9- Chapter Nine McGraw- © 2005 The McGraw-Hill Companies, Inc., All 9- Chapter Nine Estimation and Confidence Intervals GOALS When you have completed this chapter, you will be able to: ONE Define a what is meant by a point estimate TWO Define the term level of level of confidence THREE Construct a confidence interval for the population mean when the population standard deviation is known FOUR Construct a confidence interval for the population mean when the population standard deviation is unknown Goals 9- Chapter Nine continued Estimation and Confidence Intervals GOALS When you have completed this chapter, you will be able to: FIVE Construct a confidence interval for the population proportion SIX Determine the sample size for attribute and variable sampling Goals A point estimate is a single value (statistic) used to estimate a population value (parameter) An Interval Estimate states the range within which a population parameter probably lies 9- A confidence interval is a range of values within which the population parameter is expected to occur The two confidence intervals that are used extensively are the 95% and the 99% Point and Interval Estimates 9- Factors that determine the width of a confidence interval The sample size, n The desired level of confidence The variability in the population, usually estimated by s Point and Interval Estimates 9- For a 95% confidence interval about 95% of the similarly constructed intervals will contain the parameter being estimated 95% of the sample means for a specified sample size will lie within 1.96 standard deviations of the hypothesized population mean For the 99% confidence interval, 99% of the sample means for a specified sample size will lie within 2.58 standard deviations of the hypothesized population mean Interval Estimates 9- Standard Error of the Sample Mean Standard deviation of the sampling distribution of the sample means σx = σ n σ x symbol for the standard error of the sample mean σ the standard deviation of the population n is the size of the sample Standard Error of the Sample Means 9- If s is not known and n >30, the standard deviation of the sample, designated s, is used to approximate the population standard deviation X ±z s n The standard error sx = s n If the population standard deviation is known or the sample is greater than 30 we use the z distribution Standard Error of the Sample Means 9- If the population standard deviation is unknown, the underlying s population is X ±t approximately n normal, and the sample size is less than 30 we use the t distribution The value of t for a given confidence level depends upon its degrees of freedom Point and Interval Estimates Characteristics of the t distribution It is a continuous distribution There is a family of t distributions Assumption: the population is normal or nearly normal 9- 10 It is bell-shaped and symmetrical The t distribution is more spread out and flatter at the center than is the standard normal distribution, differences that diminish as n increases Point and Interval Estimates 9- 12 The Dean of the Business School wants to estimate the mean number of hours worked per week by students A sample of 49 students showed a mean of 24 hours with a standard deviation of hours What is the population mean? The value of the population mean is not known Our best estimate of this value is the sample mean of 24.0 hours This value is called a point estimate Example 9- 13 95 percent confidence interval for the population mean X ± 1.96 s n = 24.00 ± 1.96 49 = 24.00 ± 1.12 The confidence limits range from 22.88 to 25.12 About 95 percent of the similarly constructed intervals include the population parameter 9- 14 The confidence interval for a population proportion p(1 − p) p±z n Confidence Interval for a Population Proportion 9- 15 A sample of 500 executives who own their own home revealed 175 planned to sell their homes and retire to Arizona Develop a 98% confidence interval for the proportion of executives that plan to sell and move to Arizona (.35)(.65) 35 ± 2.33 = 35 ± 0497 500 Example Finite population Adjust the standard errors of the sample means and the proportion σ σx = n N −n N −1 9- 16 fixed upper bound Finite-Population Correction Factor N, total number of objects n, sample size Finite-Population Correction Factor 9- 17 Standard error of the sample proportions σp = p (1 − p ) n N −n N −1 Ignore finite-population correction factor if n/N < 05 Finite-Population Correction Factor 9- 18 95% confidence interval for the mean number of hours worked per week by the students if there are only 500 students on campus n/N = 49/500 = 098 > 05 Use finite population correction factor 500 − 49 24 ± 1.96( )( ) = 24.00 ± 1.0648 500 − 49 EXAMPLE revisited 9- 19 factors that determine the size of a sample The degree of confidence selected The maximum allowable error The variation in the population Selecting a Sample Size 9- 20 Calculating the sample size z•s n= E where E is the allowable error z the z- value corresponding to the selected level of confidence s the sample deviation of the pilot survey Selecting a Sample Size 9- 21 A consumer group would like to estimate the mean monthly electricity charge for a single family house in July within $5 using a 99 percent level of confidence Based on similar studies the standard deviation is estimated to be $20.00 How large a sample is required? (2.58)(20) n= = 107 Example 9- 22 The formula for determining the sample size in the case of a proportion is Z n = p(1 − p) E where p is the estimated proportion, based on past experience or a pilot survey z is the z value associated with the degree of confidence selected E is the maximum allowable error the researcher will tolerate Sample Size for Proportions 9- 23 The American Kennel Club wanted to estimate the proportion of children that have a dog as a pet If the club wanted the estimate to be within 3% of the population proportion, how many children would they need to contact? Assume a 95% level of confidence and that the club estimated that 30% of the children have a dog as a pet 1.96 n = (.30)(.70) = 897 03 Example 9- 24 What happens when the population has less members than the sample size calculated requires? Step One: Calculate the sample size as before no Step Two: Calculate the new sample size n= where no is the sample size calculated in step one no 1+ N Optional method, not covered in text: Sample Size for Small Populations An auditor wishes to survey employees in an organization to determine compliance with federal regulations The auditor estimates that 80% of the employees would say that the organization is in compliance The organization has 200 employees The auditor wishes to be 95% confident in the results, with a margin of error no greater than 3% How many employees should the auditor survey? 9- 25 Example Optional 9- 26 Step One Calculate the sample size as before Z n = p(1 − p) E = (.80)(.20) 1.96 03 = 683 Step Two Calculate the new sample size no n= + no N = 683 + 683 200 = 155 Example continued ... population, usually estimated by s Point and Interval Estimates 9- For a 95% confidence interval about 95% of the similarly constructed intervals will contain the parameter being estimated 95% of the sample... proportion SIX Determine the sample size for attribute and variable sampling Goals A point estimate is a single value (statistic) used to estimate a population value (parameter) An Interval Estimate... 95% and the 99% Point and Interval Estimates 9- Factors that determine the width of a confidence interval The sample size, n The desired level of confidence The variability in the population,