Question

1. suspension bridges the lengths (in feet) of the main span of the longest suspension bridges in the united states and the rest of the world are shown below. which set of data is more variable? united states 4205, 4200, 3800, 3500, 3478, 2800, 2800, 2310 world 6570, 5538, 5328, 4888, 4626, 4544, 4518, 3970

Explanation:

Step1: Calculate the mean for the United States data

Let \(x_{1},x_{2},\cdots,x_{n}\) be the data points. The mean formula is \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\).
For the United States data \(n = 7\), \(\sum_{i=1}^{7}x_{i}=4205 + 4200+3800+3500+3478+2800+2310=24293\)
\(\bar{x}_{US}=\frac{24293}{7}\approx3470.43\)

Step2: Calculate the variance for the United States data

The variance formula is \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\)
\((4205 - 3470.43)^{2}+(4200 - 3470.43)^{2}+(3800 - 3470.43)^{2}+(3500 - 3470.43)^{2}+(3478 - 3470.43)^{2}+(2800 - 3470.43)^{2}+(2310 - 3470.43)^{2}\)
\(=734.57^{2}+729.57^{2}+329.57^{2}+29.57^{2}+7.57^{2}+(- 670.43)^{2}+(-1160.43)^{2}\)
\(=539699.08 + 532272.48+108616.18+874.38+57.3+449476.38+1346603.78\)
\(\sum_{i = 1}^{7}(x_{i}-\bar{x})^{2}=3017600\)
\(s_{US}^{2}=\frac{3017600}{6}\approx502933.33\), \(s_{US}\approx709.2\)

Step3: Calculate the mean for the World data

For the World data \(n = 8\), \(\sum_{i=1}^{8}x_{i}=6570+5538+5328+4888+4626+4544+4518+3970 = 39982\)
\(\bar{x}_{World}=\frac{39982}{8}=4997.75\)

Step4: Calculate the variance for the World data

\((6570 - 4997.75)^{2}+(5538 - 4997.75)^{2}+(5328 - 4997.75)^{2}+(4888 - 4997.75)^{2}+(4626 - 4997.75)^{2}+(4544 - 4997.75)^{2}+(4518 - 4997.75)^{2}+(3970 - 4997.75)^{2}\)
\(=1572.25^{2}+540.25^{2}+330.25^{2}+(-109.75)^{2}+(-371.75)^{2}+(-453.75)^{2}+(-479.75)^{2}+(-1027.75)^{2}\)
\(=2472900.06+291870.06+109065.06+12045.06+138208.06+205889.06+229160.06+1056270.06\)
\(\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}=4524467.5\)
\(s_{World}^{2}=\frac{4524467.5}{7}\approx646352.5\), \(s_{World}\approx803.2\)

Since \(s_{World}\approx803.2>s_{US}\approx709.2\)

Answer:

The data for the rest of the world is more variable.