Problem 131 - Precalculus with Limits

Question

In Exercises 131-134, use the following definition of the arithmetic mean \( \bar{x} \) of a set of \( n \) measurements \( x_1, x_2, x_3, \dots , x_n \). \( \displaystyle \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \) Find the arithmetic mean of the six checking account balances \( \$327.15, \$785.69, \$433.04, \$265.38, \$604.12, \) and \( \$590.30 \). Use the statistical capabilities of a graphing utility to verify your result.

Step-by-Step Solution

Answer

Based on calculations, the arithmetic mean (average) of the six checking account balances is approximately \$500.95. Verification with a graphing utility should reflect the same value.

Step 1: Identify the elements

The checking account balances that need to be averaged are \$327.15, \$785.69, \$433.04, \$265.38, \$604.12, and \$590.30.

Step 2: Apply the arithmetic mean formula

Add all the account balances and divide by the total number of balances. Therefore, \( \bar{x} = \frac{{327.15 + 785.69 + 433.04 + 265.38 + 604.12 + 590.30}}{6} \). Compute this to get the arithmetic mean.

Step 3: Compute the arithmetic mean

Using the values provided in Step 2, solve for the arithmetic mean. This will be the average checking account balance.

Step 4: Verify using a graphing utility

In this step, use any statistical capabilities of a graphing utility to calculate the mean of the given data. This will verify your calculated result and ensure that the arithmetic mean was computed correctly.

Key Concepts

Descriptive Statistics

Descriptive statistics, as the name suggests, describes and summarizes data sets in a way that patterns can emerge. It is a fundamental concept in statistics used to present the main features of a collection of information succinctly. This includes measures of central tendency like mean, median, and mode, which indicate the center of a data set, and measures of variability such as range, variance, and standard deviation, which speak to how much the data varies.

These descriptive measures help to simplify complex data sets to a few indicators that represent the majority of the data points. Using the arithmetic mean from our exercise, for instance, gives us a single value that summarizes the overall financial standing of the set of checking accounts without listing each account balance individually.

Mean Calculation

The arithmetic mean, often simply called the mean, is one of the most common measures of central tendency. You calculate it by summing all the values in a data set and then dividing that sum by the count of the data points. In the formula \( \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \), \( \bar{x} \) represents the mean, \( n \) is the number of observations, and \( x_i \) represents each value in the data set. This calculation gives you a central value that can be a good indicator of the 'typical' value found in the data set, assuming the data is not skewed by outliers.

Sum of Data Set

To find the arithmetic mean, one must first calculate the sum of the data set, which is the total of all values included in the collection of numbers. This is represented by the Greek letter sigma (\(\Sigma\)) in the formula \( \sum_{i=1}^{n} x_i \). In the context of our checking account balances, each account's balance—\(\$327.15, \$785.69, \$433.04, \$265.38, \$604.12, and \$590.30\)—contributes to the total sum. After adding these together, we get the sum of the data set, which is then divided by the number of data points, again reinforcing the idea of the 'average' amount in the accounts.

Graphing Utility Statistics

Graphing utilities, which can include calculators or software programs, offer statistical capabilities that assist in the analysis of data. For instance, after data entry, these tools can calculate the arithmetic mean automatically, which serves as a verification for manual calculations. By inputting the checking account balances into the utility, it should yield the same arithmetic mean as obtained through the manual computation. By using such a utility, we can also visualize the distribution of the data, possibly identifying any outliers or skewness in the data which could affect the reliability of the mean in representing the central tendency of the entire data set.