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.

Briefly explain the meaning of an outlier. Is the mean or the median a better measure of center for a data set that contains outliers? Illustrate with the help of an example.

Which of the five measures of center (the mean, the median, the trimmed mean, the weighted mean, and the mode) can be calculated for quantitative data only, and which can be calculated for both quantitative and qualitative data? Illustrate with examples.

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: \(\begin{array}{llllll}5 & -7 & 2 & 0 & -9 & 16\end{array}\) \(\begin{array}{ll}10 & 7\end{array}\) Calculate the mean, median, and mode.

The following data give the total food expenditures (in dollars) for the past one month for a sample of 20 families. \(\begin{array}{rrrrrrrrrr}1125 & 530 & 1234 & 595 & 427 & 872 & 1480 & 699 & 1274 & 1187 \\ 933 & 1127 & 716 & 1065 & 934 & 930 & 1046 & 1199 & 1353 & 441\end{array}\) a. Calculate the mean and median for these data. b. Calculate the \(20 \%\) trimmed mean for these data.

The following data give the annual salaries (in thousand dollars) of 20 randomly selected health care workers. \(\begin{array}{llllllllll}50 & 71 & 57 & 39 & 45 & 64 & 38 & 53 & 35 & 62 \\ 74 & 40 & 67 & 44 & 77 & 61 & 58 & 55 & 64 & 59\end{array}\) a. Calculate the mean, median, and mode for these data. b. Calculate the \(15 \%\) trimmed mean for these data.

The following data represent the systolic blood pressure reading (that is, the top number in the standard blood pressure reading) in \(\mathrm{mmHg}\) for each of 20 randomly selected middle-aged males who were taking blood pressure medication. \(\begin{array}{llllllllll}139 & 151 & 138 & 153 & 134 & 136 & 141 & 126 & 109 & 144\end{array}\) \(\begin{array}{llllllllll}111 & 150 & 107 & 132 & 144 & 116 & 159 & 12.1 & 127 & 113\end{array}\) a. Calculate the mean, median, and mode for these data. b. Calculate the \(10 \%\) trimmed mean for these data.

One property of the mean is that if we know the means and sample sizes of two (or more) data sets, we can calculate the combined mean of both (or all) data sets. The combined mean for two data sets is calculated by using the formula $$ \text { Combined mean }=\bar{x}=\frac{n_{1} \bar{x}_{1}+n_{2} \bar{x}_{2}}{n_{1}+n_{2}} $$ where \(n_{1}\) and \(n_{2}\) are the sample sizes of the two data sets and \(\bar{x}_{1}\) and \(\bar{x}_{2}\) are the means of the two data sets, respectively. Suppose a sample of 10 statistics books gave a mean price of \(\$ 140\) and a sample of 8 mathematics books gave a mean price of \(\$ 160\). Find the combined mean. (Hint: For this example: \(n_{1}=10, n_{2}=8, \bar{x}_{1}=\$ 140, \bar{x}_{2}=\$ 160 .\) )

For any data, the sum of all values is equal to the product of the sample size and mean; that is, \(\Sigma x=n \bar{x}\). Suppose the average amount of money spent on shopping by 10 persons during a given week is \(\$ 105.50 .\) Find the total amount of money spent on shopping by these 10 persons.