1 / 16

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.

akina
Download Presentation

Sampling Distributions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related