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Sampling Distributions

Sampling Distributions. Estimator of the population. 3.6 2.7 5.0 2.4 Population 2.9 3.1 3.0 3.4 2.6 4.1 4.7 2.0. 5.0 4.1 Sample 3.4 4.7 2.4. Take a Sample. Parameter, Statistic.

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Sampling Distributions

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  1. Sampling Distributions

  2. Estimator of the population 3.6 2.7 5.0 2.4 Population 2.9 3.1 3.0 3.4 2.6 4.1 4.7 2.0 5.0 4.1 Sample 3.4 4.7 2.4 Take a Sample

  3. Parameter, Statistic • A parameter is a number that describes the population. In statistical practice, the value of a parameter is not known, because we cannot examine the entire population. (Census) • A statistic is a number that can be computed from the sample data without making use of any unknown parameters. In practice, we often use a statistic to estimate an unknown parameter. (Sample)

  4. Once we decide to use the sample as an estimator, the question becomes how good of an estimator is the sample? • The values of the sample changes from sample to sample. • The behavior of the sample becomes a random variable. • Study the probability distribution of the sample. • Sampling distributions for the proportion and mean.

  5. Sampling Distribution • The sampling distribution of a statistic is the distributions of values taken by the statistic in all possible samples of the same size from the same population.

  6. Take a sample of size 2 without replacement from the population {1,2,3,4,5,6} yields the table below. What does it look like?

  7. Take a sample of size 2 with replacement from the population {1,2,3,4,5,6} yields the table below. Seen as population being infinite. But what does it look like? Similar to examples on pages 567-568

  8. http://onlinestatbook.com/rvls.html

  9. Sample Proportions Let (p-hat) be the proportion of the sample having some characteristic. ?????? • The mean of the sampling distribution of all is exactly p…..the proportion in the population. • The standard deviation of the sampling distribution of all is * Remind me to show the math to this after we do samples for the mean.*

  10. Rule of Thumb • Use the recipe for the standard deviation of p-hat only when the population is at least 10 times as large as the sample: N > 10n • Introduced during binomial distribution. np>10 and n(1-p)>10 approximates the normal distribution. • This is a conservative rule of thumb…..the farther away p is from 0.5, the larger n must be to approximate using the normal distribution.

  11. Globe Toss • Thumbs only • Right hand only • Both hands • Everyone together

  12. Practice • Suppose that a particular candidate for public office is in fact favored by 48% of all registered voters in the district. A polling organization takes a random sample of 500 voters and will use p-hat the sample proportion, to estimate p. What is the approximate probability that p-hat will be greater than 0.5, causing the polling organization to incorrectly predict the results of the upcoming election?

  13. More Practice and Judgments • Newsweek (Nov. 23, 1992) reported that 40% of all U.S. employees participate in “self-insurance” health plans. • In a random sample of 100 employees, what is the probability that at least half of those in the sample participate in such a plan? • Suppose you were told that at least 60 of the 100 employees in a sample from Georgia participated in such a plan. Would you think p = 40% is true for the state of Georgia?

  14. Pages 588 - 598 • 9.19 – 9.24

  15. Binomial vs Sampling Distribution • Twenty-five percent of the customers entering a grocery store between 5 p.m. and 7 p.m. use an express checkout. Consider 80 randomly selected customers, and let x denote the number among the 80 who use the express checkout. • What is P(x = 40)? • What is P(x < 15)?

  16. Binomial vs Sampling Distribution • Twenty-five percent of the customers entering a grocery store between 5 p.m. and 7 p.m. use an express checkout. Consider 80 randomly selected customers. • What is the probability that more than 30% use the checkout? • What is the probability that our sample will be less than 10%?

  17. http://onlinestatbook.com/rvls.html

  18. From these examples we see that the sample mean equals the population mean. The variance is not equal, but if population is finite and if the population is infinite or sampled with replacement

  19. Central Limit Theorem Start with a population with a given mean µ, a standard deviation σ, and any shape distribution whatsoever. Pick n sufficiently large (30) and take all samples of size n. Compute the mean of each of these samples. Then 1. The set of all sample means is approximately normally distributed. 2. The mean of the set of sample means equals µ, the mean of the population. 3. The standard deviation of the set of sample means is approximately equal to .

  20. Serum CholesterolHow will the distribution of the samples behave?

  21. Serum CholesterolHow will the distribution of the samples behave? • Randomly select 10 samples from the list. Are the conditions ok to assume the sample will be approximately normal?

  22. Serum CholesterolHow will the distribution of the samples behave? • Randomly select 30 samples from the list. Are the conditions ok to assume the sample will be approximately normal?

  23. Student’s t-distribution The t-distribution is the proper choice whenever the population standard deviation σ is unknown.

  24. Any Questions????? • Thank You for being patient.

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