Chapter 16
Inference in Practice
Conditions for inference in practice
Any confidence interval or significance test can be trusted only under specific
conditions.
Previously in Chapters 14 and 15, when making inferences about the population
mean, , we were assuming:
(1)
Our data (observations) are a simple random sample (SRS) of size n
from the population.
(2)
Observations come from a normal distribution with parameters  and
.
(3)
The population mean μ is unknown, but the population standard
deviation σ is known.
Then we were constructing confidence interval for the population mean  based
on _________ distribution.
Let’s look at each assumption closely:
Assumption (3): This assumption is rarely satisfied in practice, i.e., the standard
deviation  is unknown. Chapter 18 will discuss how to handle this situation.
Assumption (2): The inference for the population mean  based on normal
distribution holds approximately for large samples even if the assumption (2) is
not satisfied. Why?
Assumption (1): The most important for any inference procedure is that the data
come from a process to which the laws of probability apply.
Caution: If your data don’t come from a random sample or a randomized
comparative experiment, your conclusion may be challenged.
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Cautions about confidence intervals
 The most important caution about confidence intervals in general is a
consequence of the use of a sampling distribution.
 The margin of error in a confidence interval ignores everything except the
sample-to-sample variation due to choosing the sampling randomly. The
margin of error in a confidence interval covers only random sampling errors.
 Practical difficulties such as undercoverage and nonresponse are often more
serious than random sampling error. The margin of error does not take such
difficulties into account.
Cautions about Significant Tests
 There is no sharp border between “significant” and “not significant”, only
increasingly strong evidence as the P-value decreases. There is no practical
distinction between 0.049 and 0.051. It makes no sense to treat P-value <
0.05 as a universal rule for what is significant.
 How important an effect is depends on the size of the effect as well as on its
statistical significance. Significance depends both on the size of the effect
we observe and on the size of the sample.
 Statistical significance does not tell us whether an effect is large enough to
be important, i.e., statistical significance is not the same thing as
practical significance.
Planning studies: sample size for confidence intervals
Recall: A level C confidence interval for  when population standard deviation σ is
estimate ± margin of error =
Thus, we have 3 elements affecting the width of the confidence interval:
1)
2)
3)
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We saw that we can have high degree of confidence as well as small margin of
error by
Usually researchers will have a desired confidence level and margin of error they
want to attain.
One aspect of designing any study is to decide the number of observations
needed.
Let m represent the desired margin of error. Recall the formula of margin of error:
Solving for n we get:
*****Always round up to the next higher whole number!!*****
Ex: Suppose PGSA (Poor Graduate Students Association) at the Texas state wants
to estimate the mean monthly income of SMU graduate students within $100 with
95% confidence. How many students should PGSA sample? Assume that the
standard deviation of incomes of SMU graduate students is $421.
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