Question

9/25 8:01 am sy 25 - 26 s - id.123 rv 1 question #8 the tables give the heights, in inches, of the players on the varsity and junior varsity volleyball teams. varsity team 60.0 68.2 62.4 68.9 70.1 65.5 60.8 68.0 69.1 65.7 66.0 64.3 junior varsity team 67.8 67.2 64.5 66.5 72.3 61.4 66.1 63.3 65.8 64.3 65.9 61.2 what is the approximate difference in the standard deviations of the heights of the two teams? a 3.1 inches b 3 inches c 0.3 inch d 0.2 inch

Explanation:

Step1: Calculate mean of Varsity Team

Let \(x_{1 - 12}\) be the heights of Varsity - team players. \(\bar{x}_{V}=\frac{60.0 + 68.2+62.4 + 68.9+70.1+65.5+60.8+68.0+69.1+65.7+66.0+64.3}{12}=\frac{799}{12}\approx66.58\)

Step2: Calculate variance of Varsity Team

\(s_{V}^{2}=\frac{\sum_{i = 1}^{12}(x_{i}-\bar{x}_{V})^{2}}{12 - 1}\)
\((60.0 - 66.58)^{2}+(68.2-66.58)^{2}+(62.4 - 66.58)^{2}+(68.9 - 66.58)^{2}+(70.1 - 66.58)^{2}+(65.5 - 66.58)^{2}+(60.8 - 66.58)^{2}+(68.0 - 66.58)^{2}+(69.1 - 66.58)^{2}+(65.7 - 66.58)^{2}+(66.0 - 66.58)^{2}+(64.3 - 66.58)^{2}\)
\(=(- 6.58)^{2}+(1.62)^{2}+(-4.18)^{2}+(2.32)^{2}+(3.52)^{2}+(-1.08)^{2}+(-5.78)^{2}+(1.42)^{2}+(2.52)^{2}+(-0.88)^{2}+(-0.58)^{2}+(-2.28)^{2}\)
\(=43.2964 + 2.6244+17.4724+5.3824+12.3904+1.1664+33.4084+2.0164+6.3504+0.7744+0.3364+5.1984\)
\(=\frac{129.316}{11}\approx11.76\)
\(s_{V}=\sqrt{11.76}\approx3.43\)

Step3: Calculate mean of Junior Varsity Team

Let \(y_{1 - 12}\) be the heights of Junior - Varsity team players. \(\bar{x}_{JV}=\frac{67.8+67.2+64.5+66.5+72.3+61.4+66.1+63.3+65.8+64.3+65.9+61.2}{12}=\frac{786.3}{12}\approx65.53\)

Step4: Calculate variance of Junior Varsity Team

\(s_{JV}^{2}=\frac{\sum_{i = 1}^{12}(y_{i}-\bar{x}_{JV})^{2}}{12 - 1}\)
\((67.8-65.53)^{2}+(67.2 - 65.53)^{2}+(64.5 - 65.53)^{2}+(66.5 - 65.53)^{2}+(72.3 - 65.53)^{2}+(61.4 - 65.53)^{2}+(66.1 - 65.53)^{2}+(63.3 - 65.53)^{2}+(65.8 - 65.53)^{2}+(64.3 - 65.53)^{2}+(65.9 - 65.53)^{2}+(61.2 - 65.53)^{2}\)
\(=(2.27)^{2}+(1.67)^{2}+(-1.03)^{2}+(0.97)^{2}+(6.77)^{2}+(-4.13)^{2}+(0.57)^{2}+(-2.23)^{2}+(0.27)^{2}+(-1.23)^{2}+(0.37)^{2}+(-4.33)^{2}\)
\(=5.1529+2.7889 + 1.0609+0.9409+45.8329+17.0569+0.3249+4.9729+0.0729+1.5129+0.1369+18.7489\)
\(=\frac{98.601}{11}\approx8.96\)
\(s_{JV}=\sqrt{8.96}\approx2.99\)

Step4: Calculate difference in standard deviations

\(s_{V}-s_{JV}\approx3.43 - 2.99 = 0.44\approx0.4\) (closest to 0.3)

Answer:

C. 0.3 inch