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 central tendency for a data set that contains outliers? Illustrate with the help of an example.

Which of the three measures of central tendency (the mean, the median, and the mode) can be calculated for quantitative data only, and which can be calculated for both quantitative and qualitative data? Illustrate with examples.

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.

Prices of cars have a distribution that is skewed to the right with outliers in the right tail. Which of the measures of central tendency is the best to summarize this data set? Explain.

Nixon Corporation manufactures computer monitors. The following data are the numbers of computer monitors produced at the company for a sample of 10 days. \(\begin{array}{lllllll}24 & 32 & 27 & 23 & 35 & 33 & 29\end{array}\) 40 23 28 Calculate the mean, median, and mode for these data.

Seven airline passengers in economy class on the same flight paid an average of \(\$ 361\) per ticket. Because the tickets were purchased at different times and from different sources, the prices varied. The first five passengers paid \(\$ 420, \$ 210, \$ 333, \$ 695\), and \(\$ 485\). The sixth and seventh tickets were purchased by a couple who paid identical fares. What price did each of them pay?

When studying phenomena such as inflation or population changes that involve periodic increases or decreases, the geometric mean is used to find the average change over the entire period under study. To calculate the geometric mean of a sequence of \(n\) values \(x_{1}, x_{2}, \ldots, x_{n}\), we multiply them together and then find the \(n\) th root of this product. Thus $$ \text { Geometric mean }=\sqrt[n]{x_{1} \cdot x_{2} \cdot x_{3} \cdot \ldots \cdot x_{n}} $$ Suppose that the inflation rates for the last five years are \(4 \%, 3 \%, 5 \%, 6 \%\), and \(8 \%\), respectively. Thus at the end of the first year, the price index will be \(1.04\) times the price index at the beginning of the year, and so on. Find the mean rate of inflation over the 5 -year period by finding the geometric mean of the data set \(1.04,1.03,1.05,1.06\), and \(1.08 .\) (Hint: Here, \(n=5, x_{1}=1.04, x_{2}=1.03\), and so on. Use the \(x^{1 / n}\) key on your calculator to find the fifth root. Note that the mean inflation rate will be obtained by subtracting 1 from the geometric mean.)

The range, as a measure of spread, has the disadvantage of being influenced by outliers. Illustrate this with an example.

Can the standard deviation have a negative value? Explain.