1、统计学基础(英文版第7版)课件les7e_ppt_ADA_0601Chapter Outline 6.1 Confidence Intervals for the Mean Known 6.2 Confidence Intervals for the Mean Unknown 6.3 Confidence Intervals for Population Proportions 6.4 Confidence Intervals for Variance and Standard DeviationSection 6.1Confidence Intervals for the Mean Know
2、nSection 6.1 Objectives How to find a point estimate and a margin of error How to construct and interpret confidence intervals for the population mean when is known How to determine the minimum sample size required when estimating a population meanPoint Estimate for Population muPoint Estimate A sin
3、gle value estimate for a population parameter The most unbiased point estimate of the population mean is the sample mean xExample:Point Estimate for Population muA researcher is collecting data about a college athletic conference and its student-athletes.A random sample of 40 student-athletes is sel
4、ected and their numbers of hours spent on required athletic activities for one week are recorded.Find a point estimate for the population mean,the mean number of hours spent on requiredathletic activities by all student-athletes in the conference.(Adapted from Penn Schoen Berland)Number of hours19 2
5、5 15 21 22 20 20 22 22 21 21 23 22 16 21 18 25 23 23 21 22 24 18 19 23 20 19 19 24 25 17 21 21 25 23 18 22 20 21 21Solution:Point Estimate for Population muThe sample mean of the data is 84221 140 xxn.The point estimate for the mean number of hours spent on required athletic activities by all studen
6、t-athletes in the conference is about 21.1 hours.Interval EstimateInterval estimate An interval,or range of values,used to estimate a population parameter.Before finding a margin of error for an interval estimate,first determine how confident you need to be that your interval estimate contains the p
7、opulation mean.Level of ConfidenceLevel of confidence c The probability that the interval estimate contains the population parameter,assuming that the estimation process is repeated a large number of times.Critical ValuesCritical Values Critical values are values that separate sample statistics that
8、 are probable from sample statistics that are improbable,or unusual.90 Percentages Level of Confidence If the level of confidence is 90,this means thatwe are 90confident that the interval contains thepopulation mean.cz0zcz=.c0 90 .10 05c 10.05c1.645cz1.645cz The corresponding z-scores are 1.645.Samp
9、ling ErrorSampling error The difference between the point estimate and the actual population parameter value.For:the sampling error is the difference x is generally unknown xvaries from sample to sampleMargin of ErrorMargin of error Sometimes called the maximum error of estimate or error tolerance.T
10、he greatest possible distance between the point estimate and the value of the parameter it is estimating.Denoted by E.cxcEzznMargin of error for known1.The sample is random.2.Population is normally distributed or 30n.Example:Finding the Margin of ErrorUse the data in Example 1 and a 95%confidencelev
11、el to find the margin of error for the mean number of hours spent on required athletic activities by all student-athletes in the conference.Assume the population standard deviation is 2.3 hours.Solution:Finding the Margin of Error(1 of 2)Because is known 2 3.,the sample is random,and 4030n,use the f
12、ormula for E given.The z-score that corresponds to a 95%confidencelevel is 1.96.This implies that 95%of the areaunder the standard normal curve falls within 1.96 standard deviations of the mean.Solution:Finding the Margin of Error(2 of 2)Using the values 1 96cz.,2 3.,and 40n,cEz n2 31 9640.0 7.You a
13、re 95confident that the margin of error for thepopulation mean is about 0.7 hours.Confidence Intervals for the Population MeanA c-confidence interval for the population mean xExEwhere cEzn The probability that the confidence interval contains is c,assuming that the estimation process isrepeated a la
14、rge number of times.Constructing Confidence Intervals for mu(1 of 2)Finding a Confidence Interval for a Population Mean KnownIn WordsIn Symbols1.Verify that known,sampleis random,and either the population is normally distributed or 30n.2.Find the sample statistics n andx.xxnConstructing Confidence I
15、ntervals for mu(2 of 2)In WordsIn Symbols3.Find the critical value czthat corresponds to the given level of confidence.Use Table 4,Appendix B.4.Find the margin of error E.cEzn5.Find the left and right endpoints and form the confidence interval.Left endpoint:xERight endpoint:xEInterval:xExEExample:Co
16、nstructing a Confidence Interval(1 of 3)Use the data in Example 1 to construct a 95confidence interval for the mean number of hours spent on required athletic activities by all student-athletes in the conference.Solution:Recall 21 1x.and 0 7E.Left Endpoint 21.1 0.7xE20.4Right Endpoint 21.10.7xE21.82
17、0.421.8Solution:Constructing a Confidence Interval(1 of 5)20 421 8With 95confidence,you can say that thepopulation mean number of hours spent on required athletic activities is between 20.4 and 21.8 hours.Example:Constructing a Confidence Interval(2 of 3)Use the data in Example 1 and technology to c
18、onstruct a 99confidence interval for the meannumber of hours spent on required athletic activities by all student-athletes in the conference.SolutionMinitab and StatCrunch have features that allow you to construct a confidence interval.You can construct a confidence interval by entering the original
19、 data or by using the descriptive statistics.Solution:Constructing a Confidence Interval(2 of 5)SolutionMINITABOne-Sample Z:HoursThe assumed standard deviation2.3=STATCRUNCHOne sample Z confidence interval:Mean of variableStandard deviation2.399confidence interval results:VariablenSample MeanStd.Err
20、.L.LimitU.LimitHours4021.050.3636619320.11326921.986731Solution:Constructing a Confidence Interval(3 of 5)SolutionFrom the displays,a 99%confidence interval for is(20.1,22.0).Note that this interval is rounded to the same number of decimals places as the sample mean.With 99%confidence,you can say th
21、at the populationmean number of hours spent on required athletic activities is between 20.1 and 22.0 hours.Example:Constructing a Confidence Interval(3 of 3)A college admissions director wishes to estimate the mean age of all students currently enrolled.In a random sample of 20 students,the mean age
22、 is found to be 22.9 years.From past studies,the standard deviation is known to be 1.5 years,and the population is normally distributed.Construct a 90%confidence interval of the population mean age.Solution:Constructing a Confidence Interval(4 of 5)Using 20n,22 9x.,1 5.,and 1 645cz=.,the margin of e
23、rror at the 90%confidence level is 1.51.6450.620cEzn Confidence interval:Left Endpoint:xE22.90.622.3Right Endpoint:xE22.90.623.522.323.5Solution:Constructing a Confidence Interval(5 of 5)22 323 5.With 90confidence,you can say that the mean ageof all the students is between 22.3 and 23.5 years.Interp
24、reting the Results(1 of 2)is a fixed number.It is either in the confidenceinterval or not.Incorrect:“There is a 90probability that theactual mean is in the interval(22.3,23.5).”Correct:“If a large number of samples is collected and a confidence interval is created for each sample,approximately 90of
25、these intervals willcontain .Interpreting the Results(2 of 2)The horizontal segments represent 90%confidenceintervals for different samples of the same size.In the long run,9 of every 10 such intervals will contain.Finding a Minimum Sample Size to Estimate mu Given a c-confidence level and a margin
26、of error E,the minimum sample size n needed to estimate the population mean is 2cznE.If n is not a while number,then round n up to the next whole number.If is unknown,you can estimate it using sprovided you have a preliminary sample with at least 30 members.Example:Determining a Minimum Sample SizeT
27、he researcher in Example 1 wants to estimate the mean number of hours spent on required athletic activities by all student-athletes in the conference.How many student-athletes must be included in the sample to be 95%confident that the sample meanis within 0.5 hour of the population mean?Solution:Det
28、ermining a Minimum Sample Size Using 0 95c.,1 96cz.,2 3.(from Example 2),and 0 5E.,you can solve for the minimum samplesize n.2181 29cznE.Because n is not a whole number,round up to 82.So,the researcher needs at least 82 student-athletes in the sample.The researcher has 40 student-athletes,so the sample needs 42 more members.Note that 82 is the minimum number of student-athletes to include in the sample.
