Question

Choose the statement below that is true about the given numbers. A) The number in column A is greater. (B) The number in column B is greater. (C) The two numbers are equal. (D) The relationship cannot be determined from the given information. $$\begin{array}{|c|c|}\hline \text { Column A } & \text { Column B } \\\\\hline \text { The mean of } 1,2,3,4,5,5,10 & \text { The mean of } 2,3,4,5,5,10,11 \\\\\hline\end{array}$$

Short answer

The correct answer is (B) The number in column B is greater.

Step-by-step solution

Calculate the Mean of Column A

First, calculate the mean of the numbers in column A. The mean is calculated as follows: \((1 + 2 + 3 + 4 + 5 + 5 + 10) / 7 = 30/7 = 4.29\)

Calculate the Mean of Column B

Next, calculate the mean of the numbers in column B. The mean is calculated as follows: \((2 + 3 + 4 + 5 + 5 + 10 + 11) / 7 = 40/7 = 5.71\)

Compare the Mean of Column A and B

After calculating the means, compare them. As observed, the mean of column B (5.71) is greater than the mean of column A (4.29).

Key concepts

Mean Calculation

Understanding how to calculate the mean, also known as the average, is essential in statistics and everyday life. The mean is found by adding up all the numbers in a set and then dividing by the total count of numbers. For example, if we want to calculate the mean of 1, 2, 3, 4, and 5, we would sum these numbers to get 15 and then divide by 5, since there are five numbers. This calculation results in a mean of 3.

When performing mean calculations, it is important to carefully add all the values together and to use precise division, ensuring that the division is carried out to the correct number of decimal places for an accurate result. This is crucial because the mean represents a central point in a data set, showing us the central tendency and allowing us to compare different sets of numbers, as seen in the provided exercise.

Arithmetic Mean

The arithmetic mean is the most commonly used type of mean in statistics. It's found by adding all the data points together and then dividing by the number of points. The formula for the arithmetic mean is \( \frac{{\sum {x_i}}}{{n}} \), where \( x_i \) represents each number in the dataset, and \( n \) is the total number of items.

In the context of the exercise, the arithmetic mean gives us a measure of the 'center' of a set of numbers, which is particularly useful for comparing different data sets. This aligns perfectly with the requirement in algebra and statistics to often find and compare central values, thereby establishing relationships or drawing conclusions based on numerical data.

Statistics in Algebra

The application of statistics in algebra involves various methods for describing and analyzing numerical data. One of the fundamental tasks is calculating measures of central tendency, which include mean, median, and mode. The mean is particularly significant in algebra as it is used to create equations and inequalities based on data sets.

Algebraic statistics can be used for solving real-world problems where determination of averages is essential, such as figuring out the average score needed on a test to achieve a certain grade or understanding economic or scientific data trends. In the exercise, we apply algebraic principles to calculate and compare means, demonstrating the interconnectedness of these mathematical fields.

Comparing Numerical Values

Comparing numerical values is a critical thinking skill in math that allows us to evaluate the size, value, or significance of numbers or sets of numbers. When comparing means, like in our exercise, we look at the averages of different data sets to determine which one is larger or if they are equal. Understanding how to compare these values is vital in decision-making processes across various fields, such as finance, analytics, and research.

In practice, to compare means effectively, one needs to be adept at performing accurate calculations and understanding what these calculations represent in the context of the data. Comparing the means from our exercise's Column A and Column B, we were able to determine that the average value in Column B is greater, which implies that overall, the numbers in Column B are higher when assessed as a group.