Lecture Business statistics in practice (7/e): Chapter 8 - Bowerman, O'Connell, Murphree

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Chapter 8 - Confidence intervals. After mastering the material in this chapter, you will be able to: Calculate and interpret a z-based confidence interval for a population mean when σ is known, describe the properties of the t distribution and use a t table, calculate and interpret a t-based confidence interval for a population mean when σ is unknown,...

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Nội dung Text: Lecture Business statistics in practice (7/e): Chapter 8 - Bowerman, O'Connell, Murphree

Chapter 8 Confidence Intervals McGrawHill/Irwin Copyright © 2014 by The McGrawHill Companies, Inc. All rights reserved.
Confidence Intervals 8.1 zBased Confidence Intervals for a Population Mean: σ Known 8.2 tBased Confidence Intervals for a Population Mean: σ Unknown 8.3 Sample Size Determination 8.4 Confidence Intervals for a Population Proportion 8.5 Confidence Intervals for Parameters of Finite Populations (Optional) 82
LO8-1: Calculate and interpret a z-based confidence interval for a population mean when σ is known. 8.1 zBased 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 83
LO8-1 General Confidence Interval  In general, the probability is 1 – α that the population mean μ is contained in the interval x z 2 x x z 2 n  The normal point zα/2 gives a right hand tail area under the standard normal curve equal to α/2  The normal point zα/2 gives a left hand tail area under the standard normal curve equal to a/2  The area under the standard normal curve between zα/2 and zα/2 is 1 – α 84
LO8-1 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 x z 2 x z 2 ,x z 2 n n n 85
LO8-2: Describe the properties of the t distribution and use a t table. 8.2 tBased 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 x t s n If the population is normal, then for any sample size n, this sampling distribution is called the t distribution 86
LO8-2 The t Distribution The curve of the t distribution is similar to that of the standard normal curve Symmetrical and bellshaped 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 87
LO8-3: Calculate and interpret a t-based confidence interval for a population mean when σ is unknown. tBased Confidence Intervals for a Mean: σ Unknown  If the sampled population is normally distributed with mean , then a (1 )100% confidence interval for is s x t 2 n t is the t point giving a righthand tail area of /2 /2 under the t curve having n1 degrees of freedom Figure 8.10 88
LO8-4: Determine the appropriate sample size when estimating a 8.3 Sample Size Determination (z) population mean. If σ is known, then a sample of size 2 z 2 n B so that  is within B units of , with 100(1 ) % confidence 89
LO8-5: 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 (1 a)100% confidence interval for ρ is ˆp 1 ˆp ˆp z 2 n Here, n should be considered large if both ◦n ∙ p̂ ≥ 5 ◦n ∙ (1 – p̂) ≥ 5 810
LO8-6: Determine the appropriate sample size Determining Sample Size for when estimating a population proportion. Confidence Interval for ρ  A sample size given by the formula… 2 z 2 n p1 p B will yield an estimate p̂, 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 811
LO8-7: 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 )100% confidence interval for μ is s N n x z 2 n N 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 812