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7.1b h.w: pg 429: 9, 11, 13, 17 - 20

Describing Sampling Distributions Target Goal: I can DESCRIBE the relationship between sample size and the variability of an estimator. 7.1b h.w: pg 429: 9, 11, 13, 17 - 20. Bias and Variability. Bias means that our aim is off and we consistently miss in the same direction.

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7.1b h.w: pg 429: 9, 11, 13, 17 - 20

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  1. Describing Sampling DistributionsTarget Goal:I can DESCRIBE the relationship between sample size and the variability of an estimator. 7.1b h.w:pg 429: 9, 11, 13, 17 - 20

  2. Bias and Variability Bias means that our aim is off and we consistently miss in the same direction. High variability means that repeated shots are widely scattered on the target and repeated samples do not give similar results.

  3. Variability of a Statistic • A statistic can be unbiasedand still have highvariability. • To avoid this, increase the size of the sample. • Larger samples give smaller variability.

  4. n=100 n=1000 Larger samples have a clear advantage over smaller samples. They are much more likely to produce an estimate close to the true value of the parameter.

  5. Pg. 425

  6. Ex: Survivor Fans • Survivor II: Suppose that the true proportion of adults who watched is p = 0.37. • Nielson took a 1000 SRS’s of n = 100 from the population.

  7. The values have a large spread. The range is from 0.22 to 0.54. • The center is close to 0.37 • There are no outliers or other important deviations from the overall pattern. • Samples of n=100 people do not give a trustworthy estimate of the population proportion. (too much variability)

  8. Next, this simulation takes a 1000 SRS’s of size 1000 with the same true proportion p = 0.37. The center is again close to 0.37 But, the spread is much less: from 0.321 to 0.421.

  9. Almost all samples of 1000 gave a close to the population parameter p = 0.37. • Peak at 0.37 verifies this. Rescale to see the shape better

  10. Let’s generate 500 random samples of n = 10, n = 20, and n = 30. The density histograms below display the resulting 500 x for each of the given sample sizes. What do you notice about the standard deviation of these histograms? What do you notice about the means of these histograms? What do you notice about the shape of these histograms?

  11. Exercise: Bias and Variability Which of these sampling distributions displays large or small bias(sample mean close to actual mean) and large or small variability (spread) ? Large bias and variability Small bias and variability

  12. Small bias and large variability Large bias, small variability

  13. Exercise: IRS Audits • The Internal Revenue Service Plans to examine an SRS of individual federal income tax returns from each state. • One variable of interest is the proportion of returns claiming itemized deductions. • The total number of tax returns in each state varies from almost 14 million in California to fewer than 210,000 in Wyoming.

  14. Discuss a) and b) with a partner and report back. Hint: check “conditions” (pop > 10n) • Will the sampling variability of the sample proportion change from state to state if an SRS of n = 2000 tax returns is selected in each state? Explain your answer. Since the smallest number of tax returns (i.e., the smallest population)is still more than 10 times the sample size, (210,000 > 2000(10)), thevariability will be (approximately)the same for all states.

  15. Will the sampling variability of the sample proportion change from state to state if an SRS 1% of all tax returns is selected in each state? Explain your answer. Yes, it will change. The sample size from Wyoming will be about the same size, but thesample size inCaliforniawill be considerablylarger, and therefore thevariability will decrease.

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