STA309 Final
Inference for Distributions
•
Inference for the mean of a population:
o
When σ is known:
Tests and confidence intervals for the mean m of a normal population
are based on the sample mean J of an SRS. Because of the central
limit theorem, the resulting procedures are approximately correct for
other population distributions when the sample is large.
o
The
standardized sample mean is the onesample z statistic (When we know s we use the z statistic and the standard normal
distribution.):
Assumptions:
SRS; n < 15, normal population; n ≥ 15; no outliers or strong skewness
estimate = J
SE
J
= σ / √n
critical value = z*
o
When σ is unknown:
In practice, we do not know s.
Assumptions:
SRS; n < 15, data close to normal; 15 ≤ n <40, no outliers or strong skewness; n ≥ 40; even for clearly skewed data
estimate = J
SE = s / √n.
Get the onesample
t statistic (with n 1 degrees of freedom):
Critical value = t*(n  1)
•
The t distribution
:
o
There is a t distribution for every positive integer k (degree of freedom). Call the distribution t(k).
o
The density curves of the t(k) distribution are symmetric about zero and bellshaped.
o
The curves are more spread out than the standard normal curves (due to substituting s for s).
o
*** As the degrees of freedom k increase, the t(k) density curve approaches the N(0,1) curve
o
Onesample
t
procedures
:
A level C confidence interval for
μ (look at Table D for t* using df = n – 1):
Hypothesis testing of
H
0
:
μ
=
μ
0
.
1. State the hypothesis.
2. Calculate the test statistic (solve for SE, find t* from Table D and α, plug into CI equation).
3. Compute the
P
value.
Remember to * 2 if two tailed test!
4. Compare the
P
value to α and make your conclusion.
(pvalue ≤ α
reject H
0
;
pvalue > α
accept H
0
)
•
Robustness of
t
procedures
:
a statistical inference procedure is robust if the probability calculations required are insensitive to violations of the conditions that
usually justify the procedure.
The t procedures are relatively robust when the population is nonnormal, especially for larger sample sizes. The t procedures are useful for nonnormal data when n ≥ 15 unless the data show
outliers or strong skewness.
n large and equal
•
Two Population Means:
o
Matched Pairs Comparisons:
subjects are matched in pairs and each treatment is
given to one subject in each pair.
W
e have a separate sample from each
treatment or each
population.
Subjects are matched in pairs and each treatment is
given to one subject in each pair.
To compare the responses to two treatments, apply the onesample t procedures
to the observed
differences
. We calculate the
difference
in “before” and “after” test scores for each in the pair.
Assumptions:
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 Spring '07
 Gemberling
 Central Limit Theorem, Normal Distribution

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