Download Matched pairs procedures To compare responses to two treatments

Chapter 25
1
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:
Assumptions: independently selected pairs of
measurements (hence not more than 10% of the
size of the population when selected without
replacement) form a SRS of differences d that
satisfy a Nearly Normal Condition. (View a
histogram or normal probability plot of the data to
check this condition.)
A level C confidence interval for µ d (the mean
difference) is
d ± t* €
n −1
sd
n
where s d is the sample standard deviation of
€
differences, and€t * is the appropriate critical
n −1
value depending on the level of confidence C for the
€ t-distribution with n – 1 degrees of freedom
€
Chapter 25
2
Paired sample t hypothesis test
• State hypotheses:
Null hypothesis
H0 : µ d = 0
Alternative hypothesis
HA: µ > 0, or µ < 0, or µ ≠ 0
€
• Choose the model:
An independently selected SRS of differences drawn
from a nearly normal population satisfying the 10%
Condition, so Student’s t model applies to
standardized sampling distribution for y
• Mechanics:
Compute t-statistic based on H€
0: t =
d −0
(s
d
/ n
)
.
Probability associated with appropriate HA:
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… ]
Chapter 25
3
The Sign test
A distribution-free method recharacterizes the
hypothesized mean value 0 as a hypothesized median
value, then ask what proportion of the differences
exceed or fall shy of the median value 0? This replaces
the original underlying statistic d with a proportion
pˆ = proportion of the differences greater than
(or less than) the
€ hypothesized median value 0
€
We then carry out a one-proportion z test with
null hypothesis H0: p = .50
and suitable alternative hypothesis.
While this sign test dispenses with many of the
conditions required by the t test, the t test is the more
powerful when all its conditions are met.