Question

1. find the variance and standard deviation for the seven states with the most covered bridges. oregon: 106 vermont: 121 indiana: 152 ohio: 234 pennsylvania: 347 new hampshire: 45 new jersey: 35
2. find the variance and standard deviation of the heights of five tallest skyscrapers in the united states. sears tower (willis building): 1450 feet empire state building: 1250 feet one world trade center: 1776 feet trump tower: 1388 feet 2 world trade center: 1340 feet
3. find the variance and standard deviation of the scores on the most recent reading test: 7.7, 7.4, 7.1, 7.7, 7.2, 7.3, and 7.9
4. find the variance and standard deviation of the highest temperatures recorded in eight specific states: 112, 100, 127, 120, 134, 118, 105, and 110.

Explanation:

1. For the number of covered - bridges in seven states:

Step1: Calculate the mean

Let \(x_1 = 106,x_2=121,x_3 = 347,x_4=152,x_5 = 234,x_6=45,x_7 = 35\). The number of data points \(n = 7\).
The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{106 + 121+347+152+234+45+35}{7}=\frac{1030}{7}\approx147.14\)

Step2: Calculate the squared - differences

\((x_1-\bar{x})^2=(106 - 147.14)^2=(- 41.14)^2 = 1702.50\)
\((x_2-\bar{x})^2=(121 - 147.14)^2=(-26.14)^2 = 683.30\)
\((x_3-\bar{x})^2=(347 - 147.14)^2=(199.86)^2 = 39944.02\)
\((x_4-\bar{x})^2=(152 - 147.14)^2=(4.86)^2 = 23.62\)
\((x_5-\bar{x})^2=(234 - 147.14)^2=(86.86)^2 = 7542.66\)
\((x_6-\bar{x})^2=(45 - 147.14)^2=(-102.14)^2 = 10432.58\)
\((x_7-\bar{x})^2=(35 - 147.14)^2=(-112.14)^2 = 12574.98\)

Step3: Calculate the variance

The variance \(s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}=\frac{1702.50+683.30+39944.02+23.62+7542.66+10432.58+12574.98}{6}=\frac{72803.66}{6}\approx12133.94\)

Step4: Calculate the standard deviation

The standard deviation \(s=\sqrt{s^2}=\sqrt{12133.94}\approx110.15\)

2. For the heights of five tallest skyscrapers:

Step1: Calculate the mean

Let \(x_1 = 1450,x_2=1776,x_3 = 1250,x_4=1388,x_5 = 1340\). The number of data points \(n = 5\).
The mean \(\bar{x}=\frac{1450+1776+1250+1388+1340}{5}=\frac{7204}{5}=1440.8\)

Step2: Calculate the squared - differences

\((x_1-\bar{x})^2=(1450 - 1440.8)^2=(9.2)^2 = 84.64\)
\((x_2-\bar{x})^2=(1776 - 1440.8)^2=(335.2)^2 = 112359.04\)
\((x_3-\bar{x})^2=(1250 - 1440.8)^2=(-190.8)^2 = 36404.64\)
\((x_4-\bar{x})^2=(1388 - 1440.8)^2=(-52.8)^2 = 2787.84\)
\((x_5-\bar{x})^2=(1340 - 1440.8)^2=(-100.8)^2 = 10160.64\)

Step3: Calculate the variance

The variance \(s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}=\frac{84.64+112359.04+36404.64+2787.84+10160.64}{4}=\frac{161796.8}{4}=40449.2\)

Step4: Calculate the standard deviation

The standard deviation \(s=\sqrt{s^2}=\sqrt{40449.2}\approx201.12\)

3. For the reading - test scores:

Step1: Calculate the mean

Let \(x_1 = 7.7,x_2=7.4,x_3 = 7.1,x_4=7.7,x_5 = 7.2,x_6=7.3,x_7 = 7.9\). The number of data points \(n = 7\).
The mean \(\bar{x}=\frac{7.7+7.4+7.1+7.7+7.2+7.3+7.9}{7}=\frac{51.3}{7}\approx7.33\)

Step2: Calculate the squared - differences

\((x_1-\bar{x})^2=(7.7 - 7.33)^2=(0.37)^2 = 0.14\)
\((x_2-\bar{x})^2=(7.4 - 7.33)^2=(0.07)^2 = 0.0049\)
\((x_3-\bar{x})^2=(7.1 - 7.33)^2=(-0.23)^2 = 0.0529\)
\((x_4-\bar{x})^2=(7.7 - 7.33)^2=(0.37)^2 = 0.14\)
\((x_5-\bar{x})^2=(7.2 - 7.33)^2=(-0.13)^2 = 0.0169\)
\((x_6-\bar{x})^2=(7.3 - 7.33)^2=(-0.03)^2 = 0.0009\)
\((x_7-\bar{x})^2=(7.9 - 7.33)^2=(0.57)^2 = 0.3249\)

Step3: Calculate the variance

The variance \(s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}=\frac{0.14+0.0049+0.0529+0.14+0.0169+0.0009+0.3249}{6}=\frac{0.6804}{6}=0.1134\)

Step4: Calculate the standard deviation

The standard deviation \(s=\sqrt{s^2}=\sqrt{0.1134}\approx0.34\)

4. For the highest temperatures:

Answer:

1. Variance: \(12133.94\), Standard deviation: \(110.15\)
2. Variance: \(40449.2\), Standard deviation: \(201.12\)
3. Variance: \(0.1134\), Standard deviation: \(0.34\)
4. Variance: \(127.64\), Standard deviation: \(11.29\)