Download Chapter 15 Thinking about Inference Conditions for inference

Chapter 15
Thinking about Inference
Conditions for inference in practice
Any confidence interval or significance test can be trusted only under specific
conditions.
Previously in Chapter 14, when making inferences about the population mean, μ,
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 μ and
σ.
(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 μ based
on _________ distribution.
Let’s look at each assumption closely:
Assumption (3): This assumption is rarely satisfied in practice, i.e., the standard
deviation σ is unknown. Chapter 17 will discuss how to handle this situation.
Assumption (2): The inference for the population mean μ 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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Behavior of Confidence Intervals
Recall: A level C confidence interval for μ when population standard deviation σ is
estimate ± margin of error =
What happens to the margin of error if we increase the confidence level C? Does
it increase, decrease or stay the same? (Hint: what happens to the value of z*?)
How does this affect the width of the resulting confidence interval?
Note the tradeoff: We would like to have a smaller margin of error (narrower
interval) as well as high confidence but…
Q: What could we do to get a narrower interval (smaller margin of error) without
lowering confidence?
E.g. Refer to the “Beetle cars” example in Chapter 14, the 99% confidence
interval for μ was found to be _________________ with a sample of size 49. If
instead we had taken a sample of size 100 cars and suppose their mean CO2
emission was 1.5 grams, then
How does the size of σ affect the margin of error?
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Thus we have 3 ways of reducing the width of the confidence interval:
1)
2)
3)
Planning studies: sample size for confidence intervals
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.
So 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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