Question

traveler spending the data show the traveler spending in billions of dollars for a recent year for a sample of the states. round your answers to two decimal places.
20.6 33 21.1 58.5 23.7 110.7 30.5 24.1 74.9 60.5
send data to excel
part 1 of 3
find the range.
the range is 90.1.
part: 1 / 3
part 2 of 3
find the variance.
the variance is

Explanation:

Step1: Find the mean

First, we calculate the mean ($\bar{x}$) of the data set. The data points are \(20.6, 33, 21.1, 58.5, 23.7, 110.7, 30.5, 24.1, 74.9, 60.5\). The number of data points \(n = 10\).

The sum of the data points:

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The mean \(\bar{x}=\frac{\sum x}{n}=\frac{457.6}{10} = 45.76\)

Step2: Calculate the squared differences

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

• For \(x = 20.6\): \((20.6 - 45.76)^2=(-25.16)^2 = 632.0256\)
• For \(x = 33\): \((33 - 45.76)^2=(-12.76)^2 = 162.8176\)
• For \(x = 21.1\): \((21.1 - 45.76)^2=(-24.66)^2 = 608.1156\)
• For \(x = 58.5\): \((58.5 - 45.76)^2=(12.74)^2 = 162.3076\)
• For \(x = 23.7\): \((23.7 - 45.76)^2=(-22.06)^2 = 486.6436\)
• For \(x = 110.7\): \((110.7 - 45.76)^2=(64.94)^2 = 4217.2036\)
• For \(x = 30.5\): \((30.5 - 45.76)^2=(-15.26)^2 = 232.8676\)
• For \(x = 24.1\): \((24.1 - 45.76)^2=(-21.66)^2 = 469.1556\)
• For \(x = 74.9\): \((74.9 - 45.76)^2=(29.14)^2 = 849.1396\)
• For \(x = 60.5\): \((60.5 - 45.76)^2=(14.74)^2 = 217.2676\)

Step3: Sum the squared differences

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$$

Step4: Calculate the variance

Since this is a sample (we are dealing with a sample of states), we use the sample variance formula \(s^2=\frac{\sum (x_i - \bar{x})^2}{n - 1}\)

Here, \(n=10\), so \(n - 1=9\)

$$ s^2=\frac{8037.544}{9}\approx893.06 $$

Answer:

\(893.06\)