Lecture12_ch7

# Lecture12_ch7 - Lecture 12 Stat 350 7.2 Large Sample...

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Lecture 12 Stat 350 7.2 Large Sample Confidence Intervals for a population mean ( μ )

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Point Estimate The statistic is a point estimate for population mean μ unbiased and consistent Example: Suppose we obtain a SRS of 100 plots of corn with yields (in bushels), and the mean yield is 123.8 Shortcoming: provides no information about the precision and reliability of estimation for μ x x x 123.8
Confidence Interval CI is an entire interval of plausible values Example: Suppose we observe 100 plots of corn with yields (in bushels) Sample Mean = 123.8 Sample Standard Deviation = 12.3 What are the plausible values for (population) mean yield of this variety of corn?

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Sampling Distribution n=100 (large), CLT => σ is unknown when n is large, σ s s is the sample standard deviation So, ) , ( ~ n N X s X s n 12.3 100 1.23
Sampling Distribution n=100 (large), CLT => P(-1.96 <Z<1.96) = 0.95 or P(-1.96 < <1.96) = 0.95 X N ( ,1.23) Z X   1.23 X   1.23

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Sampling Distribution P(-1.96 < <1.96) = 0.95 95% of the time, X   1.23 P ( 1.96*1.23 X   1.96 *1.23) 0.95  2.46 X   P ( 2.46) 0.95

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2 σ rule 95% of time: In general, 95% of time sample mean is within 2 standard deviations of population mean  2.46 X  
Put differently: 95% of time The random interval covers the unknown (but nonrandom) population parameter μ 95% of time. Our confidence is 95%. We need to be extremely careful when interpreting this result. X 2.46   

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Example 1 (cont’d) Suppose we observe 100 plots of corn with yields (in bushels) Sample Mean = 123.8 Sample Standard Deviation = 12.3 What are the plausible values for (population) mean yield of this variety of corn?
Example 1 (cont) : CI is an entire interval of plausible values This particular CI may contain µ or not… However, such a systematic method gives intervals covering the population mean μ in 95% of cases. Each interval is NOT 95% correct. Each interval is 100% correct or 100% wrong. It’s the method that is correct 95% of the time x 123.8 X 2.46 (121.34, 126.26)

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Confidence Intervals (CIs): Typically: estimate ± margin of error Always use an interval of the form (a, b) Confidence level (C) gives the probability that such random interval(s) will cover the true value of the parameter. It does not give us the probability that our parameter is inside the interval. In Example 1: C = 0.95, what Z gives us the middle 95%? (Look up on table) What about for other confidence levels? 90%? 99%?

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Confidence Level: Confidence level (C) gives the probability that such random intervals will cover the true value of the parameter. It does not give us the probability that our parameter is inside the interval. In Example 1: C = 0.95, what Z gives us the middle 95%? z=1.96 What about for other confidence levels?
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Lecture12_ch7 - Lecture 12 Stat 350 7.2 Large Sample...

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