1、9-1,本資料來源,Estimation and Confidence Intervals（估計與置信區(qū)間）,Chapter 9,9-3,GOALS,Define a point estimate（點估計）. Define level of confidence（置信水平）. Construct a confidence interval for the population mean when the population standard deviation is known.,9-4,GOALS,4. Construct a confidence interval for a popul

2、ation mean when the population standard deviation is unknown. 5. Construct a confidence interval for a population proportion（總體比例）. 6. Determine the sample size for attribute and variable sampling.,9-5,Point Estimates and Condence Intervals for a Mean,We will consider two cases, where （1）The populat

3、ion 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 ( ).,9-6,Population Standard Deviation () Known,A point estimate is a single statistic used to estimate a populatio

4、n parameter. Suppose Best Buy, Inc., wants to estimate the mean age of buyers of HD plasma televisions. 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. The mean of this sample is a point estimate of the

5、mean of the population.,9-7,Population Standard Deviation () Known,A point estimate is the statistic (single value), computed from sample information, which is used to estimate the population parameter. The sample mean is a point estimate of the population mean, ; p, a sample proportion, is a point

6、estimate of , the population proportion; and s, the sample standard deviation, is a point estimate of the population standard deviation.,9-8,Population Standard Deviation () Known,A confidence interval estimate is a range of values constructed from sample data so that the population parameter is lik

7、ely to occur within that range at a specified probability. The specified probability is called the level of confidence（置信水平）.,9-9,A confidence interval estimate,For reasonably large samples, the results of the central limit theorem allow us to state the following: 1. Ninety-ve percent of the sample

8、means selected from 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.,9-10,Factors Affecting Confidence Interval Estimates,The factors that determine the width of a

9、confidence interval are: 1.The sample size, n. 2.The variability in the population , usually estimated by s. 3.The desired level of confidence.,9-11,Finding z-value for 95% Confidence Interval,The area between Z = -1.96 and z= +1.96 is 0.95,9-12,Interval Estimates - Interpretation,For a 95% confiden

10、ce interval about 95% of the similarly constructed intervals will contain the parameter being estimated. Also 95% of the sample means for a specified sample size will lie within 1.96 standard deviations of the hypothesized population,9-13,Point Estimates and Confidence Intervals for a Mean Known,9-1

11、4,Point Estimates and Confidence Intervals for a Mean Known,a 95 percent condence interval : a 99 percent condence interval :,9-15,Point Estimates and Confidence Intervals for a Mean Known,a 90 percent condence interval,9-16,Example: Confidence Interval for a Mean Known,The American Management Assoc

12、iation wishes to have information 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 populati

13、on mean? What is a reasonable 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?,9-17,Example: Confidence Interval for a Mean Known,The American Management Association wishes to have information on the mea

14、n 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 reasonable value to use

15、as an estimate of the population 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.,9-18,Example: Con

16、fidence Interval for a Mean Known,The American Management Association wishes to have information 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

17、like answers to the following 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 $45,671,9-19,Example: Confidence Interval for a Mean Known,The American Managemen

18、t Association wishes to have information 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 thes

19、e results mean, i.e. what is the interpretation 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 cont

20、ain the population mean. Conversely, about 5 percent of the intervals would not contain the population mean annual income, ,9-20,Example: Confidence Interval for a Mean Unknown,9-21,Example: Confidence Interval for a Mean Unknown,9-22,Characteristics of the t-distribution,1. It is, like the z distri

21、bution, a continuous distribution. 2. It is, like the z 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 distribu

22、tion is more spread out and flatter at the center than the standard normal distribution As the sample size increases, however, the t distribution approaches the standard normal distribution,9-23,Comparing the z and t Distributions when n is small, 95% Confidence Level,9-24,Confidence Interval Estima

23、tes for the Mean,Use Z-distribution If the population standard deviation is known or the sample is at least than 30.,Use t-distribution If the population standard deviation is unknown and the sample is less than 30.,9-25,When to Use the z or t Distribution for Confidence Interval Computation,9-26,Co

24、nfidence Interval for the Mean Example using the t-distribution,A tire manufacturer 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 confid

25、ence interval for the population mean. Would it be reasonable for the manufacturer to conclude that after 50,000 miles the population mean amount of tread remaining is 0.30 inches?,9-27,Students t-distribution Table,9-28,The manager of the Inlet Square Mall, near Ft. Myers, Florida, wants to estimat

26、e the mean amount spent per shopping visit by customers. A sample of 20 customers reveals the following amounts spent.,Confidence Interval Estimates for the Mean Using Minitab,9-29,Confidence Interval Estimates for the Mean By Formula,9-30,Confidence Interval Estimates for the Mean Using Minitab,9-3

27、1,Confidence Interval Estimates for the Mean Using Excel,9-32,Using the Normal Distribution to Approximate the Binomial Distribution,To 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

28、conditions are: a. The sample data is the result of counts. b. There are only two possible 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 a

29、nd employ the standard normal distribution, to compute a confidence interval.,9-33,Confidence Interval for a Population Proportion,PROPORTION The fraction, ratio, or percent indicating the part of the sample or the population having a particular trait of interest. SAMPLE PROPORTION,9-34,Confidence I

30、nterval for a Population Proportion,To develop a condence 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. We count the number of successes in a x

31、ed number of trials.,9-35,Confidence Interval for a Population Proportion,b. There are only two possible outcomes. We usually label one of the outcomes a “success” and the other a “failure.” c. The probability of a success remains the same from one trial to the next. d. The trials are independent. T

32、his means the outcome on one trial does not affect the outcome on another.,9-36,Confidence Interval for a Population Proportion,2. 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

33、, that is, z, to complete a condence interval.,9-37,Confidence Interval for a Population Proportion,The confidence interval for a population proportion is estimated by:,9-38,Confidence Interval for a Population Proportion- Example,The union representing the Bottle Blowers of America (BBA) is conside

34、ring a proposal to merge with the Teamsters Union. According to BBA union bylaws, at least three-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

35、? Develop a 95 percent confidence interval for the population proportion. Basing your decision on this sample information, can you conclude that the necessary proportion of BBA members favor the merger? Why?,9-39,Self-Review 93,A market survey was conducted to estimate the proportion of homemakers w

36、ho would recognize the brand name of a cleanser based on the shape and the color of the container. Of the 1,400 homemakers sampled, 420 were able to identify the brand by name.,9-40,Self-Review 93,(a) Estimate the value of the population proportion. (b) Develop a 99 percent condence interval for the

37、 population proportion. (c) Interpret your ndings.,9-41,Finite-Population Correction Factor,The populations we have sampled so far have been very large or innite. What if the sampled population is not very large? We need to make some adjustments in the way we compute the standard error of the sample

38、 means and the standard error of the sample proportions.,9-42,Finite-Population Correction Factor,A population that has a xed upper bound is nite. A nite population can be rather small; It can also be very large. For a nite population, where the total number of objects or individuals is N and the nu

39、mber of objects or individuals in the sample is n, we need to adjust the standard errors in the condence interval formulas.,9-43,Finite-Population Correction Factor,This adjustment is called the nite-population correction factor. It is often shortened to FPC and is:,9-44,Finite-Population Correction

40、 Factor,A population that has a fixed upper bound is said to be finite. For a finite 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: However, if n/N .05, the finite-pop

41、ulation correction factor may be ignored.,9-45,Effects on FPC when n/N Changes,Observe that FPC approaches 1 when n/N becomes smaller,9-46,Confidence Interval Formulas for Estimating Means and Proportions with Finite Population Correction,C.I. for the Mean (),C.I. for the Proportion (),C.I. for the

42、Mean (),9-47,CI For Mean with FPC - Example,There are 250 families in Scandia, Pennsylvania. A random sample of 40 of these families revealed the mean annual church contribution was $450 and the standard deviation of this was $75. Develop a 90 percent confidence interval for the population mean. Int

43、erpret the confidence interval.,Given in Problem: N 250 n 40 s - $75 Since n/N = 40/250 = 0.16, the finite population correction factor must be used. The population standard deviation is not known therefore use the t-distribution (may use the z-dist since n30) Use the formula below to compute the co

44、nfidence interval:,9-48,CI For Mean with FPC - Example,9-49,Self-Review 94,The same study of church contributions in Scandia revealed that 15 of the 40 families sampled attend church regularly. Construct the 95 percent condence interval for the proportion of families attending church regularly. Shou

45、ld the nite-population correction factor be used? Why or why not?,9-50,Self-Review 94,9-51,Selecting 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: The degree of confidence selected. The ma

46、ximum allowable error. The variation in the population.,9-52,Selecting a Sample Size (n),The rst factor is the level of condence. Those conducting the study select the level of condence. The 95 percent and the 99 percent levels of condence are the most common, but any value between 0 and 100 percent

47、 is possible. The 95 percent level of condence corresponds to a z value of 1.96, and a 99 percent level of condence corresponds to a z value of 2.58. The higher the level of condence selected, the larger the size of the corresponding sample.,9-53,Selecting a Sample Size (n),The second factor is the

48、allowable error. The maximum allowable error, designated as E, is the amount that is added and subtracted to the sample mean (or sample proportion) to determine the endpoints of the condence interval. It is the amount of error those conducting the study are willing to tolerate. It is also one-half t

49、he width of the corresponding condence interval. A small allowable error will require a larger sample. A large allowable error will permit a smaller sample.,9-54,Selecting a Sample Size (n),The third factor in determining the size of a sample is the population standard deviation. If the population i

50、s widely dispersed, a large sample is required. On the 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.,9-55,Selecting a Sample Size (n),three suggestions for

51、nding that estimate : 1. Use a comparable study. 2. Use a range-based approach. 3. Conduct a pilot study. We can express the interaction among these three factors and the sample size in the following formula.,9-56,Sample Size Determination for a Variable,To find the sample size for a variable:,9-57,

52、A student in public administration 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 95 percent level of confidence. The student found a report by the

53、Department of Labor that estimated the standard deviation to be $1,000. What is the required sample size? Given in the problem: E, the maximum allowable error, is $100 The value of z for a 95 percent level of confidence is 1.96, The estimate of the standard deviation is $1,000.,Sample Size Determina

54、tion for a Variable-Example 1,9-58,A student in public administration 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.

55、 The student found a report by the Department of Labor that estimated the standard deviation to be $1,000. What is the required sample size? Given in the problem: E, the maximum allowable error, is $100 The value of z for a 99 percent level of confidence is 2.58, The estimate of the standard deviati

56、on is $1,000.,Sample Size Determination for a Variable-Example 2,9-59,Sample Size for Proportions,The formula for determining the sample size in the case of a proportion is:,9-60,Another Example,The American Kennel Club wanted to estimate the proportion of children that have a dog as a pet. If the c

57、lub wanted the estimate to be within 3% of the population proportion, 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.,9-61,Another Example,A study needs to estimate the proportion of cities that

58、have private refuse collectors. The investigator wants the margin of 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?,9-62,Self-Review 95,Will you assist t

59、he college registrar in determining how many transcripts to study? The 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 percent level of condence.,9-63,Self-Review 95,9-64,Chapter Summary,I. A point estimate is a single value (statistic) used to estimate a population value (parameter). II. A condence interval is a range