大二上学习统计chap009

1、The McGraw-Hill Companies, Inc. 2008McGraw-Hill/IrwinEstimation and Confidence Intervals（估计与置信区间）（估计与置信区间）Chapter 99-2GOALS1.Define a point estimate（点估计）.2.Define level of confidence（置信水平）.3.Construct a confidence interval for the population mean when the population standard deviation is known.9-3GO

2、ALS4. Construct a confidence interval for a population mean when the population standard deviation is unknown.5. Construct a confidence interval for a population proportion（总体比例）.6. Determine the sample size for attribute and variable sampling.9-4Point Estimates and Confidence Intervals for a MeanlW

3、e will consider two cases, where （1）The population standard deviation ( ) is known,（2）The population standard deviation is unknown. In this case we substitute the sample standard deviation (s) for the population standard deviation ( ).9-5Population Standard Deviation () KnownlA point estimate is a s

4、ingle statistic used to estimate a population parameter. Suppose Best Buy, Inc., wants to estimate the mean age of buyers of HD plasma televisions(高清等离子电视）. It selects a random sample of 40 recent purchasers, determines the age of each purchaser, and computes the mean age of the buyers in the sample

5、. lThe mean of this sample is a point estimate of the mean of the population.9-6Population Standard Deviation () KnownlA point estimate is the statistic (single value), computed from sample information, which is used to estimate the population parameter.is a point estimate of the population mean, ;

6、p, a sample proportion, is a point estimate of , the population proportion; and s, the sample standard deviation, is a point estimate of the population standard deviation.9-7Population mean9-8A point estimate, however, tells only part of the story. While we expect the point estimate to be close to t

7、he population parameter, we would like to measure how close it really is. A confidence interval serves this purpose.9-9Population Standard Deviation () KnownlA confidence interval estimate is a range of values constructed from sample data so that the population parameter is likely to occur within th

8、at range at a specified probability. The specified probability is called the level of confidence（置信水平）.9-109-11A confidence interval estimatelFor reasonably large samples, the results of the central limit theorem allow us to state the following:1. Ninety-five percent of the sample means selected fro

9、m a population will be within1.96 standard deviations of the population mean .2. Ninety-nine percent of the sample means will lie within 2.58 standard deviations of the population mean.9-12Finding z-value for 95% Confidence IntervalThe area betweenZ = -1.96 and z= +1.96is 0.959-13Point Estimates and

10、 Confidence Intervals for a Mean Known9-14Point Estimates and Confidence Intervals for a Mean Knownla 95 percent confidence interval :la 99 percent confidence interval :9-15Point Estimates and Confidence Intervals for a Mean Knownla 90 percent confidence interval9-16Factors Affecting Confidence Inte

11、rval EstimatesThe factors that determine the width of a confidence interval are:1.The sample size, n.2.The variability in the population , usually estimated by s.3.The desired level of confidence. 9-17Example: Confidence Interval for a Mean KnownThe American Management Association wishes to have inf

12、ormation on the mean income of middle managers in the retail industry. A random sample of 256 managers reveals a sample mean of $45,420. The standard deviation of this population is $2,050. The association would like answers to the following questions:1.What is the population mean? What is a reasona

13、ble value to use as an estimate of the population mean?2.What is a reasonable range of values for the population mean? 3.What do these results mean?9-18Example: Confidence Interval for a Mean KnownThe American Management Association wishes to have information on the mean income of middle managers in

14、 the retail industry. A random sample of 256 managers reveals a sample mean of $45,420. The standard deviation of this population is $2,050. The association would like answers to the following questions:1. What is the population mean? What is a reasonable value to use as an estimate of the populatio

15、n mean?In this case, we do not know. We do know the sample mean is $45,420. Hence, our best estimate of the unknown population value is the corresponding sample statistic. The sample mean of $45,420 is a point estimate of the unknown population mean.9-19Example: Confidence Interval for a Mean KnownT

16、he American Management Association wishes to have information on the mean income of middle managers in the retail industry(零售业）. A random sample of 256 managers reveals a sample mean of $45,420. The standard deviation of this population is $2,050. The association would like answers to the following

17、questions:2. What is a reasonable range of values for the population mean? Suppose the association decides to use the 95 percent level of confidence:The confidence limits are $45,169 and $45,6719-20Example: Confidence Interval for a Mean KnownThe American Management Association wishes to have inform

18、ation on the mean income of middle managers in the retail industry. A random sample of 256 managers reveals a sample mean of $45,420. The standard deviation of this population is $2,050. The association would like answers to the following questions:3. What do these results mean, i.e. what is the int

19、erpretation of the confidence limits $45,169 and $45,671?If we select many samples of 256 managers, and for each sample we compute the mean and then construct a 95 percent confidence interval, we could expect about 95 percent of these confidence intervals to contain the population mean. Conversely,

20、about 5 percent of the intervals would not contain the population mean annual income, 9-21Interval Estimates - InterpretationFor a 95% confidence interval about 95% of the similarly constructed intervals will contain the parameter being estimated. Also 95% of the sample means for a specified sample

21、size will lie within 1.96 standard deviations of the hypothesized population9-22Example: Confidence Interval for a Mean Unknown9-23Example: Confidence Interval for a Mean Unknown9-24Characteristics of the t-distribution1. It is, like the z distribution, a continuous distribution.2. It is, like the z

22、 distribution, bell-shaped and symmetrical.3. There is not one t distribution, but rather a family of t distributions. All t distributions have a mean of 0, but their standard deviations differ according to the sample size, n. 4. The t distribution is more spread out and flatter at the center than t

23、he standard normal distribution As the sample size increases, however, the t distribution approaches the standard normal distribution, 9-25Comparing the z and t Distributions when n is small, 95% Confidence Level9-26Confidence Interval Estimates for the MeanUse Z-distributionIf the population standa

24、rd deviation is known or the sample is at least than 30.Use t-distributionIf the population is normal and population standard deviation is unknown 9-27When to Use the z or t Distribution for Confidence Interval Computation9-28Confidence Interval for the Mean Example using the t-distributionA tire ma

25、nufacturer wishes to investigate the tread life of its tires. A sample of 10 tires driven 50,000 miles revealed a sample mean of 0.32 inch of tread remaining with a standard deviation of 0.09 inch. Construct a 95 percent confidence interval for the population mean. Would it be reasonable for the man

26、ufacturer to conclude that after 50,000 miles the population mean amount of tread remaining is 0.30 inches?9-29Students t-distribution Table9-30The manager of the Inlet Square Mall, near Ft. Myers, Florida, wants to estimate the mean amount spent per shopping visit by customers. A sample of 20 custo

27、mers reveals the following amounts spent.Confidence Interval Estimates for the Mean Using Minitab9-31Confidence Interval Estimates for the Mean By Formula9-32Confidence Interval Estimates for the Mean Using Minitab9-33Confidence Interval Estimates for the Mean Using Excel9-34Using the Normal Distrib

28、ution to Approximate the Binomial DistributionTo develop a confidence interval for a proportion, we need to meet the following assumptions.1. The binomial conditions, discussed in Chapter 6, have been met. These conditions are:a. The sample data is the result of counts.b. There are only two possible

29、 outcomes. c. The probability of a success remains the same from one trial to the next.d. The trials are independent. 2. The values n and n(1-) 5. This condition allows us to invoke the central limit theorem and employ the standard normal distribution, to compute a confidence interval.9-35Confidence

30、 Interval for a Population ProportionlPROPORTION The fraction, ratio, or percent indicating the part of the sample or the population having a particular trait of interest.lSAMPLE PROPORTION 9-36Confidence Interval for a Population ProportionlTo develop a confidence interval for a proportion, we need

31、 to meet the following assumptions.l1. The binomial conditions, discussed in Chapter 6, have been met. These conditions are:la. The sample data is the result of counts. We count the number of successes in a fixed number of trials.9-37Confidence Interval for a Population Proportionlb. There are only

32、two possible outcomes. We usually label one of the outcomes a “success” and the other a “failure.”lc. The probability of a success remains the same from one trial to the next.ld. The trials are independent. This means the outcome on one trial does not affect the outcome on another.9-38Confidence Int

33、erval for a Population Proportionl2. The values nand n(1 ) should both be greater than or equal to 5. This condition allows us to invoke the central limit theorem and employ the standard normal distribution, that is, z, to complete a confidence interval.9-39Confidence Interval for a Population Propo

34、rtion The confidence interval for a population proportion is estimated by:9-40Confidence Interval for a Population Proportion- ExampleThe union representing the Bottle Blowers of America (BBA) is considering a proposal to merge with the Teamsters Union. According to BBA union bylaws, at least three-

35、fourths of the union membership must approve any merger. A random sample of 2,000 current BBA members reveals 1,600 plan to vote for the merger proposal. What is the estimate of the population proportion? Develop a 95 percent confidence interval for the population proportion. Basing your decision on

36、 this sample information, can you conclude that the necessary proportion of BBA members favor the merger? Why?9-41Self-Review 93lA market survey was conducted to estimate the proportion of homemakers who would recognize the brand name of a cleanser based on the shape and the color of the container.

37、Of the 1,400 homemakers sampled, 420 were able to identify the brand by name.9-42Self-Review 93l(a) Estimate the value of the population proportion.l(b) Develop a 99 percent confidence interval for the population proportion.l(c) Interpret your findings. 9-43Finite-Population Correction FactorlThe po

38、pulations we have sampled so far have been very large or infinite. lWhat if the sampled population is not very large? lWe need to make some adjustments in the way we compute the standard error of the sample means and the standard error of the sample proportions.9-44Finite-Population Correction Facto

39、rlA population that has a fixed upper bound is finite. l A finite population can be rather small; It can also be very large.lFor a finite population, where the total number of objects or individuals is N and the number of objects or individuals in the sample is n, we need to adjust the standard erro

40、rs in the confidence interval formulas. 9-45Finite-Population Correction FactorlThis adjustment is called the finite-population correction factor. It is often shortened to FPC and is:9-46Finite-Population Correction FactorlA population that has a fixed upper bound is said to be finite.lFor a finite

41、population, where the total number of objects is N and the size of the sample is n, the following adjustment is made to the standard errors of the sample means and the proportion:lHowever, if n/N 30)Use the formula below to compute the confidence interval:9-50CI For Mean with FPC - Example9-51Self-R

42、eview 94lThe same study of church contributions in Scandia revealed that 15 of the 40 families sampled attend church regularly. Construct the 95 percent confidence interval for the proportion of families attending church regularly. Should the finite-population correction factor be used? Why or why n

43、ot?9-52Self-Review 949-53Selecting a Sample Size (n)There are 3 factors that determine the size of a sample, none of which has any direct relationship to the size of the population. They are:lThe degree of confidence selected. lThe maximum allowable error.lThe variation in the population. 9-54Select

44、ing a Sample Size (n)lThe first factor is the level of confidence. Those conducting the study select the level of confidence. The 95 percent and the 99 percent levels of confidence are the most common, but any value between 0 and 100 percent is possible. The 95 percent level of confidence correspond

45、s to a z value of 1.96, and a 99 percent level of confidence corresponds to a z value of 2.58. The higher the level of confidence selected, the larger the size of the corresponding sample.9-55Selecting a Sample Size (n)lThe second factor is the allowable error. The maximum allowable error, designate

46、d as E, is the amount that is added and subtracted to the sample mean (or sample proportion) to determine the endpoints of the confidence interval. It is the amount of error those conducting the study are willing to tolerate. It is also one-half the width of the corresponding confidence interval. A

47、small allowable error will require a larger sample. A large allowable error will permit a smaller sample.9-56Selecting a Sample Size (n)lThe third factor in determining the size of a sample is the population standard deviation. If the population is widely dispersed, a large sample is required. On th

48、e other hand, if the population is concentrated (homogeneous), the required sample size will be smaller. However, it may be necessary to use an estimate for the population standard deviation. 9-57Selecting a Sample Size (n)lthree suggestions for finding that estimate :1. Use a comparable study. 2. U

49、se a range-based approach. 3. Conduct a pilot study. lWe can express the interaction among these three factors and the sample size in the following formula.9-58Sample Size Determination for a VariablelTo find the sample size for a variable:9-59A student in public administration wants to determine th

50、e mean amount members of city councils in large cities earn per month as remuneration for being a council member. The error in estimating the mean is to be less than $100 with a 95 percent level of confidence. The student found a report by the Department of Labor that estimated the standard deviatio

51、n to be $1,000.What is the required sample size?Given in the problem:lE, the maximum allowable error, is $100lThe value of z for a 95 percent level of confidence is 1.96, lThe estimate of the standard deviation is $1,000. Sample Size Determination for a Variable-Example 19-60A student in public admi

52、nistration wants to determine the mean amount members of city councils in large cities earn per month as remuneration for being a council member. The error in estimating the mean is to be less than $100 with a 99 percent level of confidence. The student found a report by the Department of Labor that

53、 estimated the standard deviation to be $1,000.What is the required sample size?Given in the problem:lE, the maximum allowable error, is $100lThe value of z for a 99 percent level of confidence is 2.58, lThe estimate of the standard deviation is $1,000. Sample Size Determination for a Variable-Examp

54、le 29-61Sample Size for ProportionslThe formula for determining the sample size in the case of a proportion is:9-62Another ExampleThe American Kennel Club wanted to estimate the proportion of children that have a dog as a pet. If the club wanted the estimate to be within 3% of the population proport

55、ion, how many children would they need to contact? Assume a 95% level of confidence and that the club estimated that 30% of the children have a dog as a pet. 9-63Another ExampleA study needs to estimate the proportion of cities that have private refuse collectors. The investigator wants the margin o

56、f error to be within .10 of the population proportion, the desired level of confidence is 90 percent, and no estimate is available for the population proportion. What is the required sample size?9-64Self-Review 95lWill you assist the college registrar in determining how many transcripts to study? Th

57、e registrar wants to estimate the arithmetic mean grade point average (GPA) of all graduating seniors during the past 10 years. GPAs range between 2.0 and 4.0. The mean GPA is to be estimated within plus or minus .05 of the population mean. The standard deviation is estimated to be 0.279. Use the 99

58、 percent level of confidence. 9-65Self-Review 959-66Chapter SummarylI. A point estimate is a single value (statistic) used to estimate a population value (parameter).lII. A confidence interval is a range of values within which the population parameter is expected to occur.9-67Chapter SummarylA. The factors that determine the width of a confidence interval for a mean are:l1. The number of observations in the sample, n.l2. The variability in the population, usually es