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STA291

STA291. Statistical Methods Lecture 16. Lecture 15 Review. Assume that a school district has 10,000 6th graders. In this district, the average weight of a 6th grader is 80 pounds, with a standard deviation of 20 pounds. Suppose you draw a random sample of 50 students .

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STA291

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  1. STA291 Statistical Methods Lecture 16

  2. Lecture 15 Review • Assume that a school district has 10,000 6th graders. In this district, the average weight of a 6th grader is 80 pounds, with a standard deviation of 20 pounds. Suppose you draw a random sample of 50 students. • A)What is the probability that the average weight will be less than 75 pounds?

  3. So far … We know a little bit about: • Collecting data, and on what scale to do it • How to describe it, graphically and numerically • What to describe about it: • center and spread, if quantitative • proportion in each category, if qualitative • Probability, including: • basic rules • random variables, including discrete (binomial) and continuous (normal) examples • Sampling distributions

  4. Central Limit Theorem Thanks to the CLT … We know is approximately standard normal (for sufficiently large n, even if the original distribution is discrete, or skewed). Ditto

  5. Two primary types of statistical inference • Estimation • using information from the sample (statistic’s value, for example) to make an informed (mathematically justifiable) guess about a characteristic of the population • Hypothesis, or SignificanceTesting • using information from the sample to make an informed decision about some aspect of the population

  6. Statistical Inference: Estimation • Inferential statistical methods provide predictions about characteristics of a population, based on information in a sample from that population • For quantitative variables, we usually estimate the population mean (for example, mean household income) • For qualitative variables, we usually estimate population proportions (for example, proportion of people voting for candidate A)

  7. Two Types of Estimators • Point Estimate • A single number that is the best guess for the parameter • For example, the sample mean is usually a good guess for the population mean • Interval Estimate • A range of numbers around the point estimate • To give an idea about the precision of the estimator • For example, “the proportion of people voting for A is between 67% and 73%”

  8. A Good Estimator is … • unbiased: Centered around the true parameter • consistent: Gets closer to the true parameter as the sample size gets larger • efficient: Has a standard error that is as small as possible

  9. Unbiased • Already have two examples of unbiased estimators … • Expected Value of the ’s: m—that makes an unbiased estimator of m. • Expected Value of the ’s: p—that makes an unbiased estimator of p. • Third example:

  10. Efficiency • An estimator is efficient if its standard error is small compared to other estimators • Such an estimator has high precision • A good estimator has small standard error and small bias (or no bias at all)

  11. Bias versus Efficiency A B C D

  12. Looking back • Recap of descriptive statistics, probability work • Inferential statistics: • estimation • hypothesis testing • Considerations in estimation • bias • consistency • efficiency

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