Hypothesis Testing of Single Mean and Single Proportion: Distribution Needed for Hypothesis Testing
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Hypothesis Testing of Single Mean and Single Proportion: Distribution Needed for Hypothesis Testing
1.13
2008/06/06 17:21:50 GMT-5
2012/06/12 08:21:31.192 GMT-5
Barbara
Illowsky
Barbara Illowsky, Ph.D.
illowskybarbara@deanza.edu
Maxfield Foundation
Maxfield Foundation
cnx@cnx.org
Susan
Dean
Susan Dean
deansusan@deanza.edu
Connexions
Connexions
cnx@cnx.org
sdean billowsky
sdean billowsky cnxorg
MaxfieldFoundation
elementary
statistics
Mathematics and Statistics
en
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 n 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
~
N
(
μ
X
,
σ
X
n
)
or
t
df
The population parameter is μ. The estimated value (point estimate) for μ is x,
the sample mean.If you are testing a single population proportion, the distribution for the test is for
proportions or percentages:
P'
~
N
(
p
,
p
⋅
q
n
)
The population parameter is p. The estimated value (point estimate) for p is
p'.
p'=
xn
where x is the number of successes and n is the sample size.
Normal Distribution
A continuous random variable (RV) with pdf
f(x)=1σ2πe−(x−μ)2/2σ2 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 μ is the mean of the distribution and σ is the standard deviation. Notation: X ~ N
μ
σ
. If μ=0 and σ=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 σ 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.