Chapter 7 Estimation: Single Population Statistics for Business and Economics 8th Edition Chapter 7 Estimation: Single Population Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

ESTIMATION: AN INTRODUCTION Definition The assignment of value(s) to a population parameter based on a value of the corresponding sample statistic is called estimation. The value(s) assigned to a population parameter based on the value of a sample statistic is called an estimate. The sample statistic used to estimate a population parameter is called an estimator. Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Properties of Unbiased Point Estimators Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

POINT AND INTERVAL ESTIMATES A Point Estimate: The value of a sample statistic that is used to estimate a population parameter is called a point estimate. An Interval Estimate: In interval estimation, an interval is constructed around the point estimate, and it is stated that this interval is likely to contain the corresponding population parameter. Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

A Point Estimate Usually, whenever we use point estimation, we calculate the margin of error associated with that point estimation. The margin of error is calculated as follows: Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Interval estimation Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Point and Interval Estimates A point estimate is a single number, a confidence interval provides additional information about variability Upper Confidence Limit Lower Confidence Limit Point Estimate Width of confidence interval Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Point Estimates μ x P Mean Proportion We can estimate a Population Parameter … with a Sample Statistic (a Point Estimate) μ x Mean Proportion P Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Confidence Intervals Confidence Interval Estimator for a population parameter is a rule for determining (based on sample information) a range or an interval that is likely to include the parameter. The corresponding estimate is called a confidence interval estimate. Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

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, 7e © 2007 Pearson Education, Inc.

Confidence Level, (1-) Suppose confidence level = 95% (continued) 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 Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Point Estimate ± (Reliability Factor)(Standard Error) 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, 7e © 2007 Pearson Education, Inc.

Confidence Intervals Confidence Intervals Population Mean Population Proportion σ2 Known σ2 Unknown Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

ESTIMATION OF A POPULATION MEAN: Population Variance KNOWN Three Possible Cases Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

ESTIMATION OF A POPULATION MEAN: Population Variance NOT KNOWN Three Possible Cases Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Confidence Interval for μ (σ2 Known) ESTIMATION OF A POPULATION MEAN: Population Variance KNOWN Confidence Interval for μ (σ2 Known) Assumptions Population variance σ2 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, 7e © 2007 Pearson Education, Inc.

Point Estimate ± (Reliability Factor)(Standard Error) Margin of Error Keep in mind that in any time sampling occurs, one expects the possibility of a difference between the particular value of an estimator and the parameter's true value. The true value of an unknown parameter  might be somewhat greater or somewhat less than the value determined by even the best point estimator  . So, The confidence interval estimate for a parameter takes on the general form- Can also be written as where ME is called the margin of error (error factor) The interval width, w, is equal to twice the margin of error. w = 2(ME) The maximum distance between an estimator and the true value of a parameter is called the margin of error. Point Estimate ± (Reliability Factor)(Standard Error)

Reducing the Margin of Error The margin of error can be reduced if- The sample size is increased (n↑) the population standard deviation can be reduced (σ↓) The confidence level is decreased, (1 – ) ↓ A 95% confidence interval estimate for a population mean is determined to be 62.8 to 73.4. If the confidence level is reduced to 90%, the confidence interval for becomes narrower Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Finding the Reliability Factor, z/2 Consider a 95% confidence interval: z = -1.96 z = 1.96 Z units: Lower Confidence Limit Upper Confidence Limit X units: Point Estimate Point Estimate Find z.025 = 1.96 from the standard normal distribution table (In 1.96 we find value.9750 from table-1)(So,1-.9750=.025)(.9750-.025=.95 or,95%) In the value 0.9750 from table-1 we find 1.96, Hence, z=1.96

Common Levels of Confidence Commonly used confidence levels are 90%, 95%, and 99% Confidence Coefficient, Confidence Level Z/2 value 80% 90% 95% 98% 99% 99.8% 99.9% .80 .90 .95 .98 .99 .998 .999 1.28 1.645 1.96 2.33 2.58 3.08 3.27 Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Intervals and Level of Confidence Sampling Distribution of the Mean x Intervals extend from to x1 100(1-)% of intervals constructed contain μ; 100()% do not. x2 Confidence Intervals Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Figure 8.4 Confidence intervals. Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Example-1: Refined Sugar (Confidence Interval) A process produces bags of refined sugar. The weights of the content of these bags are normally distributed with standard deviation 1.2 ounces. The contents of a random sample of 25 bags has a mean weight of 19.8 ounces. Find the upper and lower confidence limits of a 99% confidence interval for the true mean weight for all bags of sugar produced by the process. Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Example 2 (practice) 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 90% confidence interval for the true mean resistance of the population. Here, Confidence level is 90% or .90, so The area in each tail of the normal distribution curve is α/2=(1-.90)/2=.05 In the value 0.950 from table-1 we find 1.65 Hence, z = 1.65 Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Example-2 (practice) Solution: (continued) 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: Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Example 2(practice)Interpretation We are 90% confident that the true mean resistance is between 2.0259 and 2.3741 ohms Although the true mean may or may not be in this interval, 90% of intervals formed in this manner will contain the true mean Also See Ex-7.3 from book Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Figure 8.4 Confidence intervals. Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

ESTIMATION OF A POPULATION MEAN: Population Variance NOT KNOWN Three Possible Cases Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Confidence Intervals Confidence Intervals Population Mean Population Proportion σ2 Known σ2 Unknown Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

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 follows the Student’s t distribution with (n - 1) degrees of freedom Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Confidence Interval for μ (σ2 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 The Student’s t distribution is the ratio of the standard normal distribution to the square root of the chi-square distribution divided by its degrees of freedom. Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Confidence Interval for μ (σ Unknown) (continued) Assumptions 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: where tn-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: Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

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 Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Student’s t Table-8 (appendix) Upper Tail Area Let: n = 3 df = n - 1 = 2  = .10 /2 =.05 df .10 .05 .025 1 3.078 6.314 12.706 2 1.886 2.920 4.303 /2 = .05 3 1.638 2.353 3.182 The body of the table contains t values, not probabilities t 2.920 Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Example 7.4 (book) Gasoline prices rose drastically during the early years of this century. Suppose that a recent study was conducted using truck drivers with equivalent years of experience to test run 24 trucks of a particular model over the same highway. The sample mean and standard deviation is 18.68 and 1.69526 respectively. Estimate the population mean fuel consumption for this truck model with 90% confidence interval. Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Example-3 (practice) s = 8. Form a 95% confidence interval for μ 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, 7e © 2007 Pearson Education, Inc.

Confidence Intervals Confidence Intervals Population Mean Population Proportion σ Known σ Unknown Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

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, 7e © 2007 Pearson Education, Inc.

Confidence Intervals for the Population Proportion, p (continued) 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: Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

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, 7e © 2007 Pearson Education, Inc.

Example-4 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, 7e © 2007 Pearson Education, Inc.

Example-4 (continued) 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, 7e © 2007 Pearson Education, Inc.

Ex-4 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. Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.