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 90.10 part: 1 / 3 part 2 of 3 find the variance. the variance is

Explanation:

Step1: Find the mean of the data set.

The data set is \( 20.6, 33, 21.1, 58.5, 23.7, 110.7, 30.5, 24.1, 74.9, 60.5 \). The number of data points \( n = 10 \).

First, calculate the sum of the data:
\[

$$\begin{align*} \sum x&=20.6 + 33 + 21.1 + 58.5 + 23.7 + 110.7 + 30.5 + 24.1 + 74.9 + 60.5\\ &=20.6+33 = 53.6\\ 53.6+21.1&=74.7\\ 74.7+58.5&=133.2\\ 133.2+23.7&=156.9\\ 156.9+110.7&=267.6\\ 267.6+30.5&=298.1\\ 298.1+24.1&=322.2\\ 322.2+74.9&=397.1\\ 397.1+60.5&=457.6 \end{align*}$$

\]

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

Step2: Calculate the squared differences from the mean.

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

• \( (20.6 - 45.76)^2=(-25.16)^2 = 633.0256 \)
• \( (33 - 45.76)^2=(-12.76)^2 = 162.8176 \)
• \( (21.1 - 45.76)^2=(-24.66)^2 = 608.1156 \)
• \( (58.5 - 45.76)^2=(12.74)^2 = 162.3076 \)
• \( (23.7 - 45.76)^2=(-22.06)^2 = 486.6436 \)
• \( (110.7 - 45.76)^2=(64.94)^2 = 4217.2036 \)
• \( (30.5 - 45.76)^2=(-15.26)^2 = 232.8676 \)
• \( (24.1 - 45.76)^2=(-21.66)^2 = 469.1556 \)
• \( (74.9 - 45.76)^2=(29.14)^2 = 849.1396 \)
• \( (60.5 - 45.76)^2=(14.74)^2 = 217.2676 \)

Step3: Find the sum of the squared differences.

\[

$$\begin{align*} \sum (x_i - \bar{x})^2&=633.0256+162.8176 + 608.1156+162.3076+486.6436+4217.2036+232.8676+469.1556+849.1396+217.2676\\ &=633.0256 + 162.8176=795.8432\\ 795.8432+608.1156&=1403.9588\\ 1403.9588+162.3076&=1566.2664\\ 1566.2664+486.6436&=2052.91\\ 2052.91+4217.2036&=6270.1136\\ 6270.1136+232.8676&=6502.9812\\ 6502.9812+469.1556&=6972.1368\\ 6972.1368+849.1396&=7821.2764\\ 7821.2764+217.2676&=8038.544 \end{align*}$$

\]

Step4: Calculate the variance (sample variance, since it's a sample of states).

The formula for sample variance \( s^2=\frac{\sum (x_i - \bar{x})^2}{n - 1} \)

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

\[
s^2=\frac{8038.544}{9}\approx893.17
\]

Answer:

\( 893.17 \)