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.
- Normal Distribution:
A continuous random variable (RV) with pdf
f(x)=1σ2πe−(x−μ)2/2σ2f(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 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.
