Bluman A.G. Elementary Statistics- A Step By Step Approach

Section 7–1 Confidence Intervals for the Mean When s Is Known 363SolutionStep 1 Find the mean and standard deviation for the data. Use the formulas shown inChapter 3 or your calculator. The mean 11.091. Assume the standarddeviation of the population is 14.405.Step 2 Find a2. Since the 90% confidence interval is to be used, a 1 0.90 0.10,anda2 0.102 0.05 XStep 3Step 4Find z a2 . Subtract 0.05 from 1.000 to get 0.9500. The corresponding z valueobtained from Table E is 1.65. (Note: This value is found by using the z valuefor an area between 0.9495 and 0.9505. A more precise z value obtainedmathematically is 1.645 and is sometimes used; however, 1.65 will be used inthis textbook.)Substitute in the formulaX z a 2sn m X z a 211.091 1.65 14.40530 m 11.091 1.65 14.40530 11.091 4.339 m 11.091 4.3396.752 m 15.430snHence, one can be 90% confident that the population mean of the assets of all creditunions is between $6.752 million and $15.430 million, based on a sample of 30 creditunions.Comment to Computer and Statistical Calculator UsersThis chapter and subsequent chapters include examples using raw data. If you are usingcomputer or calculator programs to find the solutions, the answers you get may varysomewhat from the ones given in the textbook. This is so because computers and calculatorsdo not round the answers in the intermediate steps and can use 12 or more decimal places forcomputation. Also, they use more-exact critical values than those given in the tables in theback of this book. These small discrepancies are part and parcel of statistics.Objective 2Determine theminimum samplesize for finding aconfidence intervalfor the mean.Sample SizeSample size determination is closely related to statistical estimation. Quite often youask, How large a sample is necessary to make an accurate estimate? The answer is notsimple, since it depends on three things: the margin of error, the population standarddeviation, and the degree of confidence. For example, how close to the true mean do youwant to be (2 units, 5 units, etc.), and how confident do you wish to be (90, 95, 99%,etc.)? For the purpose of this chapter, it will be assumed that the population standarddeviation of the variable is known or has been estimated from a previous study.7–9