Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Chapter 8 Confidence Intervals

8-2 Chapter Outline 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 8.5A Comparison of Confidence Intervals and Tolerance Intervals (Optional)

z-Based Confidence Intervals for a Mean: σ Known If a population is normally distributed with mean μ and standard deviation σ, then the sampling distribution of  is normal with mean μ  = μ and standard deviation Use a normal curve as a model of the sampling distribution of the sample mean Exactly, because the population is normal Approximately, by the Central Limit Theorem for large samples

8-4 The Empirical Rule 68.26% of all possible sample means are within one standard deviation of the population mean 95.44% of all possible observed values of x are within two standard deviations of the population mean 99.73% of all possible observed values of x are within three standard deviations of the population mean

8-5 Example 8.1 The Car Mileage Case Assume a sample size (n) of 5 Assume the population of all individual car mileages is normally distributed Assume the standard deviation (σ) is 0.8 The probability that  with be within ±0.7 (2σ  ≈0.7) of µ is

8-6 The Car Mileage Case Continued Assume the sample mean is 31.3 That gives us an interval of [31.3 ± 0.7] = [30.6, 32.0] The probability is that the interval [  ± 2σ] contains the population mean µ

8-7 Generalizing In the example, we found the probability that  is contained in an interval of integer multiples of   More usual to specify the (integer) probability and find the corresponding number of   The probability that the confidence interval will not contain the population mean  is denoted by   In the mileage example,  =

8-8 Generalizing Continued The probability that the confidence interval will contain the population mean μ is denoted by 1 -  1 –  is referred to as the confidence coefficient (1 –  )  100% is called the confidence level Usual to use two decimal point probabilities for 1 –  Here, focus on 1 –  = 0.95 or 0.99

8-9 General Confidence Interval In general, the probability is 1 –  that the population mean  is contained in the interval The normal point z  /2 gives a right hand tail area under the standard normal curve equal to  /2 The normal point - z  /2 gives a left hand tail area under the standard normal curve equal to  /2 The area under the standard normal curve between -z  /2 and z  /2 is 1 – 

8-10 Sampling Distribution Of All Possible Sample Means Figure 8.2

8-11 z-Based Confidence Intervals for a Mean with σ Known

% Confidence Interval Figure 8.3

% Confidence Interval Figure 8.4

8-14 The Effect of a on Confidence Interval Width z  /2 = z = 1.96z  /2 = z = Figures 8.2 to 8.4

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  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

8-16 The t Distribution 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 (df) For a sample of size n, there are one fewer degrees of freedom, that is, df = n – 1

8-17 Degrees of Freedom and the t-Distribution Figure 8.6

8-18 The t Distribution and Degrees of Freedom As the sample size n increases, the degrees of freedom also increases As the degrees of freedom increase, the spread of the t curve decreases As the degrees of freedom increases indefinitely, the t curve approaches the standard normal curve If n ≥ 30, so df = n – 1 ≥ 29, the t curve is very similar to the standard normal curve

8-19 t and Right Hand Tail Areas t  is the point on the horizontal axis under the t curve that gives a right hand tail equal to  So the value of t  in a particular situation depends on the right hand tail area  and the number of degrees of freedom

8-20 t and Right Hand Tail Areas Figure 8.7

8-21 Using the t Distribution Table Table 8.3 (top)

8-22 t-Based Confidence Intervals for a Mean: σ Unknown Figure 8.10

Sample Size Determination If σ is known, then a sample of size so that  is within B units of , with 100(1-  )% confidence

8-24 Example (Car Mileage Case)

8-25 Sample Size Determination (t) If σ is unknown and is estimated from s, then a sample of size so that  is within B units of , with 100(1-  )% confidence. The number of degrees of freedom for the t  /2 point is the size of the preliminary sample minus 1

8-26 Example 8.6: The Car Mileage Case

Confidence Intervals for a Population Proportion If the sample size n is large, then a (1 ­  )100% confidence interval for ρ is Here, n should be considered large if both n · p̂ ≥ 5 n · (1 – p̂) ≥ 5

8-28 Example 8.8: The Cheese Spread Case

8-29 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

A Comparison of Confidence Intervals and Tolerance Intervals (Optional) A tolerance interval contains a specified percentage of individual population measurements Often 68.26%, 95.44%, 99.73% A confidence interval is an interval that contains the population mean μ, and the confidence level expresses how sure we are that this interval contains μ Confidence level set high (95% or 99%) Such a level is considered high enough to provide convincing evidence about the value of μ

8-31 Selecting an Appropriate Confidence Interval for a Population Mean Figure 8.19