Chapter 8 Confidence Intervals

Chapter 8Confidence IntervalsCopyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/IrwinConfidence Intervals8.1 z-Based Confidence Intervals for a Population Mean: σ Known8.2 t-Based Confidence Intervals for a Population Mean: σ Unknown8.3 Sample Size Determination8.4 Confidence Intervals for a Population Proportion8.5 Confidence Intervals for Parameters of Finite Populations (Optional)8-*8.1 z-Based Confidence Intervals for a Mean: σ KnownConfidence interval for a population mean is an interval constructed around the sample mean so we are reasonable sure that it contains the population meanAny confidence interval is based on a confidence levelLO8-1: Calculate and interpret a z-based confidence interval for a population mean when σ is known.8-*General Confidence IntervalIn general, the probability is 1 – α that the population mean μ is contained in the intervalThe normal point zα/2 gives a right hand tail area under the standard normal curve equal to α/2The normal point -zα/2 gives a left hand tail area under the standard normal curve equal to a/2The area under the standard normal curve between ­zα/2 and zα/2 is 1 – αLO8-18-*General Confidence Interval ContinuedIf 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 isLO8-18-*8.2 t-Based Confidence Intervals for a Mean: σ UnknownIf σ is unknown (which is usually the case), we can construct a confidence interval for μ based on the sampling distribution of If the population is normal, then for any sample size n, this sampling distribution is called the t distributionLO8-2: Describe the properties of the t distribution and use a t table.8-*The t DistributionThe curve of the t distribution is similar to that of the standard normal curveSymmetrical and bell-shapedThe t distribution is more spread out than the standard normal distributionThe spread of the t is given by the number of degrees of freedomDenoted by dfFor a sample of size n, there are one fewer degrees of freedom, that is, df = n – 1LO8-28-*t-Based Confidence Intervals for a Mean: σ UnknownIf the sampled population is normally distributed with mean , then a (1­)100% confidence interval for  ist/2 is the t point giving a right-hand tail area of /2 under the t curve having n­1 degrees of freedomLO8-3: Calculate and interpret a t-based confidence interval for a population mean when σ is unknown.Figure 8.108-*8.3 Sample Size Determination (z) If σ is known, then a sample of sizeso that  is within B units of , with 100(1­)% confidenceLO8-4: Determine the appropriate sample sizewhen estimating a population mean.8-*8.4 Confidence Intervals for a Population ProportionIf the sample size n is large, then a (1­a)100% confidence interval for ρ isHere, n should be considered large if bothn · p̂ ≥ 5n · (1 – p̂) ≥ 5LO8-5: Calculate and interpret a large sampleconfidence interval for a population proportion.8-*Determining Sample Size for Confidence Interval for ρA sample size given by the formulawill yield an estimate p̂, precisely within B units of ρ, with 100(1-)% confidenceNote that the formula requires a preliminary estimate of pThe conservative value of p=0.5 is generally used when there is no prior information on pLO8-6: Determine the appropriate sample sizewhen estimating a population proportion.8-*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- )100% confidence interval for μ isA (1- )100% confidence interval for the population total is found by multiplying the lower and upper limits of the corresponding interval for μ by NLO8-7: Find and interpret confidence intervals for parameters of finite populations (Optional).8-*