91Ó°ÊÓ

Explain how the value of the median is determined for a data set that contains an odd number of observations and for a data set that contains an even number of observations.

Which of the three measures of central tendency (the mean, the median, and the mode) can assume more than one value for a data set? Give an example of a data set for which this summary measure assumes more than one value.

Is it possible for a (quantitative) data set to have no mean, no median, or no mode? Give an example of a data set for which this summary measure does not exist.

The following data set belongs to a population: 5 \(\begin{array}{llllll}2 & 0 & -9 & 16 & 10 & 7\end{array}\)

The following table gives the standard deductions and personal exemptions for persons filing with "single" status on their 2011 state income taxes in a random sample of 9 states. Calculate the mean and median for the data on standard deductions for these states. $$ \begin{array}{lcc} \hline \text { State } & \begin{array}{c} \text { Standard Deduction } \\ \text { (in dollars) } \end{array} & \begin{array}{c} \text { Personal Exemption } \\ \text { (in dollars) } \end{array} \\ \hline \text { Delaware } & 3250 & 110 \\ \text { Hawaii } & 2000 & 1040 \\ \text { Kentucky } & 2190 & 20 \\ \text { Minnesota } & 5450 & 3500 \\ \text { North Dakota } & 5700 & 3650 \\ \text { Oregon } & 1945 & 176 \\ \text { Rhode Island } & 5700 & 3650 \\ \text { Vermont } & 5700 & 3650 \\ \text { Virginia } & 3000 & 930 \\ \hline \end{array} $$

The following data give the 2010 gross domestic product (in billions of dollars) for all 50 states. The data are entered in alphabetical order by state (Source: Bureau of Economic Analysis). \(\begin{array}{rrrrrrrrrr}173 & 49 & 254 & 103 & 1901 & 258 & 237 & 62 & 748 & 403 \\ 67 & 55 & 652 & 276 & 143 & 127 & 163 & 219 & 52 & 295 \\ 379 & 384 & 270 & 97 & 244 & 36 & 90 & 126 & 60 & 487 \\ 80 & 1160 & 425 & 35 & 478 & 148 & 174 & 570 & 49 & 164 \\\ 40 & 255 & 1207 & 115 & 26 & 424 & 340 & 65 & 248 & 39\end{array}\) a. Calculate the mean and median for these data. Are these values of the mean and the median sample statistics or population parameters? Explain. b. Do these data have a mode? Explain.

The following data represent the numbers of tornadoes that touched down during 1950 to 1994 in the 12 states that had the most tornadoes during this period. The data for these states are given in the following order: CO, FL, IA, IL, KS, LA, MO, MS, NE, OK, SD, TX. \(\begin{array}{llllllllllll}1113 & 2009 & 1374 & 1137 & 2110 & 1086 & 1166 & 1039 & 1673 & 2300 & 1139 & 5490\end{array}\) â. Calculate the mean and median for these data. b. Identify the outlier in this data set. Drop the outlier and recalculate the mean and median. Which of these two summary measures changes by a larger amount when you drop the outlier? c. Which is the better summary measure for these data, the mean or the median? Explain.

The mean age of six persons is 46 years. The ages of five of these six persons are \(57,39,44,51\), and 37 years, respectively. Find the age of the sixth person.

The trimmed mean is calculated by dropping a certain percentage of values from each end of a ranked data set. The trimmed mean is especially useful as a measure of central tendency when a data set contains a few outliers. Suppose the following data give the ages (in years) of 10 employees of a company: \(\begin{array}{llllll}47 & 53 & 38 & 26 & 39 & 49\end{array}\) 19 \(\begin{array}{ccc}67 & 31 & 23\end{array}\) To calculate the \(10 \%\) trimmed mean, first rank these data values in increasing order; then drop \(10 \%\) of the smallest values and \(10 \%\) of the largest values. The mean of the remaining \(80 \%\) of the values will give the \(10 \%\) trimmed mean. Note that this data set contains 10 values, and \(10 \%\) of 10 is 1 . Thus, if we drop the smallest value and the largest value from this data set, the mean of the remaining 8 values will be called the \(10 \%\) trimmed mean. Calculate the \(10 \%\) trimmed mean for this data set.

When is the value of the standard deviation for a data set zero? Give one example. Calculate the standard deviation for the example and show that its value is zero.