3-1 Measures of Center 103 3-1 Beyond the Basics 35.Degrees of Freedom Five recent U.S. presidents had a mean age of 56.2 years at the time of their inauguration. Four of these ages are 64, 46, 54, and 47. a. Find the missing value. b. We need to create a list of n values that have a specific known mean. We are free to select any values we desire for some of the n values. How many of the n values can be freely assigned before the remaining values are determined? (The result is referred to as the number of degrees of freedom.) 36.Censored Data Data Set 22 “Presidents” in Appendix B lists the numbers of years that U.S. presidents lived after their first inauguration. As of this writing, five of the presidents are still alive, and after their first inauguration they have lived 44 years, 28 years, 20 years, 12 years, and 4 years so far. We might use the values of 44+, 28+, 20+, 12+, and 4+, where the positive signs indicate that the actual value is equal to or greater than the current value. (These values are said to be censored at the current time that this list was compiled.) If you use the values in Data Set 22 and ignore the presidents who are still alive, what is the mean? If you use the values given in Data Set 22 along with the additional values of 44+, 28+, 20+, 12+, and 4+, what do we know about the mean? Do the two results differ by much? 37.Trimmed Mean Because the mean is very sensitive to extreme values, we say that it is not a resistant measure of center. By deleting some low values and high values, the trimmed mean is more resistant. To find the 10% trimmed mean for a data set, first arrange the data in order, then delete the bottom 10% of the values and delete the top 10% of the values, then calculate the mean of the remaining values. Use the axial loads (pounds) of aluminum cans listed below (from Data Set 41 “Aluminum Cans” in Appendix B) for cans that are 0.0111 in. thick. An axial load is the force at which the top of a can collapses. Identify any outliers, then compare the median, mean, 10% trimmed mean, and 20% trimmed mean. 247 260 268 273 276 279 281 283 284 285 286 288 289 291 293 295 296 299 310 504 38.Harmonic Mean The harmonic mean is often used as a measure of center for data sets consisting of rates of change, such as speeds. It is found by dividing the number of values n by the sum of the reciprocals of all values, expressed as n a 1 x (No value can be zero.) a. A bicycle trip of 30 miles is traveled at a mean speed of 30 mi>h, and the return trip is traveled at a mean speed of 10 mi>h. What is the total time required to travel the 60 miles? What was the harmonic mean speed for the round trip? b. The author drove 1163 miles to a conference in Orlando, Florida. For the trip to the conference, the author stopped overnight, and the mean speed from start to finish was 38 mi>h. For the return trip, the author stopped only for food and fuel, and the mean speed from start to finish was 56 mi>h. Find the harmonic mean of 38 mi>h and 56 mi>h. How does the result compare to the (arithmetic) mean of 38 mi>h and 56 mi>h found using Formula 3-1? Which of those two means is the actual mean speed for the round trip? 39. Geometric Mean The geometric mean is often used in business and economics for finding average rates of change, average rates of growth, or average ratios. To find the geometric mean of n values (all of which are positive), first multiply the values, then find the nth root of the product. For a 6-year period, money deposited in annual certificates of deposit had annual interest rates of 0.58%, 0.29%, 0.13%, 0.14%, 0.15%, and 0.19%. Identify the single
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