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Chapter 3: Problem 34

Consider these six values a population: \(13,3,8,10,8,\) and 6 a. Determine the mean of the population. b. Determine the variance.

Short Answer

Expert verified
Mean is 8; Variance is approximately 9.67.

Step by step solution

01

Calculate the Mean

First, find the mean of the population by adding together all the values and then dividing by the number of values. Compute the sum: \[ 13 + 3 + 8 + 10 + 8 + 6 = 48 \]Now, divide by the total number of values (6):\[ \text{Mean} = \frac{48}{6} = 8 \]
02

Find Squared Deviations

Subtract the mean from each value and square the result:\[ (13 - 8)^2 = 25 \]\[ (3 - 8)^2 = 25 \]\[ (8 - 8)^2 = 0 \]\[ (10 - 8)^2 = 4 \]\[ (8 - 8)^2 = 0 \]\[ (6 - 8)^2 = 4 \]
03

Calculate Variance

For population variance, take the average of these squared deviations. Sum the squared deviations:\[ 25 + 25 + 0 + 4 + 0 + 4 = 58 \]Now, divide by the number of values (6):\[ \text{Variance} = \frac{58}{6} \approx 9.67 \]

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Key Concepts

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

Mean Calculation
When calculating the mean of a population, you're essentially finding the average value of all entries. To find the mean, add up all the numbers in your dataset, and then divide by the total number of values. Using our exercise example, we add up the numbers \( 13, 3, 8, 10, 8, \) and \( 6 \), resulting in a sum of \( 48 \).
The next step is to divide this sum by the number of values, which in this case is 6:
  • Sum of Values: \( 48 \)
  • Number of Values: \( 6 \)
Therefore, the mean \( \mu \) is given by:
\[ \mu = \frac{48}{6} = 8 \]This "8" now represents the average or central value of the set of data. It's a useful measure because it gives a single number that sums up the entire dataset.
Variance Calculation
Variance provides insight into how much the values in a population deviate, on average, from the mean. Calculating variance involves several steps. First, we need the squared deviations of each data point from the mean. Then, these squared values are averaged.
Variance is essential for understanding the spread or variability of the data.
With the squared deviations already computed in the original exercise, we find the sum:
  • Squaring the deviations of each value gives us: \( 25, 25, 0, 4, 0, 4 \).
  • Sum of Squared Deviations: \( 25 + 25 + 0 + 4 + 0 + 4 = 58 \)
  • Since we are treating this as a complete population, divide by the number of data points:
  • Population Variance: \[ \sigma^2 = \frac{58}{6} \approx 9.67 \]
Variance is a valuable statistic because it tells how far data points tend to be from the mean by this average squared amount.
Squared Deviations
The concept of squared deviations is pivotal in statistical analysis, especially when calculating variance. To find these deviations, subtract the mean from each individual value and square the result.
This squaring ensures that deviations do not cancel each other out (since squaring makes all numbers positive), maintaining accuracy in measuring variabilities.
Let's take the set of values and compute:
  • For 13: \((13 - 8)^2 = 25\)
  • For 3: \((3 - 8)^2 = 25\)
  • For 8: \((8 - 8)^2 = 0\)
  • For 10: \((10 - 8)^2 = 4\)
  • For 8: \((8 - 8)^2 = 0\)
  • For 6: \((6 - 8)^2 = 4\)
These squared deviations are crucial because they become the building blocks for calculating variance and standard deviation, statistical measures that describe how spread out the data points are in your dataset.

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Most popular questions from this chapter

Several indicators of long-term economic growth in the United States and their annual percent change are listed below. $$ \begin{array}{|lclc|} \hline \text { Economic Indicator } & \text { Percent Change } & \text { Economic Indicator } & \text { Percent Change } \\ \hline \text { Inflation } & 4.5 \% & \text { Real GNP } & 2.9 \% \\ \text { Exports } & 4.7 & \text { Investment (residential) } & 3.6 \\ \text { Imports } & 2.3 & \text { Investment (nonresidential) } & 2.1 \\ \text { Real disposable income } & 2.9 & \text { Productivity (total) } & 1.4 \\\ \text { Consumption } & 2.7 & \text { Productivity (manufacturing) } & 5.2 \\ \hline \end{array} $$ a. What is the median percent change? b. What is the modal percent change?

Sally Reynolds sells real estate along the coastal area of Northern California. Below are her total annual commissions between 2007 and 2017 . Find the mean, median, and mode of the commissions she earned for the 11 years. $$ \begin{array}{|lc|} \hline \text { Year } & \text { Amount (thousands) } \\ \hline 2007 & 292.16 \\ 2008 & 233.80 \\ 2009 & 206.97 \\ 2010 & 202.67 \\ 2011 & 164.69 \\ 2012 & 206.53 \\ 2013 & 237.51 \\ 2014 & 225.57 \\ 2015 & 255.33 \\ 2016 & 248.14 \\ 2017 & 269.11 \\ \hline \end{array} $$

The demand for the video games provided by Mid-Tech Video Games Inc. has exploded in the last several years. Hence, the owner needs to hire several new technical people to keep up with the demand. Mid-Tech gives each applicant a special test that Dr. McGraw, the designer of the test, believes is closely related to the ability to create video games. For the general population, the mean on this test is \(100 .\) Below are the scores on this test for the applicants. $$ \left.(\begin{array}{llllllllll} 95 & 105 & 120 & 81 & 90 & 115 & 99 & 100 & 130 & 10 \end{array}\right) $$ The owner is interested in the overall quality of the job applicants based on this test. Compute the mean and the median scores for the 10 applicants. What would you report to the owner? Does it seem that the applicants are better than the general population?

Calculate the (a) range, (b) arithmetic mean, and (c) variance, and (d) interpret the statistics. The Department of Statistics at Western State University offers eight sections of basic statistics. Following are the numbers of students enrolled in these sections: \(34,46,52,29,41,38,36,\) and 28.

Do the following: a. Compute the sample variance. b. Determine the sample standard deviation. Dave's Automatic Door, referred to in Exercise 29 , installs automatic garage door openers. Based on a sample, following are the times, in minutes, required to install 10 door openers: \(28,32,24,46,44,40,54,38,32,\) and 42.
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