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Chapter 3: Problem 34

If the mean of five values is 8.2 and four of the values are \(6,10,7,\) and \(12,\) find the fifth value.

Short Answer

Expert verified
The fifth value is 6.

Step by step solution

01

Understanding the Mean Formula

To find the fifth value, we need to use the formula for the mean of a set of numbers. The mean is calculated as the sum of all values divided by the number of values. Thus, we havethe equation \( \frac{6 + 10 + 7 + 12 + x}{5} = 8.2 \), where \( x \) is the fifth value we need to find.
02

Calculate the Total Sum from the Mean

Since the mean is 8.2 and there are 5 total values, the total sum of those values must be \(5 \times 8.2 = 41\).
03

Add Known Values

Next, we add up the values we know: \(6 + 10 + 7 + 12 = 35\).
04

Find the Fifth Value

To find the fifth value, subtract the sum of the known values from the total sum: \(41 - 35 = x\). Therefore, \(x = 6\).

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Key Concepts

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

Mean of a Set of Numbers
The concept of the mean is an essential part of statistics and mathematics. It provides an average or a central value for a set of numbers. Calculating the mean involves a straightforward process:
  • Add up all the numbers in your set to find the total sum.
  • Divide this sum by the number of values you have in your set.
For example, if you have the values 6, 10, 7, and 12 with a fifth unknown value, you can calculate their mean using the formula: \[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} \]In our exercise, the mean is given as 8.2 for five values, which guides us in determining the total sum of these values necessary for further calculations.
Finding the Fifth Value
To find the missing value in a set when you know the mean, you'll employ the relationship between the mean and the total sum. For a given mean (8.2 in this exercise) and known values, this calculation becomes an adventure in solving an equation. First, you multiply the mean by the number of values to find the total sum:\[ \text{Total Sum} = \text{Mean} \times \text{Number of values} \]For our exercise: \[ 8.2 \times 5 = 41 \]Now, we know the total sum of the five values is 41. With four known values (6, 10, 7, and 12) totaling 35, you can easily find the fifth value. Simply subtract the sum of the known values from the total sum to reveal the mystery number:\[ \text{Fifth Value} = 41 - 35 = 6 \]This method ensures accuracy and highlights how each piece of information fits together to give you a complete picture of the data set.
Sum of Values Calculation
The calculation of the sum is a fundamental step when dealing with mean, especially when certain values are unknown, as in this exercise. It involves:
- Gathering all known values of the dataset.- Adding them up to obtain a total.For the values 6, 10, 7, and 12, adding them gives us:\[ 6 + 10 + 7 + 12 = 35 \]
This is the sum of the known values in our scenario. Knowing this sum allows us to find the unknown value by comparing it to the calculated total sum of all five values from the mean. Thus, the logic of subtracting known from total to find the missing number becomes clear and effective in problem-solving scenarios such as this.

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Most popular questions from this chapter

The average age of U.S. astronaut candidates in the past has been \(34,\) but candidates have ranged in age from 26 to \(46 .\) Use the range rule of thumb to estimate the standard deviation of the applicants' ages.

For these situations, state which measure of central tendency - mean, median, or mode-should be used. a. The most typical case is desired. b. The distribution is open-ended. c. There is an extreme value in the data set. d. The data are categorical. e. Further statistical computations will be needed. f. The values are to be divided into two approximately equal groups, one group containing the larger values and one containing the smaller values.

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. $$\begin{array}{lr}\text { Percentage } & \text { Frequency } \\\\\hline 15.2-19.6 & 3 \\\19.7-24.1 & 15 \\\24.2-28.6 & 19 \\\28.7-33.1 & 6 \\\33.2-37.6 & 7 \\\37.7-42.1 & 0 \\\42.2-46.6 & 1\end{array}$$

Harmonic Mean The harmonic mean (HM) is defined as the number of values divided by the sum of the reciprocals of each value. The formula is $$\mathrm{HM}=\frac{n}{\Sigma(1 / X)}$$ For example, the harmonic mean of \(1,4,5,\) and 2 is $$\mathrm{HM}=\frac{4}{1 / 1+1 / 4+1 / 5+1 / 2} \approx 2.051$$ This mean is useful for finding the average speed. Suppose a person drove 100 miles at 40 miles per hour and returned driving 50 miles per hour. The average miles per hour is not 45 miles per hour, which is found by adding 40 and 50 and dividing by 2 . The average is found as shown. Since Time \(=\) distance \(\div\) rate then Time \(1=\frac{100}{40}=2.5\) hours to make the trip Time \(2=\frac{100}{50}=2\) hours to return Hence, the total time is 4.5 hours, and the total miles driven are \(200 .\) Now, the average speed is $$\text { Rate }=\frac{\text { distance }}{\text { time }}=\frac{200}{4.5} \approx 44.444 \text { miles per hour }$$ This value can also be found by using the harmonic mean formula $$\mathrm{HM}=\frac{2}{1 / 40+1 / 50} \approx 44.444$$ Using the harmonic mean, find each of these. a. A salesperson drives 300 miles round trip at 30 miles per hour going to Chicago and 45 miles per hour returning home. Find the average miles per hour. b. A bus driver drives the 50 miles to West Chester at 40 miles per hour and returns driving 25 miles per hour. Find the average miles per hour. c. A carpenter buys \(\$ 500\) worth of nails at \(\$ 50\) per pound and \(\$ 500\) worth of nails at \(\$ 10\) per pound. Find the average cost of 1 pound of nails.

The average price of an instrument at a small music store is \(\$ 325 .\) The standard deviation of the price is \(\$ 52\). If the owner decides to raise the price of all the instruments by \(\$ 20\), what will be the new mean and standard deviation of the prices?
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