1. Chapter 8 Confidence Intervals Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

2. Confidence Intervals 8.1z-Based Confidence Intervals for a Population Mean: σ Known 8.2t-Based Confidence Intervals for a Population Mean: σ Unknown 8.3Sample Size Determination 8.4Confidence Intervals for a Population Proportion 8.5Confidence Intervals for Parameters of Finite Populations (Optional) 8-2

3. 8.1 z-Based Confidence Intervals for a Mean: σ 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 LO8-1: Calculate and interpret a z-based confidence interval for a population mean when σ is known. 8-3

4. General Confidence Interval In general, the probability is 1 – α that the population mean μ is contained in the interval The normal point z α /2 gives a right hand tail area under the standard normal curve equal to α /2 The normal point -z α /2 gives a left hand tail area under the standard normal curve equal to a/2 The area under the standard normal curve between ­ z α /2 and z α /2 is 1 – α LO8-1 8-4

5. 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-  )100% confidence interval for  is LO8-1 8-5

6. 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 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 distribution LO8-2: Describe the properties of the t distribution and use a t table. 8-6

7. 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 LO8-2 8-7

8. t-Based Confidence Intervals for a Mean: σ Unknown If the sampled population is normally distributed with mean , then a (1­  )100% confidence interval for  is t  /2 is the t point giving a right-hand tail area of  /2 under the t curve having n­1 degrees of freedom LO8-3: Calculate and interpret a t-based confidence interval for a population mean when σ is unknown. Figure 8.10 8-8

9. 8.3 Sample Size Determination (z) If σ is known, then a sample of size so that  is within B units of , with 100(1­  )% confidence LO8-4: Determine the appropriate sample size when estimating a population mean. 8-9

10. 8.4 Confidence Intervals for a Population Proportion If the sample size n is large, then a (1­a)100% confidence interval for ρ is Here, n should be considered large if both ◦ n · p̂ ≥ 5 ◦ n · (1 – p̂ ) ≥ 5 LO8-5: Calculate and interpret a large sample confidence interval for a population proportion. 8-10

11. Determining Sample Size for Confidence Interval for ρ A sample size given by the formula… will yield an estimate p ̂, precisely within B units of ρ, with 100(1-  )% 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 LO8-6: Determine the appropriate sample size when estimating a population proportion. 8-11

12. 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 is A (1-  )100% confidence interval for the population total is found by multiplying the lower and upper limits of the corresponding interval for μ by N LO8-7: Find and interpret confidence intervals for parameters of finite populations (Optional). 8-12