Skip to content Skip to navigation

Connexions

You are here: Home » Content » Confidence Intervals: Confidence Interval, Single Population Mean, Population Standard Deviation Known, Normal

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual Connexions member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Endorsed by Endorsed (What does "Endorsed by" mean?)

This content has been endorsed by the organizations listed. Click each link for a list of all content endorsed by the organization.
  • CCOT Project display tagshide tags

    This module is included in aLens by: CC Open Textbook ProjectAs a part of collection:"Collaborative Statistics"

    Click the "CCOT Project" link to see all content they endorse.

    Click the tag icon tag icon to display tags associated with this content.

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • OrangeGrove display tagshide tags

    This module is included inLens: Florida Orange Grove Textbooks
    By: Florida Orange GroveAs a part of collection:"Collaborative Statistics"

    Click the "OrangeGrove" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Featured Content display tagshide tags

    This module is included inLens: Connexions Featured Content
    By: ConnexionsAs a part of collection:"Collaborative Statistics"

    Comments:

    "Collaborative Statistics was written by two faculty members at De Anza College in Cupertino, California. This book is intended for introductory statistics courses being taken by students at two- […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • Lucy Van Pelt display tagshide tags

    This module is included inLens: Lucy's Lens
    By: Tahiya MaromeAs a part of collection:"Collaborative Statistics"

    Comments:

    "Part of the Books featured on Community College Open Textbook Project"

    Click the "Lucy Van Pelt" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

  • Educational Technology Lens display tagshide tags

    This module is included inLens: Educational Technology
    By: Steve WilhiteAs a part of collection:"Collaborative Statistics"

    Click the "Educational Technology Lens" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

  • SHN CNX Workshop display tagshide tags

    This module is included inLens: Stategic Horizon Network Workshop on Alternative Couseware -- Connexions Session
    By: ConnexionsAs a part of collection:"Collaborative Statistics"

    Comments:

    "This textbook is used in many sections of community college statistics around the country. The PDF version of the book is also available in the Florida Orange Grove repository."

    Click the "SHN CNX Workshop" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

  • Bio 502 at CSUDH display tagshide tags

    This module is included inLens: Bio 502
    By: Terrence McGlynnAs a part of collection:"Collaborative Statistics"

    Comments:

    "This is the course textbook for Biology 502 at CSU Dominguez Hills"

    Click the "Bio 502 at CSUDH" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Confidence Intervals: Confidence Interval, Single Population Mean, Population Standard Deviation Known, Normal

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

User rating (How does the rating system work?)
Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

:
(0 ratings)

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

To construct a confidence interval for a single unknown population mean μμ where the population standard deviation is known, we need x¯ x as an estimate for μμ and a margin of error. Here, the margin of error is called the error bound for a population mean (abbreviated EBM). The margin of error depends on the confidence level (abbreviated CL). The confidence level is the probability that the confidence interval produced contains the true population parameter. Most often, it is the choice of the person constructing the confidence interval to choose a confidence level of 90% or higher because he wants to be reasonably certain of his conclusions.

There is another probability called alpha (αα). αα is the probability that the sample produced a point estimate that is not within the appropriate margin of error of the unknown population parameter.

Example 1

Suppose the sample mean is 7 and the error bound for the mean is 2.5.

Problem 1

x¯ = x = _______ and EBM = EBM= _______.

Solution

x¯ = x = 7 and EBM = EBM= 2.5.

The confidence interval is ( 7 - 2.5 , 7 + 2.5 ) (7-2.5,7+2.5).

If the confidence level (CL) is 95%, then we say we are 95% confident that the true population mean is between 4.5 and 9.5.

A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. Suppose we have constructed the 90% confidence interval (5, 15) where x¯ = 10 x =10 and EBM = 5 EBM=5. To get a 90% confidence interval, we must include the central 90% of the sample means. If we include the central 90%, we leave out a total of 10 % or 5% in each tail of the normal distribution. To capture the central 90% of the sample means, we must go out 1.645 standard deviations on either side of the calculated sample mean. The 1.645 is the z-score from a standard normal table that has area to the right equal to 0.05 (5% area in the right tail). The graph shows the general situation.

Normal distribution curve with values of 5 and 15 on the x-axis. Vertical upward lines from points 5 and 15 extend to the curve. The confidence interval area between these two points is equal to 0.90.

To summarize, resulting from the Central Limit Theorem,

X¯ X is normally distributed, that is, X¯ X ~ N ( μ X , σ X n ) N( μ X , σ X n )

Since the population standard deviation, σ, is known, we use a normal curve.

The confidence level, CLCL, is CL = 1 - α CL=1-α. Each of the tails contains an area equal to α 2 α 2 .

The z-score that has area to the right of α2 α2 is denoted by z α 2 z α 2 .

For example, if α 2 = 0.025 α 2=0.025, then area to the right = 0.025 =0.025 and area to the left = 1 - 0.025 = 0.975 =1-0.025=0.975 and z α 2 = z 0.025 = 1.96 z α 2 = z 0.025 =1.96 using a calculator, computer or table. Using the TI83+ or 84 calculator function, invNorm, you can verify this result. invNorm(.975,0,1)=1.96(.975,0,1)=1.96.

The error bound formula for a single population mean when the population standard deviation is known is

EBM = z α 2 ⋅ σ n EBM= z α 2 ⋅ σ n

The confidence interval has the format ( x¯ − EBM , x¯ + EBM ) ( x −EBM, x +EBM).

The graph gives a picture of the entire situation.

CL + α 2 + α 2 = CL + α = 1 CL+ α 2 + α 2 =CL+α=1.

Normal distribution curve displaying the confidence interval formulas and corresponding area formulas.

Example 2

Problem 1

Suppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of 3 points. A sample of 36 scores is taken and gives a sample mean (sample average score) of 68. Find a 90% confidence interval for the true (population) mean of statistics exam scores.

  • The first solution is step-by-step.
  • The second solution uses the TI-83+ and TI-84 calculators.

Solution A

To find the confidence interval, you need the sample mean, x¯ x , and the EBM.

  • a. x¯ = 68 x =68
  • b. EBM = z α 2 ⋅ ( σ n ) EBM= z α 2 ⋅( σ n )
  • c. σ = 3 σ=3
  • d. n = 36 n=36

CL = 0.90 CL = 0.90 so a = 1 - CL = 1 - 0.90 = 0.10 a=1-CL=1-0.90=0.10

Since α 2 = 0.05 α 2 =0.05, then z α 2 = z .05 = 1.645 z α 2 = z .05 =1.645

from a calculator, computer or standard normal table. For the table, see the Table of Contents 15. Tables.

Therefore, EBM = 1.645 ⋅ ( 3 36 ) = 0.8225 EBM=1.645⋅( 3 36 )=0.8225

This gives x¯ - EBM = 68 - 0.8225 = 67.18 x -EBM=68-0.8225=67.18

and x¯ + EBM = 68 + 0.8225 = 68.82 x +EBM=68+0.8225=68.82

The 90% confidence interval is (67.18, 68.82).

Solution B

The TI-83+ and TI-84 caculators simplify this whole procedure. Press STAT and arrow over to TESTS. Arrow down to 7:ZInterval. Press ENTER. Arrow to Stats and press ENTER. Arrow down and enter 3 for σσ, 68 for x¯ x , 36 for nn, and .90 for C-level. Arrow down to Calculate and press ENTER. The confidence interval is (to 3 decimal places) (67.178, 68.822).

We can find the error bound from the confidence interval. From the upper value, subtract the sample mean or subtract the lower value from the upper value and divide by two. The result is the error bound for the mean (EBM).

EBM = 68.822 - 68 = 0.822 EBM=68.822-68=0.822 or EBM = ( 68.822 − 67.178 ) 2 = 0.822 EBM= ( 68.822 − 67.178 ) 2 =0.822

We can interpret the confidence interval in two ways:

  1. We are 90% confident that the true population mean for statistics exam scores is between 67.178 and 68.822.
  2. Ninety percent of all confidence intervals constructed in this way contain the true average statistics exam score. For example, if we constructed 100 of these confidence intervals, we would expect 90 of them to contain the true population mean exam score.

Now for the same problem, find a 95% confidence interval for the true (population) mean of scores. Draw the graph. The sample mean, standard deviation, and sample size are:

Problem 2

  • a. x¯ = x =
  • b. σ = σ=
  • c. n = n=

Solution

  • a. x¯  =  68 x  =  68
  • b. σ  =  3 σ = 3
  • c. n  =  36 n = 36

The confidence level is CL = 0.95CL= 0.95. Graph:

The confidence interval is (use technology)

Problem 3

( x¯ - EBM , x¯ + EBM ) = (_______, _______) ( x -EBM, x +EBM)=(_______, _______). The error bound EBM =EBM = _______.

Solution

( x¯ - EBM , x¯ + EBM ) = (67.02 , 68.98) ( x -EBM, x +EBM)=(67.02 , 68.98). The error bound EBM =EBM = 0.98.

We can say that we are 95 % confident that the true population mean for statistics exam scores is between 67.02 and 68.98 and that 95% of all confidence intervals constructed in this way contain the true average statistics exam score.

Example 3

Suppose we change the previous problem.

Problem 1

Leave everything the same except the sample size. For this problem, we can examine the impact of changing nn to 100100 or changing nn to 2525.

  • a. x¯ = 68 x =68
  • b. σ = 3 σ=3
  • c. z α 2 = 1.645 z α 2 =1.645

Solution A

If we increase the sample size n n to 100, we decrease the error bound.

EBM = z α 2 ⋅ ( σ n ) = 1.645 ⋅ ( 3 100 ) = 0.4935 EBM= z α 2 ⋅( σ n )=1.645⋅( 3 100 )=0.4935

Solution B

If we decrease the sample size n n to 25, we increase the error bound.

EBM = z α 2 ⋅ ( σ n ) = 1.645 ⋅ ( 3 25 ) = 0.987 EBM= z α 2 ⋅( σ n )=1.645⋅( 3 25 )=0.987

Problem 2

Leave everything the same except for the confidence level. We increase the confidence level from 0.90 to 0.95.

  • a. x¯ = 68 x =68
  • b. σ = 3 σ=3
  • c. z α 2 z α 2 changes from 1.645 1.645 to 1.961.96.

Solution

Figure 1
(a) (b)
Normal distribution curve with 0.90 confidence interval area blocked off and corresponding residual areas.Normal distribution curve with 0.95 confidence interval area blocked off and corresponding residual areas.

The 90% confidence interval is (67.18, 68.82). The 95% confidence interval is (67.02, 68.98). The 95% confidence interval is wider. If you look at the graphs, because the area 0.95 is larger than the area 0.90, it makes sense that the 95% confidence interval is wider.

Calculating the Sample Size n

If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.

The error bound formula for a population mean when the population standard deviation is known is

  • EBM = z α 2 ⋅ σ n EBM= z α 2 ⋅ σ n
  • Solving for nn gives you an equation for the sample size.
  • n= z α 2 2 ⋅ σ 2 EBM 2 n= z α 2 2 ⋅ σ 2 EBM 2

Example 4

The population standard deviation for the age of Foothill College students is 15 years. If we want to be 95% confident that the sample mean age is within 2 years of the true population mean age of Foothill College students , how many randomly selected Foothill College students must be surveyed?

From the problem, we know that

  • σ=15σ=15
  • EBM=2EBM=2
  • z α 2 z α 2 = 1.96 =1.96 because the confidence level is 95%95%.

Using the equation for the sample size, we have

  • n= z α 2 2 ⋅ σ 2 EBM 2 n= z α 2 2 ⋅ σ 2 EBM 2
  • n= 1.96 2 ⋅ 15 2 2 2 n= 1.96 2 ⋅ 15 2 2 2
  • n=216.09n=216.09
  • Round the answer to the next higher value to ensure that the sample size is as large as it should be. Therefore, 217217 Foothill College students should be surveyed for us to be 95%95% confident that we are within 22 years of the true population age of Foothill College students.

Note:

In reality, we usually do not know the population standard deviation so we estimate it with the sample standard deviation or use some other way of estimating it (for example, some statisticians use the results of some other earlier study as the estimate).

Glossary

Confidence Level (CL):
The percent expression for the probability that the confidence interval contains the true population parameter. For example, if the CL=90%CL=90%, then in 9090 out of 100100 samples the interval estimate will enclose the true population parameter.
Error Bound for a Population Mean (EBM):
The margin of error. Depends on the confidence level, sample size, and known or estimated population standard deviation.

Content actions

Give Feedback:

E-mail the module authors | Rate module ( How does the rating system work?)

Rating system

Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

(0 ratings)

Download:

Module PDF

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.

| A lens (?)

Definition of a lens

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual Connexions member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks