chapter 7 sampling distributions. sampling distribution of the mean inferential statistics...
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Chapter 7
Sampling Distributions
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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.
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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.
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Demo of “Unbiasedness”• 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.”
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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.
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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.
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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
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Central Limit Theorem
• Extremely important.
• Given large enough sample sizes, probability distribution of x-bar is normal, regardless of probability distribution of x.
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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.
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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.
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Why create a frame / draw a sample?
• less time consuming than census
• less costly than census
• less cumbersome than census—easier, more practical
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Types of Samples
• Figure 7.5
• Nonprobability– Advantages– Disadvantages
• Probability (best)– Advantages– Disadvantages
• Simple Random Sampling
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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.