Question

listed below are prices in dollars for one night at different hotels in a certain region. find the range, variance, and standard deviation for the given sample data. include appropriate units in the results. how useful are the measures of variation for someone searching for a room? 225 180 282 252 204 185 96 223 the range of the sample data is (round to one decimal place as needed.) the standard deviation of the sample data is (round to one decimal place as needed.) the variance of the sample data is (round to one decimal place as needed.) how useful are the measures of variation for someone searching for a room? a. the measures of variation are very useful because a person does not want to buy a room where the variation is too high b. the measures of variation are not very useful because when searching for a room, low prices, location, and good accommodations are more important than the amount of variation in the area c. the measures of variation are not very useful because the values are nominal data that do not measure or count anything, so the resulting statistics are meaningless d. the measures of variation are very useful because a person does not want to buy a room where the variation is too low

Explanation:

Step1: Find the range

The range is the difference between the maximum and minimum values. The maximum value in the data set $\{225, 180, 282, 252, 204, 185, 96, 223\}$ is $282$ and the minimum is $96$.
$282 - 96=186.0$ dollars

Step2: Calculate the mean

The mean $\bar{x}=\frac{225 + 180+282+252+204+185+96+223}{8}=\frac{1647}{8}=205.875$ dollars

Step3: Calculate the squared - differences

For each data point $x_i$, calculate $(x_i-\bar{x})^2$.
$(225 - 205.875)^2=(19.125)^2 = 365.765625$
$(180 - 205.875)^2=(- 25.875)^2 = 669.515625$
$(282 - 205.875)^2=(76.125)^2 = 5794.015625$
$(252 - 205.875)^2=(46.125)^2 = 2127.515625$
$(204 - 205.875)^2=(-1.875)^2 = 3.515625$
$(185 - 205.875)^2=(-20.875)^2 = 435.765625$
$(96 - 205.875)^2=(-109.875)^2 = 12072.515625$
$(223 - 205.875)^2=(17.125)^2 = 293.265625$

Step4: Calculate the variance

The sample variance $s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}$, where $n = 8$.
$\sum_{i=1}^{8}(x_i - \bar{x})^2=365.765625+669.515625+5794.015625+2127.515625+3.515625+435.765625+12072.515625+293.265625 = 21761.875$
$s^2=\frac{21761.875}{7}\approx3108.8$ dollars$^2$

Step5: Calculate the standard deviation

The sample standard deviation $s=\sqrt{s^2}=\sqrt{3108.8}\approx55.8$ dollars

For the usefulness of the measures of variation:
The measures of variation are very useful because a person does not want to buy a room where the variation is too high. High variation may mean that there are some extremely high - priced or extremely low - priced rooms, which can affect the overall cost - estimation and decision - making process.

Answer:

The range of the sample data is $186.0$ dollars
The standard deviation of the sample data is $55.8$ dollars
The variance of the sample data is $3108.8$ dollars$^2$
A. The measures of variation are very useful because a person does not want to buy a room where the variation is too high