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Statistics 3502/6304 Prof. Eric A. Suess Chapter 5 Inferences about Population Central Values β’ Estimation and Confidence Intervals β’ Hypothesis Testing Estimation of the Population Mean β’ Point estimate of the population mean µ is the sample mean π¦. β’ The sample mean is the βbest guessβ at the value of the unknown population mean. β’ How accurately does the sample mean estimate the population mean? β’ Anyone play darts? Estimation of the Population Mean β’ If we assume the population is Normally distributed with mean µ standard deviation Ο, then the Central Limit Theorem tells us that the distribution of the sample mean π¦ is Normally distributed with mean Ο µ standard deviation (Called the standard error.) π β’ In repeated sampling, about 95% of the sample means π¦ will be within approximately 2 standard errors of the mean µ (More precisely 1.96 standard errors.) β’ Draw a picture. Estimation of the Population Mean β’ The Central Limit Theorem gives. π(π β 1.96 π π β€ π¦ β€ π + 1.96 π ) π = .95 β’ Draw a picture. β’ Or the 95% Confidence Interval for µ, when Ο known. π(π¦ β 1.96 π π β€ π β€ π¦ + 1.96 β’ Which can be written in general as π¦ ± π§πΌ 2 π π π ) π = .95 Estimation of the Population Mean β’ See Table 5.2 on page 229 for a z-values for different Confidence Levels. β’ When Ο is unknown, it will be estimated using the sample standard deviation s. See Page 250. Later. Hypothesis Testing of the Population Mean Hypothesis Testing 1. Research Hypothesis (also called the Alternative Hypothesis) π»π : 2. Null Hypothesis, π»0 : 3. Test Statistic 4. Rejection Region 5. Check assumptions and draw conclusions. Hypothesis Testing of the Population Mean β’ Examples: β’ Blood Testing β’ Jury trial β’ Errors: β’ Type I Error, False Positive, Convict an innocent prisoner β’ Type II Error, False Negative, Release a guilty prisoner β’ See Table 5.3 on page 234 Hypothesis Testing of the Population Mean β’ One-tailed Test β’ Example 5.5 page 235, 236 β’ Two-tailed Test β’ Example 5.6 page 237 β’ Summary of Hypothesis Test for µ, when Ο known. Page 238 Hypothesis Testing of the Population Mean β’ Significance Level of the Test, Ξ± β’ The significance level is usually set at 5%, limiting the Type I Error to 5%. β’ The p-value is often computed using computer software and compared to the significance level to determine if the Null Hypothesis is Rejected β’ Reject π»0 : if the p-value β€ πΌ = 0.05 Hypothesis Testing of the Population Mean β’ The p-value is defined to be the probability of obtaining a value of the test statistics that is as likely or more likely to Reject π»0 : as the actual observed value of the test statistic, assuming that the Null Hypothesis is true. β’ If the p-value is small then we will Reject π»0 : Hypothesis Testing of the Population Mean β’ The Test Statistic when Ο is know is: π¦βπ π§=π π β’ The Test Statistic when Ο is unknown is: π¦βπ π‘= π π Hypothesis Testing of the Population Mean β’ The z test statistic uses the z-table that we know. See Table 1 at the start of the book, or Table 1 at the end of the book at the bottom. β’ The t test statistic uses the t-table that we know. See Table 2 at the back of the book. Example β’ Next time we will look at a number of examples of Confidence Intervals and Hypothesis Testing.