Measures of Central Location

ANSWERS

  1. a. \(\mathrm{9}\).
    b. \(\mathrm{41}\).
    c. \(\mathrm{0}\).
    d. \(\mathrm{14}\).


  1. \(\bar{x}=2.5, \widetilde{x}=2.5\), mode \(=\{1,2,3,4\}\)


  1. \(\bar{x}=3, \widetilde{x}=2\), mode \(=2\)


  1. \(\bar{x}=3, \widetilde{x}=2\), mode \(=2\)


  1. \(\{0,0,3\}\).


  1. \(\{0,1,1,2\}\).


  1. \(\bar{x}=146.9, \widetilde{x}=147.5\)


  1. \(\bar{x}=2.6, \widetilde{x}=2\), mode \(=2\)


  1. \(\bar{x}=48.96, \widetilde{x}=49\), mode \(=49\)


  1. a. No, the survival times of the fourth and fifth mice are unknown.
    b. Yes, \(\widetilde{x}=421\)


  1. \(\bar{x}=48.96, \widetilde{x}=49\), mode \(=49\)


  1. \(\bar{x}=2.05, \widetilde{x}=2\), mode \(=1\)


  1. Mean: \(n x_{\min } \leq \sum x\) so dividing by \(n\) yields \(x_{\min } \leq \bar{x}\), so the minimum value is not above average. Median: the middle measurement, or average of the two middle measurements, \(\widetilde{x}\), is at least as large as \(x_{\min }\), so the minimum value is not above average. Mode: the mode is one of the measurements, and is not greater than itself.


  1. a. \(\bar{x}=3. \overline{18}, \widetilde{x}=3\), mode \(=5\).
    b. \(\bar{x}=6. \overline{18}, \widetilde{x}=6\), mode \(=8\)
    c. \(\bar{x}=-2. \overline{81}, \tilde{x}=-3, \operatorname{mode}=-1\)
    d. If a number is added to every measurement in a data set, then the mean, median, and mode all change by that number.


  1. a. \(\mu=1528.74\)
    b. \(\bar{x}=1502.8\)
    c. \(\bar{x}=1535.2\)


  1. a. \(\bar{x}=553.4286\) and \(\widetilde{x}=552.5\)
    b. \(\bar{x}=665.9692\) and \(\widetilde{x}=667\)
    c. \(\bar{x}=455.8933\) and \(\widetilde{x}=448\)