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Chapter 8 Confidence Intervals McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter Outline 8.1 z-Based Confidence Intervals for a Population Mean: σ Known 8.2 t-Based Confidence Intervals for a Population Mean: σ Unknown 8.3 Sample Size Determination 8.4 Confidence Intervals for a Population Proportion 8.5 A Comparison of Confidence Intervals and Tolerance Intervals (Optional) 8-2 8.1 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 x is normal with mean μx = μ and standard deviation x n 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 8-3 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 8-4 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 0.8 x 0.358 n 5 The probability that x with be within ±0.7 (2σx≈0.7) of µ is 0.9544 8-5 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 0.9544 that the interval [x ± 2σ] contains the population mean µ 8-6 Generalizing In the example, we found the probability that m is contained in an interval of integer multiples of x More usual to specify the (integer) probability and find the corresponding number of x The probability that the confidence interval will not contain the population mean m is denoted by In the mileage example, = 0.0456 8-7 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 8-8 General Confidence Interval In general, the probability is 1 – that the population mean m is contained in the interval x z 2 x x z 2 n 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 – 8-9 Sampling Distribution Of All Possible Sample Means Figure 8.2 8-10 z-Based Confidence Intervals for a Mean with σ Known x z x z n 2 2 n , x z 2 n 8-11 95% Confidence Interval x z0.025 x x 1.96 x 1.96 Figure 8.3 n , x 1.96 n n 8-12 99% Confidence Interval x z0.025 x x 2.575 x 2.575 Figure 8.4 n , x 2.575 n n 8-13 The Effect of a on Confidence Interval Width z/2 = z0.025 = 1.96 Figures 8.2 to 8.4 z/2 = z0.005 = 2.575 8-14 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 8-15 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 8-16 Degrees of Freedom and the t-Distribution Figure 8.6 8-17 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 8-18 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 8-19 t and Right Hand Tail Areas Figure 8.7 8-20 Using the t Distribution Table Table 8.3 (top) 8-21 t-Based Confidence Intervals for a Mean: σ Unknown x t 2 Figure 8.10 s n 8-22 8.3 Sample Size Determination If σ is known, then a sample of size z 2 n B 2 so that x is within B units of m, with 100(1-)% confidence 8-23 Example (Car Mileage Case) z 2 n B 1.96.8 27.32 .3 2 2 8-24 Sample Size Determination (t) If σ is unknown and is estimated from s, then a sample of size t 2 s n B 2 so that x is within B units of m, with 100(1)% confidence. The number of degrees of freedom for the t/2 point is the size of the preliminary sample minus 1 8-25 Example 8.6: The Car Mileage Case t 2 s 2.776.7583 n B .3 2 2 2 49.24 8-26 8.4 Confidence Intervals for a Population Proportion If the sample size n is large, then a (1)100% confidence interval for ρ is pˆ z 2 pˆ 1 pˆ n Here, n should be considered large if both n · p̂ ≥ 5 n · (1 – p̂) ≥ 5 8-27 Example 8.8: The Cheese Spread Case pˆ z 2 pˆ 1 pˆ .063.937 .063 1.96 n 1,000 .063 .0151 .0479,.0781 8-28 Determining Sample Size for Confidence Interval for ρ A sample size given by the formula… z 2 n p1 p B 2 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 8-29 8.5 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 8-30 Selecting an Appropriate Confidence Interval for a Population Mean Figure 8.19 8-31