Problem 22 - Algebra Form and Function

Question

In 10 packages the number of M\&M's was $$ 56,53,54,54,52,55,52,53,55,55 $$ (a) What is the mean number of M\&M's per package? (b) Is the mean a good description of the count for a package of M\&M's? Explain your reasoning.

Step-by-Step Solution

Answer

Answer: Yes, the mean is a good representation of the count for a package of M&Ms, as it provides an approximate average number expected per package and the numbers are relatively close together.

Step 1: Calculate the sum of the data

To find the mean, we first need to calculate the sum of the number of M&Ms in all packages. Add up the numbers: $$ 56 + 53 + 54 + 54 + 52 + 55 + 52 + 53 + 55 + 55 = 539 $$

Step 2: Calculate the mean

We will now divide the sum of the data by the number of packages to find the mean: $$ \text{Mean} = \frac{\text{Sum}}{\text{Number of Packages}} = \frac{539}{10} = 53.9 $$ (a) The mean number of M&Ms per package is 53.9.

Step 3: Analyze the data and discuss the mean as a representation

Now, let's take a look at the data we have: $$ 56, 53, 54, 54, 52, 55, 52, 53, 55, 55 $$ These numbers are not very far from the mean (53.9), and the range (maximum - minimum) is: $$ \text{Range} = 56 - 52 = 4 $$ The range is relatively small, showing that the numbers are quite close together. Additionally, there is no extreme value that could be impacting the mean significantly. (b) Given these observations, the mean is a good description of the count for a package of M&Ms, as it provides an approximate average number expected per package.

Key Concepts

Sum Calculation

To dive into the concept of sum calculation, let's consider the data set given:

• 56, 53, 54, 54, 52, 55, 52, 53, 55, 55

Calculating the sum involves adding together all these numbers to get a single numerical value that represents their cumulative quantity. To calculate this, each number in the data set is simply added together, step by step.For this example, the sum calculation is:\[56 + 53 + 54 + 54 + 52 + 55 + 52 + 53 + 55 + 55 = 539\]This sum (539) is essential for further statistical analysis such as determining the mean, telling us the total number of items collected across all packages.By understanding how to calculate the sum, we can get insights into the overall volume of data points present.

Data Set Range

The data set range is about understanding the spread or distribution of the numbers in a list. For the M&M package example:

• The maximum number of M&Ms in a package is 56.
• The minimum is 52.

Using these values, we calculate the range by subtracting the smallest number from the largest:\[ \text{Range} = 56 - 52 = 4 \]This range helps to understand the variance within the numbers. A small range indicates that the data points are clustered closely together. In our example, a range of 4 tells us that the number of M&Ms in any package is fairly consistent, with no large discrepancies.The range aids in assessing the reliability of the mean as a representative value for the data.

Statistical Analysis

Statistical analysis involves examining and interpreting data to make sense of it. In this context, we have a data set of M&Ms counts and want to evaluate if the mean is a proper summary.First, we identified the mean, calculated as:\[ \text{Mean} = \frac{539}{10} = 53.9 \]The mean gives us a central value that can represent the data's tendency.It is crucial to ensure the data doesn't have outliers that skew this average disproportionately. Since the range is small and the values don't have outliers, the mean of 53.9 is indeed a valid representation.Analyzing the data through these methods allows us to infer that each package approximately contains about 54 M&Ms, offering a reliable expectation.By performing these kinds of analyses, you learn to understand and communicate what your data implies effectively.