Download An unreasonable assumption A major drawback of our inference

Section 7.1
1
An unreasonable assumption
A major drawback of our inference methods so far is
that s is almost never known in practice. We can
estimate it from the sample by replacing it with the
estimate s, but this estimate is often poor, especially
when n is small.
• standard error
the statistic s n , used as an estimate for the
standard deviation s n of the sampling
distribution for the sample mean statistic x
†
†
†
Section 7.1
2
The t-distribution
When we replace s with the estimate s in the formula
for the statistic
z=
x -m
0,
s
n
we obtain a new statistic
†
x -m
0
t=
s
n
which has a distribution that is not normal, although it
†
it is bell-shaped and symmetric about 0 like the z
distribution N(0,!1). However, the t distribution
• has thicker tails (larger spread) than the standard
normal distribution (substituting s with s
introduces more variation)
• becomes closer to a normal distribution as the
number of degrees of freedom (n!–!1) increases
(increasing n leads to a closer estimate of s by s)
Section 7.1
3
One-sample t procedures
From a SRS selected from a normal population, or a
large SRS from any population, determine sample
statistics x and s.
A level
C confidence interval for m is
†
x ± t*
s
n
1-C
critical value for the
2
t-distribution with n – 1 degrees of freedom
†
where t* is the upper
†
†
Section 7.1
4
One-sample t hypothesis test
Assumptions: SRS selected from a normal population,
or a large SRS from any population
• State hypotheses:
Null hypothesis
Alternative hypothesis
H0: m = m0
Ha: m > m0, or
m < m0, or
m ≠ m0
• Calculate test statistic:
t-statistic based on H0:
x -m
0
t=
s
n
• Find P-value:
Sampling distribution probability associated with
†
appropriate H0:
P = P( T ≥ t ), or
P = P( T ≤ t ), or
P = 2P( T ≥ t )
Conclusion: assess evidence against H0 in favor of Ha
depending on how small P is.
[TI-83: STAT TESTS T-Test… ]
Section 7.1
5
Matched pairs procedures
To compare responses to two treatments in a matched
pairs design, apply the one-sample t procedures to the
observed differences of the pairs.
• robustness
statistical inference technique in which the level of
confidence or P-value change little when the
underlying assumptions are violated
Since the t-distributions have thick tails, outliers are
more common than for normal distributions, so like x,
the t procedures are strongly influenced by outliers.
†
Consequently, t procedures are robust when
• no outliers are present,
• the data are symmetrically distributed, or
• when sample sizes are large.