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DobbinChapter6,6.1,6.2,6.3RevisedSep212010

DobbinChapter6,6.1,6.2,6.3RevisedSep212010 - Chapter 6...

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Chapter 6 Sections 6.1, 6.2, 6.3 Statistical inference is the next topic we will cover in this course. We have been preparing for this by 1) describing and analyzing data (graphs and plots, descriptive statistics), 2)discussing the ways to find and/or generate data (studies, samples, experiments) and 3) we have studied sampling distributions. We are now ready for statistical inference. The purpose of statistical inference is to draw conclusions about population parameters based on data which came from a random sample or from a randomized experiment. Data (statistics) are used to infer population parameters. We will learn in this chapter the two most prominent types of statistical inference, confidence intervals for estimating the value of a population mean and tests of significance which weigh the evidence for a claim concerning the population mean. Section 6.1 Estimating the population mean, µ, with a stated confidence: A confidence interval for the population mean, µ, includes a point estimate and a margin of error . The point estimate is a single statistic calculated from a random sample of units. For example, x , the sample mean, is a point estimate of μ , the population mean. However sample means fluctuate, so we need to adjust our point estimate by adding and subtracting a margin of error, thus creating a confidence interval. The following general formula may be used when the population sigma is known: ___ % Confidence Interval: x ± margin of error , where Margin of error = * z n σ Lecture 7, Chapter 6 Page 1
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For the Normal Distribution the following values for Z* apply: For a 90% Confidence Interval Z* = 1.645 For a 95% Confidence Interval Z* = 1.960 For a 99% Confidence Interval Z* = 2.576 Example: You want to estimate the population mean SAT Math score for the high school seniors in California. You give a test to a simple random sample of 500 high school seniors in CA. The mean score for your sample, X =461. The population standard deviation is known to be σ = 100. For the SAT Math scores, a 95 % confidence interval for µ would be : This Confidence Interval was calculated using a procedure which gives a “correct” interval (Contains the Population Mean ), 95% of the times it is used. Another example: 1. Suppose we wish to estimate μ , the population mean driving time between Lafayette and Indianapolis. We select a SRS of n = 25 drivers. The observed sample mean is 1 x = 1.10 hours. Let’s assume that we know the population standard deviation of X is 0.5 σ = hours. A 95 % confidence interval for µ would be: Lecture 7, Chapter 6 Page 2
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Again, the procedure gives a “correct” interval 95% of the times it is used. Suppose we selected another sample of 25 driving times and obtained an 2 x = 1.00 hours. If we calculate a 95% confidence interval for μ based on 2 x we get (0.804, 1.196) which is a different interval estimate.
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