# Connexions

You are here: Home » Content » Collaborative Statistics » Distribution Needed for Hypothesis Testing

### 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.

Inside Collection (Textbook):

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

# Distribution Needed for Hypothesis Testing

Earlier in the course, we discussed sampling distributions. Particular distributions are associated with hypothesis testing. Perform tests of a population mean using a normal distribution or a student's-t distribution. (Remember, use a student's-t distribution when the population standard deviation is unknown and the distribution of the sample mean is approximately normal.) In this chapter we perform tests of a population proportion using a normal distribution (usually nn is large or the sample size is large).

If you are testing a single population mean, the distribution for the test is for means:

X¯ X ~ N ( μ X , σ X n ) N( μ X , σ X n ) or t df t df

The population parameter is μμ. The estimated value (point estimate) for μμ is x¯x, the sample mean.

If you are testing a single population proportion, the distribution for the test is for proportions or percentages:

P' P' ~ N ( p , p q n ) N(p, p q n )

The population parameter is pp. The estimated value (point estimate) for pp is p'p'. p'= xnp'=xn where xx is the number of successes and nn is the sample size.

## Glossary

Normal Distribution:
A continuous random variable (RV) with pdf f(x)=1σe(xμ)2/2f(x)=1σe(xμ)2/2 size 12{ ital "pdf"= { {1} over {σ sqrt {2π} } } e rSup { size 8{ - $$x - μ$$ rSup { size 6{2} } /2σ rSup { size 6{2} } } } } {}, where μμ is the mean of the distribution and σσ is the standard deviation. Notation: XX ~ N μ σ N μ σ . If μ=0μ=0 and σ=1σ=1, the RV is called the standard normal distribution.
Standard Deviation:
A number that is equal to the square root of the variance and measures how far data values are from their mean. Notation: s for sample standard deviation and σσ for population standard deviation.
Student's-t Distribution:
Investigated and reported by William S. Gossett in 1908 and published under the pseudonym Student. The major characteristics of the random variable (RV) are:
• It is continuous and assumes any real values.
• The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at the apex than the normal distribution.
• It approaches the standard normal distribution as n gets larger.
• There is a "family" of t distributions: every representative of the family is completely defined by the number of degrees of freedom which is one less than the number of data.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

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

#### 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?

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?

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