Chap 5 Inferences Based on a Single Sample Estimation ：5章基于单样本估计的推论

1、Statistics for Business and EconomicsChapter 5 Inferences Based on a Single Sample: Estimation with Confidence IntervalsContentIdentifying and Estimating the Target ParameterConfidence Interval for a Population Mean: Normal (z) StatisticConfidence Interval for a Population Mean: Students t-Statistic

2、Large-Sample Confidence Interval for a Population ProportionDetermining the Sample SizeFinite Population Correction for Simple Random SamplingSample Survey DesignsLearning ObjectivesEstimate a population parameter (means or proportion) based on a large sample selected from the populationUse the samp

3、ling distribution of a statistic to form a confidence interval for the population parameterShow how to select the proper sample size for estimating a population parameterThinking ChallengeSuppose youre interested in the average amount of money that students in this class (the population) have on the

4、m. How would you find out?Statistical MethodsStatisticalMethodsEstimationHypothesisTestingInferentialStatisticsDescriptiveStatistics5.1Identifying and Estimatingthe Target ParameterEstimation MethodsEstimationIntervalEstimationPointEstimationTarget ParameterThe unknown population parameter (e.g., me

5、an or proportion) that we are interested in estimating is called the target parameter.Target ParameterDetermining the Target ParameterParameterKey Words of PhraseType of DataMean; averageQuantitativepProportion; percentagefraction; rateQualitativePoint EstimatorA point estimator of a population para

6、meter 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.Point EstimationProvides a single valueBased on observations from one sampleGives no information about how close the value is to the unknown popula

7、tion parameterExample: Sample mean x = 3 is the point estimate of the unknown population meanInterval EstimatorAn 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.Interval EstimationProvides

8、 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% confidence5.2C

9、onfidence Interval for a Population Mean:Normal (z) StatisticEstimation ProcessMean, , is unknownPopulationSampleRandom SampleI am 95% confident that is between 40 & 60.Meanx = 50Key Elements of Interval EstimationSample statistic (point estimate)Confidence intervalConfidence limit (lower)Confidence

10、 limit (upper)A confidence interval provides a range of plausible values for the population parameter.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 IntervalThat is, we

11、 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?If sample measurements yield a value of that falls between the two lines on either

12、 side of , then the interval will contain .Confidence IntervalThe 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.The confidence coefficient is the probability that a randomly selected confi

13、dence 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 percent

14、age.Confidence CoefficientIf our confidence level is 95%, then in the long run, 95% of our confidence intervals will contain and 5% will not.95% Confidence LevelFor a confidence coefficient of 95%, the area in the two tails is .05. To choose a different confidence coefficient we increase or decrease

15、 the area (call it ) assignedto the tails. If we place /2 in each tail and z/2 is the z-value, the confidence interval with coefficient coefficient (1 ) is1.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 condit

16、ion 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 where z/2 is the z-value with an area /2 to its right and The parameter is the standard deviation of the sam

17、pled population, and n is the sample size.Note: When is unknown and n is large (n 30), the confidence interval is approximately equal toLarge-Sample (1 )% Confidence Interval for where s is the sample standard deviation.Thinking ChallengeYoure a Q/C inspector for Gallo. The for 2-liter bottles is .0

18、5 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 literConfidence Interval Solution*5.3Confidence Interval for a Population Mean:Students t-StatisticSmall Sample Unk

19、nownInstead of using the standard normal statisticuse the tstatisticin which the sample standard deviation, s, replaces the population standard deviation, .Students t-StatisticThe t-statistic has a sampling distribution very much like that of the z-statistic: mound-shaped, symmetric, with mean 0.The

20、 primary difference between the sampling distributions of t and z is that the t-statistic is more variable than the z-statistic.Degrees of FreedomThe 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

21、that the t-statistic has (n 1) degrees of freedom (df).ztStudents t Distribution0t (df = 5)Standard Normalt (df = 13)Bell-ShapedSymmetricFatter Tailst - Tablet-valueIf 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

22、 corresponding to 4 df. This entry is t.025 = 2.776. The corresponding standard normal z-score is z.025 = 1.96.Small-SampleConfidence Interval for where ta/2 is based on (n 1) degrees of freedom.Conditions Required for a Valid Small-Sample Confidence Interval for 1.A random sample is selected from t

23、he target population.2.The population has a relative frequency distribution that is approximately normal.Estimation Example Mean ( Unknown)A random sample of n = 25 has = 50 and s = 8. Set up a 95% confidence interval estimate for .Thinking ChallengeYoure a time study analyst in manufacturing. Youve

24、 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? Confidence Interval Solution*x = 3.7s = 3.8987 n = 6, df = n 1 = 6 1 = 5 t.05 = 2.0155.4Large-Sample Confidence Interval for a Population Proportion

25、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 bothThe standard deviation of the sampling distribution of is ; that is, , wher

26、e q = 1p.whereLarge-Sample Confidence Interval forNote: When n is large, can approximate the value of p in the formula for .Conditions Required for a Valid Large-Sample Confidence Interval for p1.A random sample is selected from the target population.2.The sample size n is large. (This condition wil

27、l be satisfied if both . Note that and are simply the number of successes and number of failures, respectively, in the sample.).Estimation Example ProportionA random sample of 400 graduates showed 32 went to graduate school. Set up a 95% confidence interval estimate for p.Thinking ChallengeYoure a p

28、roduction 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? Confidence Interval Solution*where is the adjusted sample proportion ofAdjusted (1 )100% Confidence Interval fo

29、r a Population Proportion, pobservations with the characteristic of interest, x is the number of successes in the sample, and n is the sample size.5.5Determining the Sample SizeIn general, we express the reliability associated with a confidence interval for the population mean by specifying the samp

30、ling 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 ErrorIn order to estimate with a sampling error (SE) and with 100(1 )% confidence, the required sample size is found as follows:

31、Sample Size Determination for 100(1 ) % Confidence Interval for The solution for n is given by the equationSample Size ExampleWhat sample size is needed to be 90% confident the mean is within 5? A pilot study suggested that the standard deviation is 45.In order to estimate p with a sampling error SE

32、 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 pThe solution for n can be written as follows:Note: Always round n up to the nearest integer value.Sample Size ExampleWhat sample s

33、ize is needed to estimate p within .03 with 90% confidence?Thinking ChallengeYou 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. W

34、hat sample size do you use?Sample Size Solution*5.6Finite Population Correction for Simple Random SampleFinite Population Correction FactorIn 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

35、 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.Rule of Thumb for Finite Population Correction FactorUse the finite population correction factor when n/N

36、 .05.Simple Random Sampling with Finite Population of Size NEstimation of the Population MeanEstimated standard error:Approximate 95% confidence interval:Simple Random Sampling with Finite Population of Size NEstimation of the Population ProportionEstimated standard error:Approximate 95% confidence

37、interval:Finite Population Correction Factor ExampleYou want to estimate a population mean, , wherex =115, s =18, N =700, and n = 60. Find an approximate 95% confidence interval for .is greater than .05 use the finite correction factorSinceFinite Population Correction Factor ExampleYou want to estim

38、ate a population mean, , wherex =115, s =18, N =700, and n = 60. Find an approximate 95% confidence interval for .5.7Sample Survey DesignsSimple Random SampleIf 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 se

39、lected, the n elements are said to be a simple random sample.Stratified Random SamplingStratified 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) wh

40、ere the within-stratum response variation is less than the variation within the entire population.Systematic SampleSometimes 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 selectin

41、g 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.Randomized Response SamplingRandomized response sampling is particularly useful when the questions of the pollsters are likely to elicit false answers. One method o