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Sampling Distributions

Sampling Distributions. Sample Statistic. Population Parameter. A Change in Emphasis. EDA … getting a feel for the data Inferential Statistics … use statistics to answer questions about parameters. The Key to Understanding Inference.

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Sampling Distributions

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  1. Sampling Distributions

  2. Sample Statistic Population Parameter A Change in Emphasis • EDA … getting a feel for the data • Inferential Statistics … use statistics to answer questions about parameters Sampling Distributions

  3. The Key to Understanding Inference • How often the result from one sample would be observed in all possible samples. • Must imagine repeatedly sampling, though only one sample will ever be taken. • Draw schematic aid. Sampling Distributions

  4. Examine Simulator in R cltSim() • How does the sampling distribution change if … • Sample size changes • The population changes (change alpha or beta) Sampling Distributions

  5. The sampling distribution depends on the … • population • sample size • variable • statistic If any one of these items changes then the sampling distribution changes. Sampling Distributions

  6. Describing Sampling Distributions • Shape – most symmetric, some right-skewed • Outliers – generally none • Center – need to discuss more (next) • Dispersion – need to discuss more (later) Sampling Distributions

  7. Center of Sampling Distribs • The center is always measured by the mean • Unbiased statistic -- the mean of the sampling distribution equals the corresponding parameter • All statistics in this class are unbiased! Thus, • The mean of the sample means is equal to ___ • The mean of the sample std. dev. is equal to ___ • The mean of the sample medians is equal to ___ m s Popn Median Sampling Distributions

  8. Unbiased • Thus the mean of the sample means equals m. m • “on average, the statistic equals the parameter” • Any given value of the statistic likely does not equal the parameter Sampling Distributions

  9. A “Good” Question • Assume that I know that m = 100 • In arandom sample of 500 individuals, I compute a sample mean of 99.1 • Is the sample mean a biased estimator of m? Sampling Distributions

  10. Dispersion of Sampling Distribs • Always use the standard deviation • However, it is called standard error of the statistic • Why SE? • SD is variability among individuals • SE is variability among statistics Sampling Distributions

  11. Mean AGE 25 35 45 55 SE decreases as n increases n = 40 n = 60 n = 80 SE of mean = 4.109 SE of mean = 3.001 SE of mean = 2.699 • Why? • Larger samples tend to be more alike (i.e., less sampling variability) Sampling Distributions

  12. because is unbiased Central Limit Theorem • Sampling Distribution of • Is approximately normal, as long as • n > 30, OR • n > 15 ANDpopulation is not strongly skewed, OR • population is approximately normal m • The mean is __ Why? • The SE is Central Limit Theorem

  13. m m+se m- se m+2 se m-3 se m-2 se m+3 se Putting All of This Together • if n >30, OR • if n > 15 AND population not strongly skewed, OR • if population is approximately normal Central Limit Theorem

  14. Explore with cltSim() • What happens to shape with increasing n? • What happens to center with increasing n? • What happens to dispersionwith increasing n? Central Limit Theorem

  15. Probability Question About an “individual” • Use population distribution • Told if the distribution is normal • Use s About a “statistic” • Use sampling distribution • CLT tells if normal • Use SE If the relevant distribution is NOT normal then the probability can NOT be computed. Sampling Distributions

  16. A company is investigating the number of sick days used by each of its employees. Assume that the population is right-skewed with a mean of 7 and a standard deviation of 2 days. • What symbol would you put on the 7? the 2? • Which distribution would be used to answer the following? • What is the probability of observing a mean of 6.4 days or less in a random sample of 25 workers? • What is the probability of observing a mean of 6.4 days or less in a random sample of 36 workers • What is the probability of observing a worker using fewer than 5 sick days? • Can each question be answered? Sampling Distributions Sampling Distributions Slide #16

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