Chap 8-1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 8 Estimation: Single Population Statistics for Business and Economics 6 th Edition

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-2 Chapter Goals After completing this chapter, you should be able to: Distinguish between a point estimate and a confidence interval estimate Construct and interpret a confidence interval estimate for a single population mean using both the Z and t distributions Form and interpret a confidence interval estimate for a single population proportion

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-3 Confidence Intervals Content of this chapter Confidence Intervals for the Population Mean, μ when Population Variance σ 2 is Known when Population Variance σ 2 is Unknown Confidence Intervals for the Population Proportion, (large samples)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-4 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

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-5 Point and Interval Estimates A point estimate is a single number, a confidence interval provides additional information about variability Point Estimate Lower Confidence Limit Upper Confidence Limit Width of confidence interval

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-6 We can estimate a Population Parameter … Point Estimates with a Sample Statistic (a Point Estimate) Mean Proportion P x μ

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-7 Unbiasedness A point estimator is said to be an unbiased estimator of the parameter if the expected value, or mean, of the sampling distribution of is, Examples: The sample mean is an unbiased estimator of μ The sample variance is an unbiased estimator of σ 2 The sample proportion is an unbiased estimator of P

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-8 is an unbiased estimator, is biased: Unbiasedness (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-9 Bias Let be an estimator of The bias in is defined as the difference between its mean and The bias of an unbiased estimator is 0

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-10 Consistency Let be an estimator of is a consistent estimator of if the difference between the expected value of and decreases as the sample size increases Consistency is desired when unbiased estimators cannot be obtained

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-11 Most Efficient Estimator Suppose there are several unbiased estimators of The most efficient estimator or the minimum variance unbiased estimator of is the unbiased estimator with the smallest variance Let and be two unbiased estimators of, based on the same number of sample observations. Then, is said to be more efficient than if The relative efficiency of with respect to is the ratio of their variances:

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-12 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

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-13 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 Gives information about closeness to unknown population parameters Stated in terms of level of confidence Can never be 100% confident

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-14 Confidence Interval and Confidence Level If P(a < < b) = 1 - then the interval from a to b is called a 100(1 - )% confidence interval of. The quantity (1 - ) is called the confidence level of the interval ( between 0 and 1) In repeated samples of the population, the true value of the parameter would be contained in 100(1 - )% of intervals calculated this way. The confidence interval calculated in this manner is written as a < < b with 100(1 - )% confidence

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-15 Estimation Process (mean, μ, is unknown) Population Random Sample Mean X = 50 Sample I am 95% confident that μ is between 40 & 60.

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-16 Confidence Level, (1- ) Suppose confidence level = 95% Also written (1 - ) = 0.95 A relative frequency interpretation: 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 (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-17 General Formula The general formula for all confidence intervals is: The value of the reliability factor depends on the desired level of confidence Point Estimate (Reliability Factor)(Standard Error)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-18 Confidence Intervals Population Mean σ 2 Unknown Confidence Intervals Population Proportion σ 2 Known

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-19 Confidence Interval for μ (σ 2 Known) Assumptions Population variance σ 2 is known Population is normally distributed If population is not normal, use large sample Confidence interval estimate: (where z /2 is the normal distribution value for a probability of /2 in each tail)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-20 Margin of Error The confidence interval, Can also be written as where ME is called the margin of error The interval width, w, is equal to twice the margin of error

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-21 Reducing the Margin of Error 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 – )

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-22 Finding the Reliability Factor, z /2 Consider a 95% confidence interval: z = -1.96z = 1.96 Point Estimate Lower Confidence Limit Upper Confidence Limit Z units: X units: Point Estimate 0 Find z.025 = 1.96 from the standard normal distribution table

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-23 Common Levels of Confidence Commonly used confidence levels are 90%, 95%, and 99% Confidence Level Confidence Coefficient, Z /2 value % 90% 95% 98% 99% 99.8% 99.9%

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-24 Intervals and Level of Confidence Confidence Intervals Intervals extend from to 100(1- )% of intervals constructed contain μ; 100( )% do not. Sampling Distribution of the Mean x x1x1 x2x2

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-25 Example A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is 0.35 ohms. Determine a 95% confidence interval for the true mean resistance of the population.

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-26 Example A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is.35 ohms. Solution: (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-27 Interpretation We are 95% confident that the true mean resistance is between and ohms Although the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true mean

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-28 Confidence Intervals Population Mean Confidence Intervals Population Proportion σ 2 Unknown σ 2 Known

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-29 Students 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 follows the Students t distribution with (n - 1) degrees of freedom

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-30 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 Confidence Interval for μ (σ 2 Unknown)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-31 Assumptions Population standard deviation is unknown Population is normally distributed If population is not normal, use large sample Use Students t Distribution Confidence Interval Estimate: where t n-1,α/2 is the critical value of the t distribution with n-1 d.f. and an area of α/2 in each tail: Confidence Interval for μ (σ Unknown) (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-32 Students 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

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-33 Students t Distribution t 0 t (df = 5) t (df = 13) t-distributions are bell- shaped and symmetric, but have fatter tails than the normal Standard Normal (t with df = ) Note: t Z as n increases

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-34 Students t Table Upper Tail Area df t The body of the table contains t values, not probabilities Let: n = 3 df = n - 1 = 2 =.10 /2 =.05 /2 =

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-35 t distribution values With comparison to the Z value Confidence t t t Z Level (10 d.f.) (20 d.f.) (30 d.f.) ____ Note: t Z as n increases

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-36 Example A random sample of n = 25 has x = 50 and s = 8. Form a 95% confidence interval for μ d.f. = n – 1 = 24, so The confidence interval is

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-37 Confidence Intervals Population Mean σ Unknown Confidence Intervals Population Proportion σ Known

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-38 Confidence Intervals for the Population Proportion, p An interval estimate for the population proportion ( P ) can be calculated by adding an allowance for uncertainty to the sample proportion ( )

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-39 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 We will estimate this with sample data: (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-40 Confidence Interval Endpoints Upper and lower confidence limits for the population proportion are calculated with the formula where z /2 is the standard normal value for the level of confidence desired is the sample proportion n is the sample size

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-41 Example A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-42 Example A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers. (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-43 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 to 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.

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-44 PHStat Interval Options options

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-45 Using PHStat (for μ, σ unknown) A random sample of n = 25 has X = 50 and S = 8. Form a 95% confidence interval for μ

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-46 Chapter Summary Introduced the concept of confidence intervals Discussed point estimates Developed confidence interval estimates Created confidence interval estimates for the mean (σ 2 known) Introduced the Students t distribution Determined confidence interval estimates for the mean (σ 2 unknown) Created confidence interval estimates for the proportion