Statistics for Business and Economics Chapter 7 Estimation: Single Population Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-1
Confidence Intervals Contents of this chapter: Confidence Intervals for the Population Mean, µ when Population Variance σ is Known when Population Variance σ is Unknown Confidence Intervals for the Population Proportion, pˆ (large samples) Confidence interval estimates for the variance of a normal population Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-
7.1 Definitions An estimator of a population parameter is a random variable that depends on sample information... whose value provides an approximation to this unknown parameter A specific value of that random variable is called an estimate Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-3
Point and Interval Estimates A point estimate is a single number, a confidence interval provides additional information about variability Lower Confidence Limit Point Estimate Width of confidence interval Upper Confidence Limit Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-4
Point Estimates We can estimate a Population Parameter Mean µ x Variance σ with a Sample Statistic (a Point Estimate) s Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-5
Unbiasedness A point estimator θˆ is said to be an unbiased estimator of the parameter θ if the expected value, or mean, of the sampling distribution of θˆ is θ, Examples: E( θ) ˆ = The sample mean x is an unbiased estimator of µ The sample variance s is an unbiased estimator of σ The sample proportion pˆ is an unbiased estimator of P θ Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-6
Unbiasedness θˆ 1 θˆ is an unbiased estimator, is biased: (continued) θˆ 1 θˆ θ θˆ Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-7
Bias Let θˆ be an estimator of θ The bias in θˆ is defined as the difference between its mean and θ Bias( θ) ˆ = E(θ) ˆ θ The bias of an unbiased estimator is 0 Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-8
Most Efficient Estimator Suppose there are several unbiased estimators of θ The most efficient estimator or the minimum variance unbiased estimator of θ is the unbiased estimator with the smallest variance Let θˆ 1 and θˆ be two unbiased estimators of θ, based on the same number of sample observations. Then, θˆ is said to be more efficient than if Var( θˆ 1) < Var(θˆ 1 θˆ ) The relative efficiency of θˆ 1 with respect to θˆ is the ratio of their variances: Relative Efficiency = Var(θˆ Var(θˆ Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-9 1 ) )
7. Confidence Intervals How much uncertainty is associated with a point estimate of a population parameter? An interval estimate provides more information about a population characteristic than does a point estimate Such interval estimates are called confidence intervals Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-10
Confidence Interval Estimate An interval gives a range of values: Takes into consideration variation in sample statistics from sample to sample Based on observation from 1 sample Stated in terms of level of confidence Can never be 100% confident Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-11
Confidence Interval and Confidence Level If P(a < θ < b) = 1 - α then the interval from a to b is called a 100(1 - α)% confidence interval of θ. The quantity (1 - α) is called the confidence level of the interval (α between 0 and 1) In repeated samples of the population, the true value of the parameter θ would be contained in 100(1 - α)% of intervals calculated this way. The confidence interval is written as LCL < θ < UCL with 100(1 - α)% confidence Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-1
Estimation Process Population (mean, µ, is unknown) Random Sample Mean X = 50 I am 95% confident that µ is between 40 & 60. Sample Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-13
Confidence Level, (1-α) Suppose confidence level = 95% (1 - α) = 0.95 (continued) From repeated samples, 95% of all the confidence intervals that can be constructed will contain the unknown true parameter A specific interval either will contain or will not contain the true parameter No probability involved in a specific interval Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-14
General Formula The general formula for all confidence intervals is: Point Estimate ± (Reliability Factor)(Standard Error) The value of the reliability factor depends on the desired level of confidence Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-15
Confidence Intervals Confidence Intervals Population Mean Population Proportion Population Variance σ Known σ Unknown Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-16
7. Confidence Interval for µ (σ Known) Assumptions Population variance σ is known Population is normally distributed If population is not normal, use large sample Confidence interval estimate: σ x z < µ < x + z α/ α/ n σ n (where z α/ is the normal distribution value for a probability of α/ in each tail) Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-17
Margin of Error The confidence interval, σ x zα/ < µ < x + zα/ n σ n Can also be written as x ±ME where ME is called the margin of error ME = zα/ σ n The interval width, w, is equal to twice the margin of error Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-18
Reducing the Margin of Error ME = zα/ σ n The margin of error can be reduced if the population standard deviation can be reduced (σ ) The sample size is increased (n ) The confidence level is decreased, (1 α) Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-19
Finding the Reliability Factor, z α/ Consider a 95% confidence interval: 1 α =.95 α =.05 α =.05 Z units: X units: z = -1.96 z = 1.96 Lower Confidence Limit 0 Point Estimate Upper Confidence Limit Find z.05 = ±1.96 from the standard normal distribution table Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-0
Common Levels of Confidence Commonly used confidence levels are 90%, 95%, and 99% Confidence Level 80% 90% 95% 98% 99% 99.8% 99.9% Confidence Coefficient, 1 α.80.90.95.98.99.998.999 Z α/ value 1.8 1.645 1.96.33.58 3.08 3.7 Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-1
Intervals and Level of Confidence Sampling Distribution of the Mean LCL Intervals extend from = x z to UCL = x + z σ n σ n α / 1 α α/ µ x = µ x x 1 Confidence Intervals x 100(1-α)% of intervals constructed contain µ; 100(α)% do not. Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-
Example A sample of 7 light bulb from a large normal population has a mean life length of 1478 hours. We know that the population standard deviation is 36 hours. Determine a 95% confidence interval for the true mean length of life in the population. Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-3
Example (continued) Solution: x ± z σ n =1478 ± 1.96 (36/ 7) =1478 ± 13.58 1464.4 < µ < 1491.58 Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-4
Interpretation We are 95% confident that the true mean life time is between 1464.4 and 1491.58 Although the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true mean Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-5
7.3 Confidence Intervals Confidence Intervals Population Mean Population Proportion Population Variance σ Known σ Unknown Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-6
Student s t Distribution Consider a random sample of n observations with mean x and standard deviation s from a normally distributed population with mean µ Then the variable t = x µ s/ n follows the Student s t distribution with (n - 1) degrees of freedom Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-7
Confidence Interval for µ (σ Unknown) If the population standard deviation σ is unknown, we can substitute the sample standard deviation, s This introduces extra uncertainty, since s is variable from sample to sample So we use the t distribution instead of the normal distribution Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-8
Confidence Interval for µ Assumptions (σ Unknown) Population standard deviation is unknown Population is normally distributed If population is not normal, use large sample Use Student s t Distribution Confidence Interval Estimate: s x t n -1,α/ < µ < x + t n-1, α/ n s n (continued) where t n-1,α/ is the critical value of the t distribution with n-1 d.f. and an area of α/ in each tail: P(t α/ n 1 > tn 1, ) = α/ Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-9
Margin of Error The confidence interval, s x t n -1,α/ < µ < x + t n-1, α/ n s n Can also be written as x ±ME where ME is called the margin of error: ME = tn-1, α/ σ n Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-30
Student s t Distribution The t is a family of distributions The t value depends on degrees of freedom (d.f.) Number of observations that are free to vary after sample mean has been calculated d.f. = n - 1 Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-31
Student s t Distribution Note: t Z as n increases Standard Normal (t with df = ) t-distributions are bellshaped and symmetric, but have fatter tails than the normal t (df = 13) t (df = 5) 0 t Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-3
Student s t Table Upper Tail Area df.10.05.05 1 3.078 6.314 1.706 Let: n = 3 df = n - 1 = α =.10 α/ =.05 1.886.90 4.303 3 1.638.353 3.18 α/ =.05 The body of the table contains t values, not probabilities 0.90 t Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-33
t distribution values With comparison to the Z value Confidence t t t Z Level (10 d.f.) (0 d.f.) (30 d.f.).80 1.37 1.35 1.310 1.8.90 1.81 1.75 1.697 1.645.95.8.086.04 1.960.99 3.169.845.750.576 Note: t Z as n increases Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-34
Example A random sample of n = 5 has x = 50 and s = 8. Form a 95% confidence interval for µ d.f. = n 1 = 4, so t 1,α/ = t4,.05 n =.0639 The confidence interval is s s x t n-1,α/ < µ < x + t n-1, α/ n n 8 50 (.0639) < µ < 50 + (.0639) 5 46.698 < µ < 53.30 8 5 Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-35
7.4 Confidence Intervals Confidence Intervals Population Mean Population Proportion Population Variance σ Known σ Unknown Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-36
Confidence Intervals for the Population Proportion An interval estimate for the population proportion ( p ) can be calculated by adding an allowance for uncertainty to the sample proportion ( ) pˆ Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-37
Confidence Intervals for the Population Proportion, p Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation σ p = p(1 p) n We will estimate this with sample data: (continued) pˆ (1 p) ˆ n Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-38
Confidence Interval Endpoints Upper and lower confidence limits for the population proportion are calculated with the formula pˆ(1 pˆ) pˆ zα/ < p < pˆ + zα/ n pˆ(1 n pˆ) where z α/ is the standard normal value for the level of confidence desired pˆ is the sample proportion n is the sample size Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-39
Example A random sample of 100 people shows that 5 are left-handed. Form a 95% confidence interval for the true proportion of left-handers Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-40
5 100 Example (continued) A random sample of 100 people shows that 5 are left-handed. Form a 95% confidence interval for the true proportion of left-handers. pˆ(1 pˆ) pˆ zα/ < p < pˆ + zα/ n 1.96.5(.75) 100 0.1651 < < p p < < 5 + 1.96 100 0.3349 pˆ(1 pˆ) n.5(.75) 100 Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-41
Interpretation We are 95% confident that the true percentage of left-handers in the population is between 16.51% and 33.49%. Although the interval from 0.1651 to 0.3349 may or may not contain the true proportion, 95% of intervals formed from samples of size 100 in this manner will contain the true proportion. Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-4
7.5 Confidence Intervals Confidence Intervals Population Mean Population Proportion Population Variance σ Known σ Unknown Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-43
Confidence Intervals for the Population Variance Goal: Form a confidence interval for the population variance, σ The confidence interval is based on the sample variance, s Assumed: the population is normally distributed Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8-44
Confidence Intervals for the Population Variance The random variable χ n 1 = (n 1)s σ follows a chi-square distribution with (n 1) degrees of freedom (continued) Where the chi-square value χn 1, α denotes the number for which P( χ χ = n 1 > n 1, α ) α Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8-45
Confidence Intervals for the Population Variance (continued) The (1 - α)% confidence interval for the population variance is (n 1)s χ n 1, α/ < σ < (n 1)s χ n 1,1- α/ Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8-46
Example You are testing the speed of a batch of computer processors. You collect the following data (in Mhz): Sample size 17 Sample mean 3004 Sample std dev 74 Assume the population is normal. Determine the 95% confidence interval for σ x Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8-47
Finding the Chi-square Values n = 17 so the chi-square distribution has (n 1) = 16 degrees of freedom α = 0.05, so use the the chi-square values with area 0.05 in each tail: χ χ n 1,1- α/ n 1, α/ = = χ χ 16, 0.975 16, 0.05 = = 6.91 8.85 probability α/ =.05 probability α/ =.05 χ 16,0.975 = 6.91 χ 16,0.05 = 8.85 χ 16 Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8-48
Calculating the Confidence Limits The 95% confidence interval is (n 1)s χ n 1, α/ < σ < (n 1)s χ n 1,1- α/ (17 1)(74) 8.85 3037 < < σ σ < < (17 1)(74) 6.91 1683 Converting to standard deviation, we are 95% confident that the population standard deviation of CPU speed is between 55.1 and 11.6 Mhz Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8-49