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ioe265f11-Lec17 - Lec17 Confidence Intervals for Mean and...

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Lec17 - Confidence Intervals for Mean and Variance IOE 265 F11 1 1 Statistical Confidence Intervals: Mean and Variance of a Normal Population Distribution 2 Topics I. Properties of t Distributions II. One Sample Intervals III. Confidence Intervals for Variance Using the Chi-Square Distribution

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Lec17 - Confidence Intervals for Mean and Variance IOE 265 F11 2 3 I. Properties of t Distributions If n is small, we may not rely on the CLT to identify CI. However, we may derive CI based on assumptions about the distribution of data. If we assume normality, the t distribution provides a useful tool to evaluate means with smaller sample sizes. In other words, X 1 .. X n are iid normal with unknown and . 4 Family of t-distributions Theorem: When X-bar is the mean of a random sample of size n from a normal distribution with mean , then we can define: T = T is a random variable that follows a t distribution with n-1 degrees of freedom (df). n S X /
Lec17 - Confidence Intervals for Mean and Variance IOE 265 F11 3 5 Properties of the t-Distribution Let t be the density curve for df, then: Each t is bell-shaped, centered at zero, As increases, the spread of the curve decreases -- if df = infinity then t curve approaches z curve (standard normal distribution) Z curve 6 t critical values The area under the curve to the right of the t critical value is .

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ioe265f11-Lec17 - Lec17 Confidence Intervals for Mean and...

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