Conditions for inference on means
One mean.
Modified simple conditions: Our modification, based on our understanding of the Central Limit Theorem:
1. Simple Random Sample
2. Pop’n distribution normal is normal OR the sample size is large enough so that the sampling dist’n of
X will be approximately normally distributed. (Almost always n  30 is large enough, and often even
smaller is OK.)
3. Pop’n standard deviation  is known.
Additional complication introduced by not knowing the population standard deviation:
If the population is skewed, we have a greater chance of obtaining a sample with one or more scores which
are quite separated from the rest. Those may/will have a large effect on the sample standard deviation and
so the results of the analysis may/will depend strongly on whether the sample has an outlier in it. Thus, we
should be cautious of applying t-procedures when the sample has an outlier in it, particularly if the sample
size is not large. (See exercises 18.32 and 18.34.)
One mean.
Professional Conditions:
1. Simple Random Sample
2. There are different conditions needed about the distribution of the population, depending on the sample
size. Fill in the blanks here.
a. Sample size is (small) _______________. Use t-procedures only if the sample data appear to
come from a normal distribution (roughly symmetric, single peak, no outliers.) If the sample
data are clearly skewed or if outliers are present, do not rely on the t-procedures.
b. Sample size (medium)_________________. Use t-procedures unless your data has outliers or
strong skewness. In some professional situations, you might report the results of the analysis
both with and without the outlier(s.)
c. Sample size (large) __________________. The t-procedures can be used even for clearly
skewed distributions.
Two means – from a matched pairs design.
Because we are using the t-procedures for one mean on the differences, the sample is the set of differences. Use
the conditions above on the set of differences.
Two means - Two independent samples:
Professional Conditions:
1. Randomization. The data must be collected with suitable randomization techniques. For a survey, two
independent separate SRS’s from two separate pop’ns. For an experiment, an appropriate randomized
design.
2. Shape of the pop’n distributions.
It’s best to have equal or approximately equal samples from the two populations. If so, then if
a.
the two dist’ns have the similar shapes, it is acceptable to use t-procedures even for sample sizes
as small as _________.
b. In any case, it is acceptable to _________________ and then use the criteria for t-procedures of
one mean on the ______________________.
Conditions for inference on proportions
One proportion:
Conditions:
1. Simple Random Sample
2. The sample size has to be large enough for the dist’n of p to be approximately normal.
Professional Conditions: How large depends on the value of the population proportion and what
technique we’re using.
a. Large-sample confidence interval. Both the number of successes and the number of failures in
the sample must both be ______________.
b. Accurate confidence intervals – plus-four. If the confidence level is _____________, then OK
to use this procedure if the total sample size is _________ with any counts of successes and
failures.
c.
Significance tests. OK if both _____________and ________________
3. The sample size has to be large enough for the dist’n of p to be approximately normal.
Simple Condition: OK if _________________________________________
Two proportions:
Conditions:
1. Randomization. The data must be collected with suitable randomization techniques. For a survey,
either two independent separate SRS’s from two pop’ns OR one SRS from one pop’n, which is then
divided into two groups. For an experiment, an appropriate randomized design.
2. The sample size has to be large enough for the sampling dist’n of pB  pA to be approximately normal.
Professional Conditions: How large depends on the value of the population proportion and what
technique we’re using.
a. Large-sample confidence interval. OK if the number of successes and the number of failures in
each of the two samples must be _______________.
b. Accurate confidence intervals – plus-four. If the confidence level is 90% or higher, then OK to
use this procedure if each sample size is ____________________, with any counts of successes
and failures.
c. Significance tests. OK if the number of successes and the number of failures in each of the two
samples is ___________________.
3. The sample size has to be large enough for the dist’n of pB  pA to be approximately normal.
Simple conditions:
OK if _________________________________________
________________________________________________________________________