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1、统计学基础(英文版第7版)课件les7e_ppt_ADA_0601,统计学基础(英文版第7版)课件les7e_ppt_AD,Chapter Outline,6.1 Confidence Intervals for the Mean,6.2 Confidence Intervals for the Mean,6.3 Confidence Intervals for Population Proportions6.4 Confidence Intervals for Variance and Standard Deviation,Chapter Outline6.1 Confidence,Sect
2、ion 6.1,Confidence Intervals for the Mean,Section 6.1Confidence Interval,Section 6.1 Objectives,How to find a point estimate and a margin of errorHow to construct and interpret confidence intervals for the population mean when,is known,How to determine the minimum sample size required when estimatin
3、g a population mean,Section 6.1 ObjectivesHow to f,Point Estimate for Population mu,Point EstimateA single value estimate for a population parameterThe most unbiased point estimate of the population mean,is the sample mean,Point Estimate for Population,Example: Point Estimate for Population mu,A res
4、earcher is collecting data about a college athletic conference and its student-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 ho
5、urs spent on required,athletic activities by all student-athletes in the conference. (Adapted from Penn Schoen Berland),Number of hours,19 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 21,Example: Point Estimate for Po,Solution: Poi
6、nt Estimate for Population mu,The sample mean of the data is,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.,Solution: Point Estimate for P,Interval Estimate,Interval estimate An interval, or range o
7、f 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 population mean,.,Interval EstimateInterval esti,Level of Confidence,Level of confidence c The probabili
8、ty that the interval estimate contains the population parameter, assuming that the estimation process is repeated a large number of times.,Level of ConfidenceLevel of co,Critical Values,Critical ValuesCritical values are values that separate sample statistics that are probable from sample statistics
9、 that are improbable, or unusual.,Critical ValuesCritical Values,90 Percentages Level of Confidence,If the level of confidence is, this means that,we are,confident that the interval contains the,population mean,.,The corresponding z-scores are,.,90 Percentages Level of Confid,Sampling Error,Sampling
10、 error The difference between the point estimate and the actual population parameter value.For,:,the sampling error is the difference,is generally unknown,varies from sample to sample,Sampling ErrorSampling error :,Margin of Error,Margin of error Sometimes called the maximum error of estimate or err
11、or tolerance.The greatest possible distance between the point estimate and the value of the parameter it is estimating. Denoted by E.,Margin of error for,The sample is random.Population is normally distributed or,.,Margin of ErrorMargin of error,Example: Finding the Margin of Error,Use the data in E
12、xample 1 and a,confidence,level 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.,Example: Finding the Margin of,Solution: Finding the Margin of Error (1 of
13、2),Because,is known, the sample is random,and, use the formula for E given.,The z-score that corresponds to a,confidence,level is 1.96. This implies that,of the area,under the standard normal curve falls within 1.96 standard deviations of the mean.,Solution: Finding the Margin o,Solution: Finding th
14、e Margin of Error (2 of 2),Using the values, and,You are,confident that the margin of error for the,population mean is about 0.7 hours.,Solution: Finding the Margin o,Confidence Intervals for the Population Mean,A c-confidence interval for the population mean,where,The probability that the confidenc
15、e interval contains,is c, assuming that the estimation process is,repeated a large number of times.,Confidence Intervals for the P,Constructing Confidence Intervals for mu (1 of 2),Finding a Confidence Interval for a Population Mean,In Words,In Symbols,Verify that,known, sample,is random, and either
16、 the population is normally distributed or,.,Find the sample statistics n and,.,Constructing Confidence Interv,Constructing Confidence Intervals for mu (2 of 2),In Words,In Symbols,Find the critical value,that corresponds to the given level of confidence.,Use Table 4, Appendix B.,Find the margin of
17、error E.,Find the left and right endpoints and form the confidence interval.,Left endpoint:,Right endpoint:,Interval:,Constructing Confidence Interv,Example: Constructing a Confidence Interval (1 of 3),Use the data in Example 1 to construct a,confidence interval for the mean number of hours spent on
18、 required athletic activities by all student-athletes in the conference.,Solution: Recall,and,Left Endpoint,Right Endpoint,Example: Constructing a Confid,Solution: Constructing a Confidence Interval (1 of 5),With,confidence, you can say that the,population mean number of hours spent on required athl
19、etic activities is between 20.4 and 21.8 hours.,Solution: Constructing a Confi,Example: Constructing a Confidence Interval (2 of 3),Use the data in Example 1 and technology to construct a,confidence interval for the mean,number of hours spent on required athletic activities by all student-athletes i
20、n 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 data or by using the descriptive statistics.,Example: Constructing a Confid,Solution: Constructing a Confidence Interval
21、(2 of 5),Solution,MINITAB,One-Sample Z: Hours,The assumed,STATCRUNCH,One sample Z confidence interval:,: Mean of variable,confidence interval results:,Solution: Constructing a Confi,Solution: Constructing a Confidence Interval (3 of 5),SolutionFrom the displays, a,confidence interval for,is,(20.1, 2
22、2.0). Note that this interval is rounded to the same number of decimals places as the sample mean.,With,confidence, you can say that the population,mean number of hours spent on required athletic activities is between 20.1 and 22.0 hours.,Solution: Constructing a Confi,Example: Constructing a Confid
23、ence 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 population is normally distr
24、ibuted. Construct a,confidence interval of the population mean age.,Example: Constructing a Confid,Solution: Constructing a Confidence Interval (4 of 5),Using, and,the margin of error at the,confidence level is,Confidence interval:,Left Endpoint:,Right Endpoint:,Solution: Constructing a Confi,Soluti
25、on: Constructing a Confidence Interval (5 of 5),With,confidence, you can say that the mean age,of all the students is between 22.3 and 23.5 years.,Solution: Constructing a Confi,Interpreting the Results (1 of 2),is a fixed number. It is either in the confidence,interval or not.,Incorrect: “There is
26、a,probability that the,actual 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,of these intervals will,contain,.,Interpreting the Results (1 of,Interpreting the Results (2 of 2),The horizont
27、al segments represent,confidence,intervals for different samples of the same size.In the long run, 9 of every 10 such intervals will contain,.,Interpreting the Results (2 of,Finding a Minimum Sample Size to Estimate mu,Given a c-confidence level and a margin of error E, the minimum sample size n nee
28、ded to estimate the population mean,is,.,If n is not a while number, then round n up to the next whole number.If,is unknown, you can estimate it using s,provided you have a preliminary sample with at least 30 members.,Finding a Minimum Sample Size,Example: Determining a Minimum Sample Size,The resea
29、rcher 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,confident that the sample mean,is within 0.5 hour of the population mean?,Example: Determining
30、 a Minimum,Solution: Determining a Minimum Sample Size,Using,(from Example 2),and, you can solve for the minimum sample,size n.,.,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.,Solution: Determining a Minimu,感谢聆听,感谢聆听,
