Slides Prepared by JOHN S. LOUCKS St. Edwards University, 2002 South-Western /Thomson Learning,- -,- -,- -,Chapter 8 Interval Estimation,Interval Estimation of a Population Mean:Large-Sample Case Interval Estimation of a Population Mean: Small-Sample Case Determining the Sample Size Interval Estimation of a Population Proportion,Interval Estimation of a Population Mean: Large-Sample Case,Sampling Error Probability Statements about the Sampling Error Constructing an Interval Estimate:Large-Sample Case with Known Calculating an Interval Estimate:Large-Sample Case with Unknown,Sampling Error,The absolute value of the difference between an unbiased point estimate and the population parameter it estimates is called the sampling error. For the case of a sample mean estimating a population mean, the sampling error isSampling Error =,Probability Statements About the Sampling Error,Knowledge of the sampling distribution of enables us to make probability statements about the sampling error even though the population mean is not known. A probability statement about the sampling error is a precision statement.,Precision Statement There is a 1 - probability that the value of a sample mean will provide a sampling error of or less.,Probability Statements About the Sampling Error,/2,/2,1 - of allvalues,Samplingdistributionof,Interval Estimate of a Population Mean: Large-Sample Case (n 30),With Knownwhere: is the sample mean1 - is the confidence coefficientz/2 is the z value providing an area of/2 in the upper tail of the standard normal probability distributions is the population standard deviationn is the sample size,Interval Estimate of a Population Mean: Large-Sample Case (n 30),With UnknownIn most applications the value of the population standard deviation is unknown. We simply use the value of the sample standard deviation, s, as the point estimate of the population standard deviation.,National Discount has 260 retail outlets throughout the United States. National evaluates each potential location for a new retail outlet in part on the mean annual income of the individuals in the marketing area of the new location.Sampling can be used to develop an interval estimate of the mean annual income for individuals in a potential marketing area for National Discount. A sample of size n = 36 was taken. The sample mean, , is $21,100 and the sample standard deviation, s, is $4,500. We will use .95 as the confidence coefficient in our interval estimate.,Example: National Discount, Inc.,Precision StatementThere is a .95 probability that the value of a sample mean for National Discount will provide a sampling error of $1,470 or less. determined as follows:95% of the sample means that can be observed are within + 1.96 of the population mean . If , then 1.96 = 1,470.,Example: National Discount, Inc.,Example: National Discount, Inc.,Interval Estimate of the Population Mean: UnknownInterval Estimate of is:$21,100 + $1,470or $19,630 to $22,570We are 95% confident that the interval contains thepopulation mean.,Interval Estimation of a Population Mean: Small-Sample Case (n 30 and use the large-sample interval-estimation procedures. Population is Normally Distributed and is Known The large-sample interval-estimation procedure can be used. Population is Normally Distributed and is Unknown The appropriate interval estimate is based on a probability distribution known as the t distribution.,t Distribution,The t distribution is a family of similar probability distributions. A specific t distribution depends on a parameter known as the degrees of freedom. As the number of degrees of freedom increases, the difference between the t distribution and the standard normal probability distribution becomes smaller and smaller. A t distribution with more degrees of freedom has less dispersion. The mean of the t distribution is zero.,Interval Estimation of a Population Mean: Small-Sample Case (n 30) with Unknown,Interval Estimatewhere 1 - = the confidence coefficientt/2 = the t value providing an area of /2 in the upper tail of a t distributionwith n - 1 degrees of freedoms = the sample standard deviation,Example: Apartment Rents,Interval Estimation of a Population Mean:Small-Sample Case (n 30) with Unknown A reporter for a student newspaper is writing an article on the cost of off-campus housing. A sample of 10 one-bedroom units within a half-mile of campus resulted in a sample mean of $550 per month and a sample standard deviation of $60.Let us provide a 95% confidence interval estimate of the mean rent per month for the population of one-bedroom units within a half-mile of campus. Well assume this population to be normally distributed.,