/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Find the population variance and... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 7

Find the population variance and standard deviation or the sample variance and standard deviation as indicated. $$ \text { Population: } 3,6,10,12,14 $$

Short Answer

Expert verified
Variance = 16, Standard Deviation = 4

Step by step solution

01

Calculate the Mean of the Population

First, find the mean (average) of the population values. The values are 3, 6, 10, 12, and 14. The mean is calculated by the formula: \[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} \ \text{Mean} = \frac{3 + 6 + 10 + 12 + 14}{5} = \frac{45}{5} = 9 \]
02

Calculate Each Deviation from the Mean

Subtract the mean from each value to find the deviation of each value: \[ 3 - 9 = -6, \ 6 - 9 = -3, \ 10 - 9 = 1, \ 12 - 9 = 3, \ 14 - 9 = 5 \]
03

Square Each Deviation

Square each of the deviations: \[ (-6)^2 = 36, \ (-3)^2 = 9, \ 1^2 = 1, \ 3^2 = 9, \ 5^2 = 25 \]
04

Calculate the Mean of the Squared Deviations

Find the mean of these squared deviations to get the population variance: \[ \text{Variance} = \frac{36 + 9 + 1 + 9 + 25}{5} = \frac{80}{5} = 16 \]
05

Calculate the Population Standard Deviation

To find the population standard deviation, take the square root of the variance: \[ \text{Standard Deviation} = \root 2 \/16 = 4 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

calculate mean
To start any variance or standard deviation calculations, you first need to find the mean (average) of your dataset.
Simply, the mean is the sum of all values divided by the number of values.
The formula to calculate the mean is: \[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} \]
For example, consider the population values: 3, 6, 10, 12, and 14.
First, sum them up: \[ 3 + 6 + 10 + 12 + 14 = 45 \]
Then, divide by the number of values (which is 5): \[ \text{Mean} = \frac{45}{5} = 9 \]
So, the mean of these values is 9.
deviation from mean
Next, we move on to finding how each value differs from the mean.
These differences are called deviations.
To find the deviation of each value, subtract the mean from each value in the dataset.
Let's go through it with our dataset:
1. Calculate for 3: \[ 3 - 9 = -6 \]
2. For 6: \[ 6 - 9 = -3 \]
3. For 10: \[ 10 - 9 = 1 \]
4. For 12: \[ 12 - 9 = 3 \]
5. For 14: \[ 14 - 9 = 5 \]
So, the deviations are -6, -3, 1, 3, and 5 respectively.
These deviations tell us how far each value lies from the mean.
squared deviations
After finding the deviations, the next step is to square each of these deviations.
Why square them? To ensure all values are positive and give more weight to larger differences.
Squaring a number means multiplying it by itself.
Let's square our deviations:
1. For -6: \[ (-6)^2 = 36 \]
2. For -3: \[ (-3)^2 = 9 \]
3. For 1: \[ 1^2 = 1 \]
4. For 3: \[ 3^2 = 9 \]
5. For 5: \[ 5^2 = 25 \]
So, the squared deviations are 36, 9, 1, 9, and 25 respectively.
mean of squared deviations
Now, we need to find the average of these squared deviations.
This average is known as the variance.
Use the same process as finding the mean but with the squared deviations instead.
Sum up the squared deviations: \[ 36 + 9 + 1 + 9 + 25 = 80 \]
Then, divide by the number of values: \[ \text{Variance} = \frac{80}{5} = 16 \]
So, the variance of this population is 16.
population standard deviation
Finally, to find the population standard deviation, take the square root of the variance.
This step reverses the squaring we did earlier and returns the value to the original unit.
Use the square root function: \[ \text{Standard Deviation} = \root 2 \/16 = 4 \]
Therefore, the population standard deviation is 4.
This value tells us how spread out the values are in the population.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Explain what each quartile represents.

True or False: When comparing two populations, the larger the standard deviation, the more dispersion the distribution has. provided that the variable of interest from the two populations has the same unit of measure.

The data in the next column represent exam scores in a statistics class taught using traditional lecture and a class taught using a "flipped" classroom. The "flipped" classroom is one where the content is delivered via video and watched at home, while class time is used for homework and activities. $$ \begin{array}{llllllll} \hline \text { Traditional } & 70.8 & 69.1 & 79.4 & 67.6 & 85.3 & 78.2 & 56.2 \\\ & 81.3 & 80.9 & 71.5 & 63.7 & 69.8 & 59.8 & \\ \hline \text { Flipped } & 76.4 & 71.6 & 63.4 & 72.4 & 77.9 & 91.8 & 78.9 \\ & 76.8 & 82.1 & 70.2 & 91.5 & 77.8 & 76.5 & \end{array} $$ (a) Determine the mean and median score for each class. Comment on any differences. (b) Suppose the score of 59.8 in the traditional course was incorrectly recorded as \(598 .\) How does this affect the mean? the median? What property does this illustrate?

What does the five-number summary consist of?

You are negotiating a contract for the Players Association of the NBA. Which measure of central tendency will you use to support your claim that the average player's salary needs to be increased? Why? As the chief negotiator for the owners, which measure would you use to refute the claim made by the Players Association?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.