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Unformatted text preview: Chapter 6: Confidence Intervals and Hypothesis Testing Readings: Sections 6.16.3 September 26, 2009 1 Introduction • We have examined data and arrived at conclusions many times previously. • Formal inference adds on emphasis on supporting our conclusions by probability calcula tions. • Two most common types of formal statistical inference: – Confidence intervals : used when we want to estimate a population parameter. – Significance tests : used when we want to assess the evidence provided by the data in favor of some claim about the population (yes/no questions about the population). 2 Confidence Intervals • Confidence intervals allow us to estimate population parameters (for example, popu lation mean μ , or population proportion p , etc) using a range of values (i.e., an interval). • Any confidence interval looks like: estimate ± margin of error . • The level C Confidence interval for a population mean μ : ¯ x ± z * σ √ n , where σ is the population standard deviation, n is the sample size, and z * is the value on the standard normal curve with area C between z * and z * . • Several most commonly used z * ’s: 1 C 90% 95% 99% z * 1.645 1.960 2.576 Example 1 : Suppose that the seed weight of the princess bean Phaseotus vulgaris has an unknown mean and standard deviation 100mg. A random sample of 40 seeds were taken and the average weight is 450mg. a. Give a 90% confidence interval for the mean seed weight of the princess beans. Interpretation of CIs We are 90% confident that the mean seed weight of the princess bean is between and . – This means: the method that we used to construct the 90% confidence interval captures the true mean μ 90% of the time. – Here is a picture of 50 different 90% CIs for μ , each based on a sample of size 40... μ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2 • Note that four (or 8%) of the intervals failed to capture μ . • If we repeated this many more times, 10% of the intervals would fail to capture μ and 90% of them would cover μ . Example 1 (cont’d): b. Is it true that weights of 90% of the seeds lie in the interval you found in part a)? No. The confidence interval is for population mean, not for individual population members. If we take many random samples, each consisting 40 seeds, and make a confidence interval for each sample, 90% of these confidence intervals will contain the true population mean weight. Margin of error = z * σ √ n • If you increase the sample size n , you decrease the margin of error (hence shorter CI)....
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 Spring '08
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 Statistics, Null hypothesis, Statistical hypothesis testing

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