Bài giảng Statistical Techniques in Business and Economics - Chapter 9 Estimation and Confidence Intervals

Chapter 9Estimation and Confidence IntervalsChapter GoalsDefine a point estimator, a point estimate, and desirable properties of a point estimator such as unbiasedness, efficiency, and consistency. Define an interval estimator and an interval estimateDefine a confidence interval, confidence level, margin of error, and a confidence interval estimateConstruct a confidence interval for the population mean when the population standard deviation is knownWhen you have completed this chapter, you will be able to:and...1234Chapter GoalsConstruct a confidence interval for the population mean when the population is normally distributed and the population standard deviation is unknownConstruct a confidence interval for a population proportionDetermine the sample size for attribute and variable sampling7658Construct a confidence interval for the population variance when the population is normally distributedTerminologyPoint Estimateis a single value (statistic) used to estimate a population value (parameter)Confidence Intervalis a range of values within which the population parameter is expected to occurInterval Estimate states the range within which a population parameter probably liesDesirable properties of a point estimator unbiased possible values are concentrated close to the value of the parameterunbiased when the expected value equals the value of the population parameter being estimated. Otherwise, it is biased!values are distributed evenly on both sides of the value of the parameter efficient consistentStandard error of the sample mean is the standard deviation of the sampling distribution of the sample meansIt is computed bywhereis the symbol for the standard error of the sample meanis the standard deviation of the populationnis the size of the sampleTerminologysxsn=ssxStandard Error of the MeansIf  is not known and n > 30, the standard deviation of the sample(s) is used to approximate the population standard deviationn=sxsComputed by 1. The sample size, n 2. The variability in the population, usually estimated by s 3. The desired level of confidenceFactors that determine the width of a confidence interval are:IN GENERAL, A confidence interval for a mean is computed by:Constructing Confidence IntervalsInterpretingz±xnsα/2Interpreting Confidence IntervalsThe GlobeSuppose that you read that“the average selling price of a family home in York Region is $200 000 +/- $15000 at 95% confidence!”This meanswhat?Interpreting Confidence IntervalsIn statistical terms, this means:that we are 95% sure that the interval estimate obtained contains the value of the population mean. Lower confidence limit is $185 000 Upper confidence limit is $215 000The Globe“the average selling price of a family home in York Region is $200 000 +/- $15 000 at 95% confidence!”Also($200 000 - $15 000)($200 000 + $15 000)Interpreting Confidence IntervalsThe Globe“the mean time to sell a family home in York Region is 40 days. Your newspaper also reports thatYou select a random sample of 36 homes sold during the past year, and determine a 90% confidence interval estimate for the population mean to be (31-39) days.Do your sample results support the paper’s claim?Interpreting Confidence IntervalsYou select a random sample of 36 homes sold during the past year, and determine a 90% confidence interval estimate for the population mean to be (31-39) days.There is a 10% chance (100%-90%) that the interval estimate does not contain the value of the population mean! Lower confidence limit is 31 days Upper confidence limit is 39 daysOur evidence does not support the statement made by the newspaper, i.e., the population mean is not 40 days, when using a 90% interval estimate 3139i.e.  = 0.10.05.0590% 10% chance of falling outside this intervalInterpreting Confidence Intervals90% Confidence Intervalor, focus on tail areas  is the probability of a value falling outside the confidence intervalTry it!Find the appropriate value of z:0 1.75Locate Area on the normal curve1This is a 92% confidence interval2Look up a= 0.46 in Table to get the corresponding z-scoreSearch in the centre of the table for the area of 0.46Z = +/- 1.75 -1.750.92Constructing Confidence Intervals95% C.I. for the mean:Common Confidence Intervals99% C.I. for the mean:About 95% of the constructed intervals will contain the parameter being estimated. Also, 95% of the sample means for a specified sample size will lie within 1.96 standard deviations of the hypothesized population mean.z±xnsα/2Interval EstimatesUse the z tablez±xnsα/2Use the t-tablet±xnsα/2If the population standard deviation is known or n > 30If the population standard deviation is unknown and n.05, use the correction factor)127002502700)(2504.(645. 18.1--±04.08.1±90% CL =FormulaN -nN - 1nsZα/2X±Selecting the Sample Size 1. The degree of confidence selected 2. The maximum allowable error 3. The variation in the populationthat determine the sample size are:FactorsE is the allowable errorZ is the z-score for the chosen level of confidenceS is the sample deviation of the pilot surveySelecting the Sample Sizeữ2ứửỗốổEszα/2n =Formulawhere 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. Selecting the Sample SizeHow large a sample is required?QuestionSelecting the Sample Size 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. = (10.32)2= 106.5 A minimum of 107 homes must be sampled.90% CL =25.00202.58ỗốổã=ỗốổSolutionFormulaữ2ứửỗốổEszα/2Selecting the Sample SizeThe Kennel Club wants to estimate the proportion of children that have a dog as a pet. Assume a 95% level of confidence and that the club estimates that 30% of the children have a dog as a pet. If the club wants the estimate to be within 3% of the population proportion, how many children would they need to contact? QuestionThe Kennel Club wants to estimate the proportion of children that have a dog as a pet. Assume a 95% level of confidence and that the club estimates that 30% of the children have a dog as a pet. QuestionSelecting the Sample SizeNewFormulanppZE=-ổốỗửứữ()12 203.96.1)3.1(3.ữứửỗốổ-=()233.65)21(.=n = 896.4 A minimum of 897 children must be sampled.Test your learning www.mcgrawhill.ca/college/lindClick onOnline Learning Centrefor quizzesextra contentdata setssearchable glossaryaccess to Statistics Canada’s E-Stat dataand much more!This completes Chapter 9