\&M's Plain had 444 candies with the following breakdown by color: 130 brown, 98 yellow, 96 red, 35 orange, 52 blue, and 33 green. Develop a chart depicting this information and write a paragraph summarizing the results.

Step-by-Step Solution

Answer

The 14-ounce bag contains mostly brown, yellow, and red M&M's, with brown being the most prevalent color at approximately 29.28%.

Step 1: Gather Data

To create a chart depicting the M&M's color distribution, we start by collecting the given data. We have the total number of M&M's as 444, and the breakdown by color: Brown - 130, Yellow - 98, Red - 96, Orange - 35, Blue - 52, Green - 33.

Step 2: Calculate Proportions

For a more insightful chart, we calculate the proportion of each color relative to the total number of candies. The proportion for each color is calculated as follows:\[\text{Proportion} = \frac{\text{Number of M&M's of a color}}{\text{Total number of M&M's}} \times 100\]Let's calculate this for each color:

Step 2.1: Brown Proportion

For brown M&M's, the proportion is:\(\frac{130}{444} \times 100 \approx 29.28\%\).

Step 2.2: Yellow Proportion

For yellow M&M's, the proportion is:\(\frac{98}{444} \times 100 \approx 22.07\%\).

Step 2.3: Red Proportion

For red M&M's, the proportion is:\(\frac{96}{444} \times 100 \approx 21.62\%\).

Step 2.4: Orange Proportion

For orange M&M's, the proportion is:\(\frac{35}{444} \times 100 \approx 7.88\%\).

Step 2.5: Blue Proportion

For blue M&M's, the proportion is:\(\frac{52}{444} \times 100 \approx 11.71\%\).

Step 2.6: Green Proportion

For green M&M's, the proportion is:\(\frac{33}{444} \times 100 \approx 7.43\%\).

Step 3: Create the Chart

With the proportions calculated, create a chart, such as a pie chart or a bar graph, to visually represent the M&M color distribution. Label each section of the chart with the color and its corresponding percentage.

Step 4: Write a Summary

The chart clearly depicts the distribution of M&M colors in a 14-ounce bag containing 444 candies. The most prevalent colors are brown, yellow, and red, comprising more than 70% of the total, with brown being the most abundant (approximately 29.28%). The least common colors are orange and green, making up less than 16% combined. This distribution suggests a potential preference or manufacturing decision towards the colors brown, yellow, and red.

Key Concepts

Data Representation

When analyzing data, representing it in an organized manner is crucial to understand and communicate the information effectively. In the case of M&M's color distribution, the data representation involves listing all the colors and their respective counts.
This gives us a complete picture of the situation. Here, we have 444 M&M's distributed among six colors: brown, yellow, red, orange, blue, and green. Each color count helps us see which colors are more or less prevalent.
Clear representation of data facilitates further analysis, such as calculating proportions and creating visual aids. It serves as the foundation for deeper insights.

Proportion Calculation

Calculating proportions is a key step when you want to understand the relative size of parts compared to a whole. For M&M's, we calculate the proportion of each color to visualize this comparison more easily.
The formula used is:

• \( \text{Proportion} = \frac{\text{Number of M&M's of a color}}{\text{Total number of M&M's}} \times 100 \)

Using this formula for each color lets us transform raw counts into percentages.
With brown at approximately 29.28%, yellow at 22.07%, and so on, we quickly grasp which color dominates or trails behind. Such calculations are essential in many statistical analyses, offering a clear numerical perspective.

Chart Creation

Creating a chart is a fantastic way to present data visually, making complex information easier to digest. Upon calculating the proportions of M&M's colors, the next step is to create a chart that displays these findings effectively.
Using charts like a pie chart or bar graph offers visual clarity, with each segment's size correlating directly with its percentage. For instance, a pie chart would fill nearly a third with brown, reflecting its dominance.
Such visual tools not only clarify distributions but also engage viewers, making data interpretation quicker and more intuitive. Charts are thus an invaluable aid in data analysis.

Color Distribution Analysis

Color distribution analysis involves interpreting data to determine what it suggests about the preferences or decisions behind the figures. In our M&M example, the analysis shows that certain colors are more prevalent.
Brown, yellow, and red together dominate the candy count at over 70%. This could indicate a strategic decision or consumer preference. Conversely, the lower percentages for orange and green suggest a lower priority or selection among buyers or manufacturers.
Understanding these distributions helps draw conclusions and make informed predictions about trends, choices, or market strategies surrounding a product.