Chapter 3 - Section 3.4 - Assess Your Understanding - Applying the Concepts - Page 173: 30

(1) Let x be the travel time of the students. The population mean of the sample provided in Problem 22 of Section 3.1 is calculated as 26.44, and the population standard deviation of the sample provided in Problem 20 of Section 3.2 is calculated as 12.84. The z-scores of 9 students are calculated as follows: The z-score of Amanda when the travel time is 39 is calculated as \[\begin{align} & Z=\frac{x-\mu }{\sigma } \\ & =\frac{39-26.44}{12.84} \\ & =0.9781 \end{align}\] The z-score of Amberwhen the travel time is 21 is calculated as \[\begin{align} & Z=\frac{x-\mu }{\sigma } \\ & =\frac{21-26.44}{12.84} \\ & =-\ 0.4236 \end{align}\] The z-score of Tim when the travel time is 9 is calculated as \[\begin{align} & Z=\frac{x-\mu }{\sigma } \\ & =\frac{9-26.44}{12.84} \\ & =-\ 1.3582 \end{align}\] The z-score of Mike when the travel time is 32 is calculated as \[\begin{align} & Z=\frac{x-\mu }{\sigma } \\ & =\frac{32-26.44}{12.84} \\ & =0.4330 \end{align}\] The z-score of Nicole when the travel time is 30 is calculated as \[\begin{align} & Z=\frac{x-\mu }{\sigma } \\ & =\frac{30-26.44}{12.84} \\ & =0.2772 \end{align}\] The z-score of Scot when travel time is 45 is calculated as \[\begin{align} & Z=\frac{x-\mu }{\sigma } \\ & =\frac{45-26.44}{12.84} \\ & =1.445 \end{align}\] The z-score of Erica when travel time is 11 is calculated as \[\begin{align} & Z=\frac{x-\mu }{\sigma } \\ & =\frac{11-26.44}{12.84} \\ & =-\ 1.202 \end{align}\] The z-score of Tiffany when travel time is 12 is calculated as \[\begin{align} & Z=\frac{x-\mu }{\sigma } \\ & =\frac{12-26.44}{12.84} \\ & =-\ 1.1246 \end{align}\] The z-score of Glenn when travel time is 39 is calculated as \[\begin{align} & Z=\frac{x-\mu }{\sigma } \\ & =\frac{39-26.44}{12.84} \\ & =0.9781 \end{align}\] (2) The mean of the z-scores is calculated as \[\begin{align} & \text{Mean}=\frac{0.9781+\left( -0.4236 \right)+\cdots +\left( -1.1246 \right)+0.9781}{9} \\ & =0.0003 \end{align}\] The standard deviation of the z-scores is calculated as \[\begin{align} & \text{S}\text{.D}\text{.}=\sqrt{\frac{{{\left( 0.9781-0.0003 \right)}^{2}}+{{\left( -0.4236-0.0003 \right)}^{2}}+\cdots +{{\left( -1.1246-0.0003 \right)}^{2}}+{{\left( 0.9781-0.0003 \right)}^{2}}}{9}} \\ & =\text{1}\text{.0001} \end{align}\]

Given above.