Chapter 10:
Introduction to Statistical Inference
• Recall that the purpose of descriptive statistics is to
make the collected data more easily comprehensible
and understandable.
• Some tools we’ve examined in descriptive statistics
include frequency distributions, measures of central
tendency, and measure of dispersion.
• Because it is not always possible to address every
member of the population, we take samples.
• The statistical question that needs to be answered is
whether or not the characteristics observed in the
sample are likely to reflect the true characteristics of
the larger population from which the sample was
taken.
• Inferential statistics provide us with the tools we
need to answer this question.
• In inferential statistics, the goal is to make
statements about the characteristics of a
population based on what we have learned
from the sample data.
• Inferential statistics has two broad
applications: estimation and hypothesis
testing.
• Estimation uses information contained in a
sample to make a “guess” of the population
value.
• Hypothesis testing determines whether or not
a hypothesized value or relationship in the
population is likely to be true.
• Recall that in a random sample, every member
of the population has an equal chance of
being selected.
• Also recall that a descriptive measure
calculated from a sample is statistic.
• We use statistics as a way to estimate a
population parameter.
• Just how accurately does a sample statistic
estimate a population parameter???
• Typically we usually draw only one sample from a
population and use that sample statistic
calculated as an estimate of the population
parameter.
• If we drew a different sample, our estimate for
the population would be slightly different.
• So if we calculated a mean, we would end up with
a slightly different mean.
• If we took 6 samples from the same population,
we would likely have 6 different means.
• A sampling distribution is the distribution of
numbers, obtained by calculating a sample
statistic, for all possible samples of a given size
drawn from the same population.
• Let’s say we did have a population and pulled
six samples. The means of those six samples
are 20, 23, 24, 21, 22, and 25. Which one
would you report?
• You might want to report the mean of those
sample means.
• A sample statistic will not always equal the
population parameter (in most cases it won’t).
• Random sampling error is the measure of the
extent to which the sample statistic differs
from the population parameter, due to
random chance.
Parameter = statistic + random sampling error
• Note: Random sampling error and standard error
are interchangeable terms.
Sample Size and Standard Error
• As sample size increases, standard error
decreases.
The Central Limit Theorem
• Just how large does n have to be?
• The rule of thumb is that n has to be 30 or more.
• Once we know we are dealing with a Normal
distribution, we can utilize the Empirical Rule and
the standard Normal table to help us attain
information about our population.
Back to Z-Scores
• When dealing with a sample mean, calculate
its z-score by:
• When the population standard deviation is
unknown:
• These z-scores will measure how many standard
deviations the sample mean deviates from the
population mean.
Example: Calculate and interpret the z-score for
the following data.
Interpretation: The sample mean of 50 is 2 standard deviations
above the population mean.
Example: Calculate and interpret the z-score for
the following data.
Interpretation: The sample mean of 85 is 1.5
standard deviations below the population mean.
.6368-.2451=.3917
.9582