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Chapter 8 Confidence Intervals McGraw-Hill/Irwin Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Confidence Intervals 8.1 8.2 8.3 8.4 8.5 8.6 z-Based Confidence Intervals for a Population Mean: σ Known t-Based Confidence Intervals for a Population Mean: σ Unknown Sample Size Determination Confidence Intervals for a Population Proportion Confidence Intervals for Parameters of Finite Populations (Optional) A Comparison of Confidence Intervals and Tolerance Intervals (Optional) 8-2 LO 1: Calculate and interpret a z-based confidence interval for a population mean when is known. 8.1 z-Based Confidence Intervals for a Mean: s Known Confidence interval for a population mean is an interval constructed around the sample mean so we are reasonable sure that it contains the population mean Any confidence interval is based on a confidence level 8-3 LO1 General Confidence Interval In general, the probability is 1 – a that the population mean m is contained in the interval x za 2 s x x za 2 s n The normal point za/2 gives a right hand tail area under the standard normal curve equal to a/2 The normal point - za/2 gives a left hand tail area under the standard normal curve equal to a/2 The area under the standard normal curve between -za/2 and za/2 is 1 – a 8-4 LO1 General Confidence Interval Continued If a population has standard deviation σ (known), and if the population is normal or if sample size is large (n 30), then … … a (1-a)100% confidence interval for m is x za x za n s 2 s 2 n , x za 2 s n 8-5 LO 2: Describe the properties of the t distribution and use a t table. 8.2 t-Based Confidence Intervals for a Mean: σ Unknown If σ is unknown (which is usually the case), we can construct a confidence interval for m based on the sampling distribution of t x m s n If the population is normal, then for any sample size n, this sampling distribution is called the t distribution 8-6 LO2 The t Distribution The curve of the t distribution is similar to that of the standard normal curve Symmetrical and bell-shaped The t distribution is more spread out than the standard normal distribution The spread of the t is given by the number of degrees of freedom Denoted by df For a sample of size n, there are one fewer degrees of freedom, that is, df = n – 1 8-7 LO2 Degrees of Freedom and the t-Distribution As the number of degrees of freedom increases, the spread of the t distribution decreases and the t curve approaches the standard normal curve 8-8 LO2 Using the t Distribution Table Rows correspond to the different values of df Columns correspond to different values of a See Table 8.3, Tables A.4 and A.20 in Appendix A and the table on the inside cover Table 8.3 and A.4 gives t points for df 1 to 30, then for df = 40, 60, 120 and ∞ On the row for ∞, the t points are the z points Table A.20 gives t points for df from 1 to 100 For df greater than 100, t points can be approximated by the corresponding z points on the bottom row for df = ∞ Always look at the accompanying figure for guidance on how to use the table 8-9 LO 3: Calculate and interpret a t-based confidence interval for a population mean when σ is unknown. t-Based Confidence Intervals for a Mean: σ Unknown If the sampled population is normally distributed with mean m, then a (1a)100% confidence interval for m is x ta s 2 n ta/2 is the t point giving a right-hand tail area of a/2 under the t curve having n1 degrees of freedom 8-10 LO 4: Determine the appropriate sample size when estimating a population mean. 8.3 Sample Size Determination (z) If σ is known, then a sample of size za 2s n B 2 so that x is within B units of m, with 100(1a)% confidence 8-11 LO 5: Calculate and interpret a large sample confidence interval for a population proportion. 8.4 Confidence Intervals for a Population Proportion If the sample size n is large, then a (1a)100% confidence interval for ρ is p̂ z a 2 p̂1 p̂ n Here, n should be considered large if both n · p̂ ≥ 5 n · (1 – p̂) ≥ 5 8-12 LO 6: Determine the appropriate sample size when estimating a population proportion. Determining Sample Size for Confidence Interval for ρ A sample size given by the formula… za 2 n p1 p B 2 will yield an estimate p̂, precisely within B units of ρ, with 100(1-a)% confidence. Note that the formula requires a preliminary estimate of p. The conservative value of p=0.5 is generally used when there is no prior information on p. 8-13 LO 7: Find and interpret confidence intervals for parameters of finite populations (optional). 8.5 Confidence Intervals for Parameters of Finite Populations (Optional) For a large (n ≥ 30) random sample of measurements selected without replacement from a population of size N, a (1- a)100% confidence interval for μ is x za 2 s n N n N A (1- a)100% confidence interval for the population total is found by multiplying the lower and upper limits of the corresponding interval for μ by N 8-14 LO 8: Distinguish between confidence intervals and tolerance intervals (optional). Tolerance interval: contains specified percentage of individual population measurements 8.6 A Comparison of Confidence Intervals and Tolerance Intervals Often 68.26%, 95.44%, 99.73% Confidence interval: interval containing the population mean μ, and the confidence level expresses how sure we are that this interval contains μ Often level is set high (e.g., 95% or 99%) Such a level is considered high enough to provide convincing evidence about the value of μ 8-15