5.6 Exercises 69However, the data suggest that the distributions are not normal, and
the sample sizes are too small for the central limit theorem to take
effect. Also, the equal variance assumption is highly suspect. If we use
a well - known t - distribution approximation, the approximate 95%
confi dence interval is [ − 2361.99, 158.99]. This interval includes 0.
Based on these assumptions, we cannot conclude that the means
are different. For detailed calculations see Chernick and Friis ( 2003 ,
pp. 163 – 166).
Because of the small sample size and apparent nonnormality, a
nonparametric or bootstrap confi dence interval would be more appro-
priate. Chernick and Friis ( 2003 ) also generate a bootstrap confi dence
interval using the percentile method.
Now, as in the case of Chernick and Friis ( 2003 ), let us compare a
95% confi dence interval for the treatment mean is [572.89, 1598.91]
based on the parametric method that uses the t - distribution. Using
Resampling Stats software and generating 10,000 bootstrap samples,
the bootstrap percentile method 95% confi dence interval is [727.1,
1558.9]. This is quite different from the parametric interval, and is a
tighter interval. The difference is another indication that the treatment
data is not normal, and neither is the sample mean. The same type of
result could be shown for the control group and for the mean
difference.
Other bootstrap confi dence intervals can be generated and are
called bootstrap t , double bootstrap, BCa, and tilted bootstrap. See
Chernick ( 2007 ) for details on these methods.
5.6 EXERCISES
- D e fi ne the following
(a) Inferential statistics
(b) Point estimate of a population parameter
(c) Confi dence interval for a population parameter
(d) Bias of an estimate
(e) Mean square error - What are the two most important properties for an estimator?
- What is the disadvantage of just providing a point estimate?
- What is the standard error of the mean?