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Graded Homework Assignment:

Graded Homework Assignment:. 5.21; 5.24; 5.46; 5.62 Due in Lab Sections Oct 12-13. Last Time: i.e., the Class before Exam 1:. The Binomial Distribution. Population. Repeat simple experiment n many times independently. Binomial Random Variable. X : number of successes

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Graded Homework Assignment:

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  1. Graded Homework Assignment: 5.21; 5.24; 5.46; 5.62 Due in Lab Sections Oct 12-13

  2. Last Time:i.e., the Class before Exam 1: The Binomial Distribution

  3. Population Repeat simple experiment n many times independently.

  4. Binomial Random Variable X: number of successes in nmany independent repetitions of an experiment, each repetition having a probability p of success

  5. Binomial Distribution Formula: TABLE C Pages T-6 to T-10 in the book

  6. BINOMIAL COEFFICIENTS:

  7. PROOF:

  8. The Sample Proportion Unbiased Estimator

  9. Normal Approximation of/to the Binomial

  10. Normal Approximation: Let’s check it out! Standardizing:

  11. Normal Approximation: Let’s check it out!

  12. Normal Approximation: Let’s check it out! Are we stuck with a bad approximation??

  13. Today: The Sampling Distribution Of the Sample Mean

  14. Useful Fact for Independent Normal Random Variables:

  15. Important Distinction: UNKNOWNUNCERTAIN Population Mean Sample Mean This is a number This is a random variable Today, we pretend that we “know” the truth (the population parameters) and see what should happen when we collect some data. That assumption will be dropped when we move to estimation. In estimation we do not claim to know the truth, instead we try to infer the truth from data.

  16. Population with some characteristic X X has some (possibly unknown) distribution X has an expected value and variance (which are possibly unknown) Taking a sample of size n: Pick a member of the population X1 Pick a member of the population X2 etc. Pick a member of the population Xn Each Xi is uncertain thus a random variable

  17. A random sample of your age

  18. Another random sample of your age

  19. The sample mean is a random variable

  20. What we did is unusual: We collected a random sample of n=5 twice. Normally, one would just collect a single random sample of size n.

  21. The sample mean is a random variable As we already know: This kind of random variable is called a Sample Statistic

  22. Expected value of `X On average (over many samples) sample mean is population mean The sample mean is an unbiased estimator of the population mean

  23. On average (over many samples) sample mean is population mean How close is the sample mean to the true population mean?

  24. Variance of `X The larger the sample size (n), the smaller the deviation of`X from X

  25. Summary about `X

  26. Central Limit Theorem (CLT) Often, n  25 is enough to get a good approximation. If population has a normal distribution, n=1 is large enough. (Why?)

  27. Note: WHY?

  28. Note: WHY?

  29. Another sample statistic:the sample variance careful!

  30. On average, the sample variance is a good approximation of the population variance It is for this purpose that we use n-1 in the denominator of the sample variance. It would, otherwise, not be an unbiased estimator.

  31. Sampling (proportions) Example: Want to find out % of people supporting Mr. Notax for President

  32. A sample survey of n voters A randomly sampled voter has probability p of being a supporter

  33. A wild guess at p

  34. Summary of results (sample proportion)(see also last class)

  35. Excursion: Sampling with or without replacement?

  36. Finite population correction (FPC) FPC can be ignored for very large N (marketing) FPC matters in other domains like quality control

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