Skip to content Skip to navigation Skip to collection information

Connexions

You are here: Home » Content » Collaborative Statistics » Comparing Two Independent Population Means with Known Population Standard Deviations

Navigation

Table of Contents

Lenses

What is a lens?

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to 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 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 ...

In these lenses

  • Lucy Van Pelt display tagshide tags

    This collection is included inLens: Lucy's Lens
    By: Tahiya Marome

    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.

  • Bio 502 at CSUDH display tagshide tags

    This collection is included inLens: Bio 502
    By: Terrence McGlynn

    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.
 

Comparing Two Independent Population Means with Known Population Standard Deviations

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

Summary: This module provides an overview of hypothesis testing in situations where there are both two independent population means and known population standard deviations in statistics.

Even though this situation is not likely (knowing the population standard deviations is not likely), the following example illustrates hypothesis testing for independent means, known population standard deviations. The sampling distribution for the difference between the means is normal and both populations must be normal. The random variable is X 1 ¯ - X 2 ¯ X 1 - X 2 . The normal distribution has the following format:

Normal distribution

X 1 ¯ − X 2 ¯ ~ N [ u 1 − u 2 , ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2 ] X 1 ¯ − X 2 ¯ ~N[ u 1 − u 2 , ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2 ]
(1)

The standard deviation is:

( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2 ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2
(2)

The test statistic (z-score) is:

z = ( x 1 ¯ − x 2 ¯ ) − ( μ 1 − μ 2 ) ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2 z= ( x 1 ¯ − x 2 ¯ ) − ( μ 1 − μ 2 ) ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2
(3)

Example 1

independent groups, population standard deviations known: The mean lasting time of 2 competing floor waxes is to be compared. Twenty floors are randomly assigned to test each wax. Both populations have a normal distribution. The following table is the result.

Table 1
Wax Sample Mean Number of Months Floor Wax Last Population Standard Deviation
1 3 0.33
2 2.9 0.36

Problem 1

Does the data indicate that wax 1 is more effective than wax 2? Test at a 5% level of significance.

Solution

This is a test of two independent groups, two population means, population standard deviations known.

Random Variable: X 1 ¯ - X 2 ¯ = X 1 - X 2 = difference in the mean number of months the competing floor waxes last.

H o : μ 1 ≤ μ 2 H o : μ 1 ≤ μ 2

H a : μ 1 > μ 2 H a : μ 1 > μ 2

The words "is more effective" says that wax 1 lasts longer than wax 2, on the average. "Longer" is a ">"">" symbol and goes into HaHa. Therefore, this is a right-tailed test.

Distribution for the test: The population standard deviations are known so the distribution is normal. Using the formula above, the distribution is:

X 1 ¯ − X 2 ¯ ~ N ( 0 , 0.33 2 20 + 0.36 2 20 ) X 1 ¯ − X 2 ¯ ~N ( 0 , 0.33 2 20 + 0.36 2 20 )

Since μ 1 ≤ μ 2 μ 1 ≤ μ 2 then μ 1 − μ 2 ≤ 0 μ 1 − μ 2 ≤0 and the mean for the normal distribution is 0.

Calculate the p-value using the normal distribution: p-value = 0.1799

Graph:

Figure 1
Normal distribution curve of difference in the average number of months the competing floor waxes last with values of 0 and 0.1 on the x-axis. A vertical upward line extends from 0.1 to the curve and the p-value points to the area to the right of 0.1.

x 1 ¯ - x 2 ¯ = 3 - 2.9 = 0.1 x 1 - x 2 =3-2.9=0.1

Compare α and the p-value: α = 0.05α=0.05 and p-value = 0.1799. Therefore, α <α< p-value.

Make a decision: Since α <α< p-value, do not reject H o H o .

Conclusion: At the 5% level of significance, from the sample data, there is not sufficient evidence to conclude that the mean time wax 1 lasts is longer (wax 1 is more effective) than the mean time wax 2 lasts.

Note:
TI-83+ and TI-84: Press STAT. Arrow over to TESTS and press 3:2-SampZTest. Arrow over to Stats and press ENTER. Arrow down and enter .33 for sigma1, .36 for sigma2, 3 for the first sample mean, 20 for n1, 2.9 for the second sample mean, and 20 for n2. Arrow down to μμ1: and arrow to > μμ2. Press ENTER. Arrow down to Calculate and press ENTER. The p-value is p = 0.1799 and the test statistic is 0.9157. Do the procedure again but instead of Calculate do Draw.

Collection Navigation

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add:

Collection to:

My Favorites (?)

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

| A lens I own (?)

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to 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 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

Module to:

My Favorites (?)

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

| A lens I own (?)

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to 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 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