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Chapter 3: Problem 33

If the mean of five values is \(64,\) find the sum of the values.

Short Answer

Expert verified
The sum of the values is 320.

Step by step solution

01

Understand the Formula for Mean

The mean (or average) of a set of numbers is calculated by adding all the numbers together and then dividing by the number of numbers in the set. In mathematical terms, the mean \( \bar{x} \) is calculated as \( \bar{x} = \frac{\text{sum of values}}{\text{number of values}} \).
02

Plug Values into the Mean Formula

We are given that the mean \( \bar{x} \) of the five values is 64. Therefore, according to the formula \( \bar{x} = \frac{\text{sum of values}}{5} \), we can substitute the values we know: \( 64 = \frac{\text{sum of values}}{5} \).
03

Solve for the Sum of Values

To find the sum of the values, we need to isolate it in the equation. Multiply both sides of the equation \( 64 = \frac{\text{sum of values}}{5} \) by 5 to solve for the sum:\[ 64 \times 5 = \text{sum of values} \]
04

Calculate the Sum

Now perform the multiplication:\[ 64 \times 5 = 320 \]Therefore, the sum of the five values is 320.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum of values
When you're dealing with the mean of a set of numbers, understanding the concept of the "Sum of values" is pivotal. Imagine you have several numbers and each one represents a piece of a larger puzzle. Adding these numbers together gives you the puzzle's complete picture. This total is known as the "sum of values." Whether for exams, personal curiosities, or practical applications, knowing how to find this sum is incredibly handy. The essential step is first identifying all the individual numbers you're working with and then adding them together.
  • This total is the building block for calculating the mean.
  • It assembles all contributions into one cumulative figure.
  • In cases like our exercise, knowing the mean allows backward calculation to find this sum.
Understanding this concept aids in demystifying more complex mathematical scenarios and brings clarity to processes involving averages.
Mathematical formula
The phrase "Mathematical formula" can sometimes feel intimidating, but it simply refers to a recipe for calculations. Think of it as a specific guideline or instruction set to achieve the desired numerical result. In the context of mean, this formula is straightforward but powerful. To calculate the mean, we use the formula: \[\bar{x} = \frac{\text{sum of values}}{\text{number of values}}\]This concise expression tells you that the mean is the outcome of the sum of the given numbers divided by how many numbers there are.
  • "\(\bar{x}\)" represents the mean, highlighting the outcome of interest.
  • The top part of the fraction, "sum of values," signifies the total from combining all numbers.
  • The bottom part indicates the "number of values." This dictates division by the total count of numbers.
Applying this formula simplifies daunting numerical tasks into manageable steps.
Arithmetic mean
The "Arithmetic mean" is a way to describe the average you'd obtain by leveling out numbers evenly across several pieces. It's one of the most commonly used forms to find the center of a data set. By equally distributing values, this average reflects a central tendency, making it ideal for straightforward evaluations.
  • The arithmetic mean is helpful in everyday decisions like budgeting, where finding a middle ground simplifies planning.
  • Unlike median and mode, which focus on other aspects of data distribution, the arithmetic mean offers a balanced perspective.
  • Our exercise calculating a mean of 64 demonstrates how this average represents an equivalency point across numbers.
By recognizing patterns and leveling the contributions, arithmetic mean offers clarity and is widely applicable in both academic settings and practical life.

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Most popular questions from this chapter

Below are the percentages of the population over 25 years of age who have completed 4 years of college or more for the 50 states and the District of Columbia. Find the mean and modal class. $$\begin{array}{lr}\text { Percentage } & \text { Frequency } \\\\\hline 15.2-19.6 & 3 \\\19.7-24.1 & 15 \\\24.2-28.6 & 19 \\\28.7-33.1 & 6 \\\33.2-37.6 & 7 \\\37.7-42.1 & 0 \\\42.2-46.6 & 1\end{array}$$

The data show the population (in thousands) for a recent year of a sample of cities in South Carolina. \(\begin{array}{llllll}29 & 26 & 15 & 13 & 17 & 58 \\ 14 & 25 & 37 & 19 & 40 & 67 \\ 23 & 10 & 97 & 12 & 129 & \\ 27 & 20 & 18 & 120 & 35 & \\ 66 & 21 & 11 & 43 & 22 & \end{array}\) Find the data value that corresponds to each percentile. a. 40 th percentile b. 75 th percentile c. 90th percentile d. 30th percentile Using the same data, find the percentile corresponding to the given data value. e. 27 f. 40 g. 58 h. 67

The mean and standard deviation of the number of hours the employees work in the music store per week are, respectively, 18.6 and 3.2 hours. If the owner increases the number of hours each employee works per week by 4 hours, what will be the new mean and standard deviation of the number of hours worked by the employees?

A sample of 12 drivers shows the time that they spent (in minutes) stopped in rush-hour traffic on a specific snowy day last winter. Find the range, variance, and standard deviation for the data. \(\begin{array}{lll}52 & 56 & 53 \\ 61 & 49 & 51\end{array}\) \(\begin{array}{lll}53 & 58 & 53 \\ 60 & 71 & 58\end{array}\)

The data show the amount of sales tax paid in Denver County, Colorado. Find the first and third quartiles for the data. $$ \begin{array}{lclc} \text { Month } & \text { Sales Tax } & \text { Month } & \text { Sales Tax } \\\ \hline \text { Jan } & 363,061 & \text { July } & 518,868 \\ \text { Feb } & 358,208 & \text { August } & 554,013 \\ \text { March } & 418,500 & \text { September } & 506,809 \\ \text { April } & 266,771 & \text { October } & 341,421 \\ \text { May } & 399,814 & \text { November } & 349,026 \\ \text { June } & 453.698 & \text { December } & 532.545 \end{array} $$
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