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Chapter 7

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.

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Chapter 7

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  1. Chapter 7 Sampling Distributions

  2. 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.

  3. 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.

  4. 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.”

  5. 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.

  6. 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.

  7. 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

  8. Central Limit Theorem • Extremely important. • Given large enough sample sizes, probability distribution of x-bar is normal, regardless of probability distribution of x.

  9. 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.

  10. 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.

  11. Why create a frame / draw a sample? • less time consuming than census • less costly than census • less cumbersome than census—easier, more practical

  12. Types of Samples • Figure 7.5 • Nonprobability • Advantages • Disadvantages • Probability (best) • Advantages • Disadvantages • Simple Random Sampling

  13. 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.

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