Lecture6.chapt6 - Lecture 6 Chapter 6 Statistical inference...

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Lecture 6, Chapter 6 Statistical inference is the next topic we will cover in this course. We have been preparing for this by describing and analyzing data (graphs and plots, descriptive statistics); discussing the means to find and/or generate data (studies, samples, experiments); and we have defined sampling distributions. We are now ready for statistical inference. The purpose of statistical inference is to draw conclusions from data. It adds to the graphing and analyzing because we substantiate our conclusions by probability calculations. Formal inference is based on the long-run regular behavior that probability describes. When you use statistical inference, you determine some truth about a population parameter based on data taken from a random sample or a randomized experiment. We will learn in this chapter the two most prominent types of formal statistical inference: 1) Confidence intervals for estimating the value of a population parameter and 2) Tests of significance which assess the evidence for a claim. Section 6.1 Estimating with confidence: A confidence interval for a population parameter includes a point estimate and a margin of error. The point estimate is single statistic calculated from a random sample of units. For example, x , the sample mean, is a point estimate of μ , the population mean. Point estimates give us very little information. So we add to our point estimate a margin of error making up a confidence interval. Lecture 6, Chapter 6 Page 1
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Example: 1. You want to estimate the mean SAT Math score for high school seniors in California. At considerable effort and expense, you give the test to a simple random sample of 500 high school seniors. The mean score for your sample is x = 461 points. The standard deviation of the SAT Math test is a known 100 σ = points. Questions: 1. Can we include a measure of the precision associated with the point estimate? 2. Can we include a measure of our confidence in our results? Answer: Yes, we can construct a confidence interval for μ . A confidence interval is calculated from the sample data and it represents an interval estimate of the population parameter. A confidence interval includes: 1. an interval computed from the sample. (The interval is a measure of the variability of our point estimate). 2. a confidence level. (This confidence level measures the confidence that our inference is correct). In this lesson we want to find a confidence interval for a population mean, μ . Given a SRS of n units from the population, we can calculate the sample mean x . This single, calculated value, x , represents an outcome in the sampling distribution of x . Lecture 6, Chapter 6 Page 2
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To obtain a 95% confidence interval for μ based on this single observed value, we treat our observed outcome, x , as though it is the true mean of the sampling distribution of x . We then construct our interval about this observed value x .
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