When you perform a hypothesis test of a single population mean μμ using a
Student's-t distribution (often called a t-test), there are fundamental assumptions
that need to be met in order for the test to work properly. Your data should be a
simple random sample that comes from a population that is approximately
normally distributed. You use the sample standard deviation to approximate the
population standard deviation. (Note that if the sample size is sufficiently large, a t-test will work even if the population is not approximately normally distributed).
When you perform a hypothesis test of a single population mean μμ using a
normal distribution (often called a z-test), you take a simple random sample from
the population. The population you are testing is normally distributed or your sample size is sufficiently large. You know the value of the population
standard deviation.
When you perform a hypothesis test of a single population proportion pp, you
take a simple random sample from the population. You must meet the conditions
for a binomial distribution which are there are a certain number nn of independent
trials, the outcomes of any
trial are success or failure, and each trial has the same probability of a success pp. The shape of the binomial distribution needs to be
similar to the shape of the normal distribution. To ensure this, the quantities npnp
and nqnq must both be greater than five (np>5np>5 and nq>5nq>5). Then the binomial distribution of sample (estimated)
proportion can be approximated by the normal distribution with
μ=pμ=p
and
σ=
p
⋅
q
n
σ=
p
⋅
q
n
.
Remember that q=1-pq=1-p.
- Binomial Distribution:
A discrete random variable (RV) which arises from Bernoulli trials. There are a fixed number, nn, of independent trials. “Independent” means that the result of any trial (for example, trial 1) does not affect the results of the following trials, and all trials are conducted under the same conditions. Under these circumstances the binomial RV
XX size 12{X} {} is defined as the number of successes in nn trials. The notation is:
XX~ B
(
n
,
p
)B(n,p). The mean is μ=np
μ
np
and the standard deviation is
σ
=
npq
σ=npq. The probability of exactly xx successes in nn trials is
P
(
X
=
x
)
=
n
x
p
x
q
n
−
x
P(X=x)=
n
x
p
x
q
n
−
x
.
- 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-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.
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