Measures of Central Location

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

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

5. $\bar{x}=3, \widetilde{x}=2$, mode $=2$
   

7. $\bar{x}=3, \widetilde{x}=2$, mode $=2$
   

9. $\{0,0,3\}$.
   

11. $\{0,1,1,2\}$.
    

13. $\bar{x}=146.9, \widetilde{x}=147.5$
    

15. $\bar{x}=2.6, \widetilde{x}=2$, mode $=2$
    

17. $\bar{x}=48.96, \widetilde{x}=49$, mode $=49$
    

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

21. $\bar{x}=48.96, \widetilde{x}=49$, mode $=49$

23. $\bar{x}=2.05, \widetilde{x}=2$, mode $=1$
    

25. 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.
    

27. 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.

29. a. $\mu=1528.74$
    b. $\bar{x}=1502.8$
    c. $\bar{x}=1535.2$
    

31. 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 $\widetilde{x}=448$