chap 5 inferences based on a single sample estimation ：5章基于单样本估计的推论

2011 Pearson Education, Inc, 2011 Pearson Education, Inc,Statistics for Business and Economics,Chapter 5 Inferences Based on a Single Sample: Estimation with Confidence Intervals, 2011 Pearson Education, Inc,Content,Identifying and Estimating the Target ParameterConfidence Interval for a Population Mean: Normal (z) StatisticConfidence Interval for a Population Mean: Students t-StatisticLarge-Sample Confidence Interval for a Population ProportionDetermining the Sample SizeFinite Population Correction for Simple Random SamplingSample Survey Designs, 2011 Pearson Education, Inc,Learning Objectives,Estimate a population parameter (means or proportion) based on a large sample selected from the populationUse the sampling distribution of a statistic to form a confidence interval for the population parameterShow how to select the proper sample size for estimating a population parameter, 2011 Pearson Education, Inc,Thinking Challenge,Suppose youre interested in the average amount of money that students in this class (the population) have on them. How would you find out?, 2011 Pearson Education, Inc,Statistical Methods,StatisticalMethods,Estimation,HypothesisTesting,InferentialStatistics,DescriptiveStatistics, 2011 Pearson Education, Inc,5.1,Identifying and Estimatingthe Target Parameter, 2011 Pearson Education, Inc,Estimation Methods, 2011 Pearson Education, Inc,Target Parameter,The unknown population parameter (e.g., mean or proportion) that we are interested in estimating is called the target parameter., 2011 Pearson Education, Inc,Target Parameter,Determining the Target ParameterParameterKey Words of PhraseType of DataMean; averageQuantitativepProportion; percentagefraction; rateQualitative, 2011 Pearson Education, Inc,Point Estimator,A point estimator of a population parameter is a rule or formula that tells us how to use the sample data to calculate a single number that can be used as an estimate of the target parameter., 2011 Pearson Education, Inc,Point Estimation,Provides a single valueBased on observations from one sampleGives no information about how close the value is to the unknown population parameter, 2011 Pearson Education, Inc,Interval Estimator,An interval estimator (or confidence interval) is a formula that tells us how to use the sample data to calculate an interval that estimates the target parameter., 2011 Pearson Education, Inc,Interval Estimation,Provides a range of values Based on observations from one sampleGives information about closeness to unknown population parameter Stated in terms of probabilityKnowing exact closeness requires knowing unknown population parameterExample: Unknown population mean lies between 50 and 70 with 95% confidence, 2011 Pearson Education, Inc,5.2,Confidence Interval for a Population Mean:Normal (z) Statistic, 2011 Pearson Education, Inc,Estimation Process,Mean, , is unknown,Population, 2011 Pearson Education, Inc,Key Elements of Interval Estimation,Sample statistic (point estimate),A confidence interval provides a range of plausible values for the population parameter., 2011 Pearson Education, Inc,According to the Central Limit Theorem, the sampling distribution of the sample mean is approximately normal for large samples. Let us calculate the interval estimator:,Confidence Interval,That is, we form an interval from 1.96 standard deviations below the sample mean to 1.96 standard deviations above the mean. Prior to drawing the sample, what are the chances that this interval will enclose , the population mean?, 2011 Pearson Education, Inc,If sample measurements yield a value of that falls between the two lines on either side of , then the interval will contain .,Confidence Interval,The area under the normal curve between these two boundaries is exactly .95. Thus, the probability that a randomly selected interval will contain is equal to .95., 2011 Pearson Education, Inc,The confidence coefficient is the probability that a randomly selected confidence interval encloses the population parameter - that is, the relative frequency with which similarly constructed intervals enclose the population parameter when the estimator is used repeatedly a very large number of times. The confidence level is the confidence coefficient expressed as a percentage.,Confidence Coefficient, 2011 Pearson Education, Inc,If our confidence level is 95%, then in the long run, 95% of our confidence intervals will contain and 5% will not.,95% Confidence Level,For a confidence coefficient of 95%, the area in the two tails is .05. To choose a different confidence coefficient we increase or decrease the area (call it ) assigned,to the tails. If we place /2 in each tail and z/2 is the z-value, the confidence interval with coefficient coefficient (1 ) is, 2011 Pearson Education, Inc,1.A random sample is selected from the target population.2.The sample size n is large (i.e., n 30). Due to the Central Limit Theorem, this condition guarantees that the sampling distribution of is approximately normal. Also, for large n, s will be a good estimator of .,Conditions Required for a Valid Large-SampleConfidence Interval for , 2011 Pearson Education, Inc,where z/2 is the z-value with an area /2 to its right and The parameter is the standard deviation of the sampled population, and n is the sample size.Note: When is unknown and n is large (n 30), the confidence interval is approximately equal to,Large-Sample (1 )% Confidence Interval for ,where s is the sample standard deviation., 2011 Pearson Education, Inc,Thinking Challenge,Youre a Q/C inspector for Gallo. The for 2-liter bottles is .05 liters. A random sample of 100 bottles showed x = 1.99 liters. What is the 90% confidence interval estimate of the true mean amount in 2-liter bottles?,2 liter, 1984-1994 T/Maker Co.,2 liter, 2011 Pearson Education, Inc,Confidence Interval Solution*, 2011 Pearson Education, Inc,5.3,Confidence Interval for a Population Mean:Students t-Statistic, 2011 Pearson Education, Inc,Small Sample Unknown,Instead of using the standard normal statistic,use the tstatistic,in which the sample standard deviation, s, replaces the population standard deviation, ., 2011 Pearson Education, Inc,Students t-Statistic,The t-statistic has a sampling distribution very much like that of the z-statistic: mound-shaped, symmetric, with mean 0.,The primary difference between the sampling distributions of t and z is that the t-statistic is more variable than the z-statistic., 2011 Pearson Education, Inc,Degrees of Freedom,The actual amount of variability in the sampling distribution of t depends on the sample size n. A convenient way of expressing this dependence is to say that the t-statistic has (n 1) degrees of freedom (df)., 2011 Pearson Education, Inc,z,t,Students t Distribution,0,t (df = 5),Standard Normal,t (df = 13),Bell-ShapedSymmetricFatter Tails, 2011 Pearson Education, Inc,t - Table, 2011 Pearson Education, Inc,t-value,If we want the t-value with an area of .025 to its right and 4 df, we look in the table under the column t.025 for the entry in the row corresponding to 4 df. This entry is t.025 = 2.776. The corresponding standard normal z-score is z.025 = 1.96., 2011 Pearson Education, Inc,Small-SampleConfidence Interval for ,where ta/2 is based on (n 1) degrees of freedom., 2011 Pearson Education, Inc,Conditions Required for a Valid Small-Sample Confidence Interval for ,1.A random sample is selected from the target population.2.The population has a relative frequency distribution that is approximately normal., 2011 Pearson Education, Inc,Estimation Example Mean ( Unknown),A random sample of n = 25 has = 50 and s = 8. Set up a 95% confidence interval estimate for ., 2011 Pearson Education, Inc,Thinking Challenge,Youre a time study analyst in manufacturing. Youve recorded the following task times (min.): 3.6, 4.2, 4.0, 3.5, 3.8, 3.1. What is the 90% confidence interval estimate of the population mean task time?, 2011 Pearson Education, Inc,Confidence Interval Solution*,x = 3.7s = 3.8987,n = 6, df = n 1 = 6 1 = 5 t.05 = 2.015, 2011 Pearson Education, Inc,5.4,Large-Sample Confidence Interval for a Population Proportion, 2011 Pearson Education, Inc,The mean of the sampling distribution of is p; that is, is an unbiased estimator of p.,Sampling Distribution of,For large samples, the sampling distribution of is approximately normal. A sample size is considered large if both,The standard deviation of the sampling distribution of is ; that is, , where q = 1p., 2011 Pearson Education, Inc,where,Large-Sample Confidence Interval for,Note: When n is large, can approximate the value of p in the formula for ., 2011 Pearson Education, Inc,Conditions Required for a Valid Large-Sample Confidence Interval for p,1.A random sample is selected from the target population.,2.The sample size n is large. (This condition will be satisfied if both . Note that and are simply the number of successes and number of failures, respectively, in the sample.)., 2011 Pearson Education, Inc,Estimation Example Proportion,A random sample of 400 graduates showed 32 went to graduate school. Set up a 95% confidence interval estimate for p., 2011 Pearson Education, Inc,Thinking Challenge,Youre a production manager for a newspaper. You want to find the % defective. Of 200 newspapers, 35 had defects. What is the 90% confidence interval estimate of the population proportion defective?, 2011 Pearson Education, Inc,Confidence Interval Solution*, 2011 Pearson Education, Inc,where is the adjusted sample proportion of,Adjusted (1 )100% Confidence Interval for a Population Proportion, p,observations with the characteristic of interest, x is the number of successes in the sample, and n is the sample size., 2011 Pearson Education, Inc,5.5,Determining the Sample Size, 2011 Pearson Education, Inc,In general, we express the reliability associated with a confidence interval for the population mean by specifying the sampling error within which we want to estimate with 100(1 )% confidence. The sampling error (denoted SE), then, is equal to the half-width of the confidence interval.,Sampling Error, 2011 Pearson Education, Inc,In order to estimate with a sampling error (SE) and with 100(1 )% confidence, the required sample size is found as follows:,Sample Size Determination for 100(1 ) % Confidence Interval for ,The solution for n is given by the equation, 2011 Pearson Education, Inc,Sample Size Example,What sample size is needed to be 90% confident the mean is within 5? A pilot study suggested that the standard deviation is 45., 2011 Pearson Education, Inc,In order to estimate p with a sampling error SE and with 100(1 )% confidence, the required sample size is found by solving the following equation for n:,Sample Size Determination for 100(1 ) % Confidence Interval for p,The solution for n can be written as follows:,Note: Always round n up to the nearest integer value., 2011 Pearson Education, Inc,Sample Size Example,What sample size is needed to estimate p within .03 with 90% confidence?, 2011 Pearson Education, Inc,Thinking Challenge,You work in Human Resources at Merrill Lynch. You plan to survey employees to find their average medical expenses. You want to be 95% confident that the sample mean is within $50. A pilot study showed that was about $400. What sample size do you use?, 2011 Pearson Education, Inc,Sample Size Solution*, 2011 Pearson Education, Inc,5.6,Finite Population Correction for Simple Random Sample, 2011 Pearson Education, Inc,Finite Population Correction Factor,In some sampling situations, the sample size n may represent 5% or perhaps 10% of the total number N of sampling units in the population. When the sample size is large relative to the number of measurements in the population (see the next slide), the standard errors of the estimators of and p should be multiplied by a finite population correction factor., 2011 Pearson Education, Inc,Rule of Thumb for Finite Population Correction Factor,Use the finite population correction factor when n/N .05., 2011 Pearson Education, Inc,Simple Random Sampling with Finite Population of Size N,Estimation of the Population MeanEstimated standard error:Approximate 95% confidence interval:, 2011 Pearson Education, Inc,Simple Random Sampling with Finite Population of Size N,Estimation of the Population ProportionEstimated standard error:Approximate 95% confidence interval:, 2011 Pearson Education, Inc,Finite Population Correction Factor Example,You want to estimate a population mean, , wherex =115, s =18, N =700, and n = 60. Find an approximate 95% confidence interval for ., 2011 Pearson Education, Inc,Finite Population Correction Factor Example,You want to estimate a population mean, , wherex =115, s =18, N =700, and n = 60. Find an approximate 95% confidence interval for ., 2011 Pearson Education, Inc,5.7,Sample Survey Designs, 2011 Pearson Education, Inc,Simple Random Sample,If n elements are selected from a population in such a way that every set of n elements in the population has an equal probability of being selected, the n elements are said to be a simple random sample., 2011 Pearson Education, Inc,Stratified Random Sampling,Stratified random sampling is used when the sampling units (i.e., the units that are sampled) associated with the population can be physically separated into two or more groups of sampling units (called strata) where the within-stratum response variation is less than the variation within the entire population., 2011 Pearson Education, Inc,Systematic Sample,Sometimes it is difficult or too costly to select random samples. For example, it would be easier to obtain a sample of student opinions at a large university by systematically selecting every hundredth name from the student directory. This type of sample design is called a systematic sample. Although systematic samples are usually easier to select than other types of samples, one difficulty is the possibility of a systematic sampling bias., 2011 Pearson Education, Inc,Randomized Response Sampling,Randomized response sampling is particularly useful when the questions of the pollsters are likely to elicit false answers. One method of coping w