Transcript Sampling Distributions

Chapter 7
Sampling Distributions
Sampling Distribution of the Mean
• Inferential statistics
– conclusions about population
• Distributions
– if you examined every possible sample, you
could put the results into a sampling
distribution.
• “Cereal Filling” is an excellent story about
inferential statistics.
Review Central Tendency
• Many measures.
• Arithmetic Mean is best, IF data or
population probability distribution is
normal or approximately normal.
• “Unbiased”
– a property of statistics
– if you take all possible sample means for a
given sample size, the average of the sample
means will equal µ.
Demo of “Unbiasedness”
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Table 7.1
RV = ?
Finite population for demo purposes
µ=? σ=?!
Say that you take a sample, n = 2, with
replacement. How many different x-bars
are there?
• If you average all of them, the average = μ.
• This demonstrates “unbiasedness.”
Unbiased Estimator
• Statistics are used to estimate parameters.
• Some statistics are better estimators than
others.
• We want unbiased estimators.
• X-bar is an unbiased estimator of µ.
Standard Error of the Mean
• Our estimator of µ is x-bar.
• X-bar changes from sample to sample,
that is, x-bar varies.
• The variation of x-bar is described by
the standard deviation of x-bar,
otherwise known as the standard error
of the mean.
Sampling from Normally
Distributed Populations
• If your population is Normally distributed
(ie. You are dealing with a RV that
conforms to a normal probability
distribution), with parameters µ and σ,
• and you are sampling with replacement,
• then the sampling distribution will be
normally distributed with mean= µ and
standard error = σ/n
Central Limit Theorem
• Extremely important.
• Given large enough sample sizes,
probability distribution of x-bar is
normal, regardless of probability
distribution of x.
7.3 Sampling Distribution of the Proportion
• Given a nominal random variable with two
values (e.g. favor, don’t favor, etc.), code
(or score) one of the values as a 1 and code
the other as a 0.
• By adding all of the codes (or scores) and
dividing by n, you can find the sample
proportion.
Population Proportion
• The sample proportion is an unbiased estimator
of the population proportion.
• The standard error of the proportion appears in
formula 7.7, page 239.
• The sampling distribution of the proportion is
binomial; however, it is well approximated by
the normal distribution if np and n(1-p) both
are at least 5.
• The appropriate z-score appears in formula 7.8,
page 240.
Why create a frame / draw a sample?
• less time consuming than census
• less costly than census
• less cumbersome than census—
easier, more practical
Types of Samples
• Figure 7.5
• Nonprobability
– Advantages
– Disadvantages
• Probability (best)
– Advantages
– Disadvantages
• Simple Random Sampling
Ethical Issues
• Purposefully excluding particular groups
or members from the “frame.”
• Knowingly using poor design.
• Leading questions.
• Influencing the respondent.
• Respondent falsifying answers.
• Incorrect generalization to the
population.