Question

1. find the variance of the sample for the five states with the most covered bridges:

oregon: 106
vermont: 121
indiana: 152
ohio: 234
pennsylvania: 347

Explanation:

Step 1: Find the mean (\(\bar{x}\)) of the data set

The data values are \(106\), \(121\), \(152\), \(224\), \(347\). The formula for the mean is \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n = 5\) (number of data points) and \(x_{i}\) are the individual data values.

First, calculate the sum of the data values:
\(\sum_{i=1}^{5}x_{i}=106 + 121+152 + 224+347\)
\(106+121 = 227\); \(227+152=379\); \(379 + 224=603\); \(603+347 = 950\)

Then, the mean \(\bar{x}=\frac{950}{5}=190\)

Step 2: Calculate the squared differences from the mean

For each data point \(x_{i}\), calculate \((x_{i}-\bar{x})^{2}\):

• For \(x_{1}=106\): \((106 - 190)^{2}=(-84)^{2}=7056\)
• For \(x_{2}=121\): \((121 - 190)^{2}=(-69)^{2}=4761\)
• For \(x_{3}=152\): \((152 - 190)^{2}=(-38)^{2}=1444\)
• For \(x_{4}=224\): \((224 - 190)^{2}=(34)^{2}=1156\)
• For \(x_{5}=347\): \((347 - 190)^{2}=(157)^{2}=24649\)

Step 3: Find the sum of the squared differences

\(\sum_{i = 1}^{5}(x_{i}-\bar{x})^{2}=7056+4761 + 1444+1156+24649\)

Calculate step by step:
\(7056+4761=11817\); \(11817+1444 = 13261\); \(13261+1156=14417\); \(14417+24649 = 39066\)

Step 4: Calculate the sample variance (\(s^{2}\))

The formula for sample variance is \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\), where \(n = 5\), so \(n-1=4\)

\(s^{2}=\frac{39066}{4}=9766.5\)

Answer:

\(9766.5\)