# Finding Confidence Intervals with the Normal Distribution

## Standard Deviation

Reality often differs from theory; in the real world we seldom know what the true population standard deviation is. The **standard deviation** refers to the variability of individual observations around their mean. But for this lesson we are going to pretend that the **population standard deviation** denoted by the symbol sigma, is known to us and we are going to use that to help us construct the confidence interval for the **population mean**, which itself is denoted by the symbol mu.

A **confidence interval** is a range of values that expresses the uncertainty associated with a parameter, like the population mean.

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

## You must cCreate an account to continue watching

### Register to view this lesson

As a member, you'll also get unlimited access to over 84,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Get unlimited access to over 84,000 lessons.

Try it now*It only takes a few minutes to setup and you can cancel any time.*

###### Already registered? Log in here for access

Back*Coming up next:*Determining the Sample Size to Estimate Confidence Intervals: Definition & Process

### You're on a roll. Keep up the good work!

### Just checking in. Are you still watching?

Yes! Keep playing.## Cases Where This Applies

There are three possible cases where this can be applied.

**Case 1**: the population standard deviation is known. The sample size is small (n<30). In other words, n, the sample size, is less than 30. And, the population is normally distributed.

**Case II**: Again, the population standard deviation is known. But, this time around, the sample size is large (n>=30). This means that n is greater than or equal to 30.

In **Case III**: Again, the population standard deviation is known. The sample size is small (n<30). And, the population is not normally distributed or we don't know its distribution.

The third case uses nonparametric methods to find the confidence interval for mu, meaning we use inferential methods that are not concerned with parameters- such as the population mean or population standard deviation.

## Calculating the Confidence Interval

In this lesson, we are going to focus on the first two cases where we use the normal distribution to make the confidence interval for mu. In the first two cases, we would calculate the confidence interval for mu using the following equations:

- Where x bar denotes the value of the sample mean
- Sigma refers to the population standard deviation
- And n refers to the sample size

The value for z is found from standard normal distribution tables for a given confidence level right here. The quantity of z times sigma x bar is the margin of error and it is denoted by the symbol E. In other words, E = z times (x) sigma x bar

## Example

Simply put, the **margin of error (E)** is the quantity we subtract or add to x bar to obtain a confidence interval for mu. Let's build on this to solidify your knowledge of all this crazy terminology with an actual example.

A tech company has just come out with a new cell phone. It needs to figure out the price at which to sell this phone by first figuring out the average price of all similar cell phones available on the market. The company's market research department takes a sample of 16 comparable cell phones to find they have a mean price of $500. The market research department also knows that the population standard deviation of the prices for all such cell phones is $100. Assume the population is normally distributed. Construct a 90% confidence interval for the mean price for all similar cell phones.

First, let's just figure out what we know. We know that the sample size, n = 16. The sample mean, x bar is $500. And the population standard deviation (sigma) is $100. The standard deviation of x bar is simply sigma, divided by the square root of n, using the equation shown before. In our case, that's simply 100/4, which is equal to $25. Using the tables right here, you'd find the value for z for a confidence level of 90% is 1.65. At this point, you have all the values you possibly need to figure out the appropriate confidence interval.

Remember our equations from before; the 90% confidence interval for mu is equal to x bar + - z times sigma sub x bar. Just plug and chug to get 500 + - 1.65 (25). That's equal to 500 + - 41.25. So, we get $458.75 to $541.25. This is our confidence interval. In other words, we are 90% confident that the mean price of all such cell phones is between $458.75 and $541.25.

## Lesson Summary

Now you know how to construct confidence intervals from normal populations when the population standard deviation is known. The **standard deviation** is the variability of individual observations around their mean. The **population standard deviation** in our equations was denoted by the symbol sigma, while the **population mean** was denoted by the symbol mu. The **margin of error (E)** is the quantity we subtract or add to x bar to obtain a confidence interval for mu.

Using the equations we went over, you should now be able to use them and the tables on this page to construct confidence intervals in the two cases we went over.

To unlock this lesson you must be a Study.com Member.

Create your account

### Register to view this lesson

### Unlock Your Education

#### See for yourself why 30 million people use Study.com

##### Become a Study.com member and start learning now.

Become a MemberAlready a member? Log In

Back*Finding Confidence Intervals with the Normal Distribution*

Related Study Materials

- Math Courses
- Statistics Courses
- Transferable Credit Courses
- Certificates Courses
- Study Courses
- College Algebra Textbook
- Business Math Textbook
- High School Geometry: Homework Help Resource
- High School Geometry: Help and Review
- High School Geometry: Tutoring Solution
- High School Trigonometry: Help and Review
- High School Trigonometry: Homework Help Resource
- High School Trigonometry: Tutoring Solution
- Holt McDougal Algebra 2: Online Textbook Help
- ACT Math Prep: Review & Practice
- CBEST Math: Practice & Study Guide
- Math 105: Precalculus Algebra
- Introduction to Statistics: Tutoring Solution

##### Browse by Courses

- Practice Problem Set for Rational Expressions
- Practice Problem Set for Radical Expressions & Functions
- Practice Problem Set for Exponentials and Logarithms
- Practice Problem Set for Probability Mechanics
- Practice Problem Set for Sequences and Series
- Quiz & Worksheet - How to Apply L'Hopital's Rule to Complex Cases
- Quiz & Worksheet - How to Apply L'Hopital's Rule to Simple Cases
- Quiz & Worksheet - Function Properties from Derivatives for Data Mining
- Quiz & Worksheet - How to Identify Functions From Derivative Graphs with Data Mining
- Quiz & Worksheet - L'Hopital's Rule
- ACT English - Grammar and Usage: Tutoring Solution
- ACT English - Rhetorical Strategy: Tutoring Solution
- ACT English - Organization: Tutoring Solution
- ACT English - Style: Tutoring Solution
- ACT Math - Overview: Tutoring Solution

##### Browse by Lessons

- Biology 202L: Anatomy & Physiology II with Lab
- Biology 201L: Anatomy & Physiology I with Lab
- California Sexual Harassment Refresher Course: Supervisors
- California Sexual Harassment Refresher Course: Employees
- Sociology 110: Cultural Studies & Diversity in the U.S.
- Classroom Management for Special Education
- Enzymes in the Human Body
- Biology 202L Labs
- Biology 201L Labs
- Significant Art in Texas and the US
- Addressing Cultural Diversity in Distance Learning
- New Hampshire Homeschooling Laws
- Setting Student Expectations for Distance Learning
- COVID-19 Education Trends that are Here to Stay
- What to Do with a COVID-19 College Gap Year
- Active Learning Strategies for the Online Classroom
- How to Promote Online Safety for Students in Online Learning

##### Latest Courses

- To Kill a Mockingbird: Characters, Setting & Author
- Australian Tertiary, Quaternary & Quinary Industry Growth
- James Joyce's Araby: Tone & Theme
- Bradford Protein Assay: Advantages & Disadvantages
- El Sur by Jorge Luis Borges: Author, Summary & Theme
- Impact of Global Media on Social Values
- Klipspringer & Owl Eyes in The Great Gatsby
- Quiz & Worksheet - Nahuas of Mexico & El Salvador
- Quiz & Worksheet - Symbolism in A Tale of Two Cities
- Quiz & Worksheet - Don Quixote Chapter 8 Overview
- Quiz & Worksheet - Characteristics of Transvestites
- Flashcards - Real Estate Marketing Basics
- Flashcards - Promotional Marketing in Real Estate
- What is STEM Education?
- Pronoun Worksheets

##### Latest Lessons

- LSAT Prep: Tutoring Solution
- SAT Physics: Help and Review
- CBEST Reading: Practice & Study Guide
- PSAT Prep: Help and Review
- Technical Writing: Help and Review
- Complex Numbers: Homeschool Curriculum
- Earthquakes: Homework Help
- Quiz & Worksheet - GRE Text Completion Question Format
- Quiz & Worksheet - Directly Proportional
- Quiz & Worksheet - Gravitropism
- Quiz & Worksheet - Functions of Auxins
- Quiz & Worksheet - Structure & Functions of Flagella

##### Popular Courses

- Multiplier in Economics: Definition, Effect & Formula
- How to Study for CSET Social Science
- Fun Math Games for 3rd Grade
- How to Pass the Life & Health Insurance Exam
- NBCRNA Recertification Requirements
- School Closures in Massachusetts: Online Learning in MA During the COVID-19 Outbreak
- FTCE Biology 6-12: Passing Score
- Free English Language Courses
- Books for Guided Reading
- How to See If Your School Accepts Study.com Credit
- Meiosis Lesson Plan
- How to Take a Study.com Proctored Exam

##### Popular Lessons

##### Math

##### Social Sciences

##### Science

##### Business

##### Humanities

##### Education

##### History

##### Art and Design

##### Tech and Engineering

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

##### Health and Medicine

- Jobs and productivity! How do retail stores rate? One way to answer this question is to examine annual profits per employee. The following data give annual profits per employee (in units of one thous
- The average zinc concentration taken from a sample of zinc measurements in 36 different locations is found to be 3.0 ounces per gallon. Find the 90% confidence interval for the mean zinc concentration
- Assume that the average daily price change for a share of XYZ corporation is 0.05% obtained from a sample of 50 days. Also, assume that the standard deviation of the population (the population deviati
- An existing inventory for a test measuring self-esteem indicates that the scores have a standard deviation of 8. A psychologist gave the self-esteem test to a random sample of 70 individuals, and thei
- Based on a random sample of 1,160 adults, the mean amount of sleep per night is 7.92 hours. Assuming the population standard deviation for amount of sleep per night is 2.1 hours, construct and interpr
- Based on a sample of repair records, an engineer calculates a 95% confidence interval for the mean cost to repair a fiber-optic component to be ($140, $160). A supervisor summarizes this result in a r
- Consumer Reports would like to conduct a hypothesis test to determine if average battery life for an iPad is different from the average battery life for a Kindle Fire. The following data represents th
- In constructing a confidence interval estimate of the population mean you decide to select 49 random observations to get your point estimate of the mean (sample mean). Your friend is also constructing
- A random sample of six cars from a particular model year had the following fuel consumption figures (in miles per gallon). Sample data: |18.7 |19.9 |21 |19.8 |20.4 |18 Find the 90% confidence inte
- Suppose that a one-day 97.5% VaR is estimated at $13 million from 2,000 observations. The one-day changes are approximately normal with mean zero and standard deviation of $6 million. Estimate a 99% c

#### Explore our library of over 84,000 lessons

- Create a Goal
- Create custom courses
- Get your questions answered

**Premium**to add all these features to your account!

**Premium**to add all these features to your account!