1、 Slide 1Elementary StatisticsSeventh EditionChapter 6Confidence IntervalsCopyright 2019,2015,2012,Pearson Education,Inc.Copyright 2019,2015,2012,Pearson Education,Inc.Slide 2Chapter Outline 6.1 Confidence Intervals for the Mean Known 6.2 Confidence Intervals for the Mean Unknown 6.3 Confidence Inter
2、vals for Population Proportions 6.4 Confidence Intervals for Variance and Standard DeviationCopyright 2019,2015,2012,Pearson Education,Inc.Slide 3Section 6.1Confidence Intervals for the Mean KnownCopyright 2019,2015,2012,Pearson Education,Inc.Slide 4Section 6.1 Objectives How to find a point estimat
3、e 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 meanCopyright 2019,2015,2012,Pearson Education,Inc.Slide 5Point Estimate for Population muPoint Estimate A
4、single value estimate for a population parameter The most unbiased point estimate of the population mean is the sample mean xCopyright 2019,2015,2012,Pearson Education,Inc.Slide 6Example:Point Estimate for Population muA researcher is collecting data about a college athletic conference and its stude
5、nt-athletes.A random sample of 40 student-athletes is selected 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 confere
6、nce.(Adapted from Penn Schoen Berland)Number of hours19 25 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 21Copyright 2019,2015,2012,Pearson Education,Inc.Slide 7Solution:Point Estimate for Population muThe sample mean of the data is 84
7、221 140 xxn.The point estimate for the mean number of hours spent on required athletic activities by all student-athletes in the conference is about 21.1 hours.Copyright 2019,2015,2012,Pearson Education,Inc.Slide 8Interval EstimateInterval estimate An interval,or range of values,used to estimate a p
8、opulation 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 population mean.Copyright 2019,2015,2012,Pearson Education,Inc.Slide 9Level of ConfidenceLevel of confidence c The probability that the
9、interval estimate contains the population parameter,assuming that the estimation process is repeated a large number of times.Copyright 2019,2015,2012,Pearson Education,Inc.Slide 10Critical ValuesCritical Values Critical values are values that separate sample statistics that are probable from sample
10、statistics that are improbable,or unusual.Copyright 2019,2015,2012,Pearson Education,Inc.Slide 1190 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 correspo
11、nding z-scores are 1.645.Copyright 2019,2015,2012,Pearson Education,Inc.Slide 12Sampling 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 sampleCopyright 2019
12、,2015,2012,Pearson Education,Inc.Slide 13Margin of ErrorMargin of error Sometimes called the maximum error of estimate or error tolerance.The 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 sampl
13、e is random.2.Population is normally distributed or 30n.Copyright 2019,2015,2012,Pearson Education,Inc.Slide 14Example:Finding the Margin of ErrorUse the data in Example 1 and a 95%confidencelevel to find the margin of error for the mean number of hours spent on required athletic activities by all s
14、tudent-athletes in the conference.Assume the population standard deviation is 2.3 hours.Copyright 2019,2015,2012,Pearson Education,Inc.Slide 15Solution:Finding the Margin of Error(1 of 2)Because is known 2 3.,the sample is random,and 4030n,use the formula for E given.The z-score that corresponds to
15、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.Copyright 2019,2015,2012,Pearson Education,Inc.Slide 16Solution:Finding the Margin of Error(2 of 2)Using the values 1 96cz.,2 3.,and 40n,cEz n2 31 9640.0 7.Y
16、ou are 95confident that the margin of error for thepopulation mean is about 0.7 hours.Copyright 2019,2015,2012,Pearson Education,Inc.Slide 17Confidence Intervals for the Population MeanA c-confidence interval for the population mean xExEwhere cEzn The probability that the confidence interval contain
17、s is c,assuming that the estimation process isrepeated a large number of times.Copyright 2019,2015,2012,Pearson Education,Inc.Slide 18Constructing Confidence Intervals for mu(1 of 2)Finding a Confidence Interval for a Population Mean KnownIn WordsIn Symbols1.Verify that known,sampleis random,and eit
18、her the population is normally distributed or 30n.2.Find the sample statistics n andx.xxnCopyright 2019,2015,2012,Pearson Education,Inc.Slide 19Constructing Confidence Intervals for mu(2 of 2)In WordsIn Symbols3.Find the critical value czthat corresponds to the given level of confidence.Use Table 4,
19、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:xExECopyright 2019,2015,2012,Pearson Education,Inc.Slide 20Example:Constructing a Confidence Interval(1 of 3)Use the data in Example 1 to constru
20、ct 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 21.10.7xE21.820.421.8Copyright 2019,2015,2012,Pearson Education,Inc.Slide 21Solution:Const
21、ructing 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.Copyright 2019,2015,2012,Pearson Education,Inc.Slide 22Example:Constructing a Confidence Interval(2 of 3)Use the dat
22、a in Example 1 and technology to construct 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
23、 interval by entering the original data or by using the descriptive statistics.Copyright 2019,2015,2012,Pearson Education,Inc.Slide 23Solution:Constructing a Confidence Interval(2 of 5)SolutionMINITABOne-Sample Z:HoursThe assumed standard deviation2.3=STATCRUNCHOne sample Z confidence interval:Mean
24、of variableStandard deviation2.399confidence interval results:VariablenSample MeanStd.Err.L.LimitU.LimitHours4021.050.3636619320.11326921.986731Copyright 2019,2015,2012,Pearson Education,Inc.Slide 24Solution:Constructing a Confidence Interval(3 of 5)SolutionFrom the displays,a 99%confidence interval
25、 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 that the populationmean number of hours spent on required athletic activities is between 20.1 and 22.0 hours.Copyright 2019,2015,2012,Pearson Education,Inc.Sl
26、ide 25Example: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 is found to be 22.9 years.From past studies,the standard deviation is known to be 1.5 years,and the
27、population is normally distributed.Construct a 90%confidence interval of the population mean age.Copyright 2019,2015,2012,Pearson Education,Inc.Slide 26Solution:Constructing a Confidence Interval(4 of 5)Using 20n,22 9x.,1 5.,and 1 645cz=.,the margin of error at the 90%confidence level is 1.51.6450.6
28、20cEzn Confidence interval:Left Endpoint:xE22.90.622.3Right Endpoint:xE22.90.623.522.323.5Copyright 2019,2015,2012,Pearson Education,Inc.Slide 27Solution: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 yea
29、rs.Copyright 2019,2015,2012,Pearson Education,Inc.Slide 28Interpreting 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 co
30、nfidence interval is created for each sample,approximately 90of these intervals willcontain.Copyright 2019,2015,2012,Pearson Education,Inc.Slide 29Interpreting the Results(2 of 2)The horizontal segments represent 90%confidenceintervals for different samples of the same size.In the long run,9 of ever
31、y 10 such intervals will contain.Copyright 2019,2015,2012,Pearson Education,Inc.Slide 30Finding a Minimum Sample Size to Estimate mu Given a c-confidence level and a margin 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
32、 to the next whole number.If is unknown,you can estimate it using sprovided you have a preliminary sample with at least 30 members.Copyright 2019,2015,2012,Pearson Education,Inc.Slide 31Example:Determining a Minimum Sample SizeThe researcher in Example 1 wants to estimate the mean number of hours sp
33、ent 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?Copyright 2019,2015,2012,Pearson Education,Inc.Slide 32Solution:Determining a Minimu
34、m 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.
