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? 145 79 99 106 256 127 113 145 the range of the sample data is 177.0 dollars (round to one decimal place as needed.) the standard deviation of the sample data is 54.3 dollars (round to one decimal place as needed.) the variance of the sample data is 2940.9 dollars² (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 low. b. the measures of variation are very useful because a person does not want to buy a room where the variation is too high. 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 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.

Explanation:

Step1: Calculate the range

The range is the difference between the maximum and minimum values in the data - set. The maximum value is 256 and the minimum value is 79. So, $Range = 256 - 79=177.0$ dollars.

Step2: Calculate the mean

The data - set is \(x = \{145,79,99,106,256,127,113,145\}\). The mean \(\bar{x}=\frac{145 + 79+99+106+256+127+113+145}{8}=\frac{1070}{8}=133.75\) dollars.

Step3: Calculate the variance

The variance formula for a sample is \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\).
\((145-133.75)^{2}=126.5625\), \((79 - 133.75)^{2}=2997.5625\), \((99-133.75)^{2}=1207.5625\), \((106-133.75)^{2}=770.0625\), \((256-133.75)^{2}=14945.5625\), \((127-133.75)^{2}=45.5625\), \((113-133.75)^{2}=430.5625\), \((145-133.75)^{2}=126.5625\).
\(\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}=126.5625+2997.5625+1207.5625+770.0625+14945.5625+45.5625+430.5625+126.5625 = 20649.5\).
\(s^{2}=\frac{20649.5}{7}\approx2949.9\) dollars².

Step4: Calculate the standard deviation

The standard deviation \(s=\sqrt{s^{2}}\). So, \(s=\sqrt{2949.9}\approx54.3\) dollars.

Step5: Evaluate the usefulness of measures of variation

When searching for a room, low prices, location, and good accommodations are more important than the amount of variation in the area. So the measures of variation are not very useful.

Answer:

The range of the sample data is 177.0 dollars.
The variance of the sample data is 2949.9 dollars².
The standard deviation of the sample data is 54.3 dollars.
D. 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.