Question

listed below are the amounts (dollars) it costs for marriage proposal packages at different baseball stadiums. find the range, variance, and standard deviation for the given sample data. include appropriate units in the results. are there any outliers, and are they likely to have much of an effect on the measures of variation?
36 45 56 60 70 75 85 150 195 215 250 375 450 1750 2500
the range of the sample data is 2462 dollars.
(type an integer or a decimal. do not round.)
the standard deviation of the sample data is
(round to one decimal place as needed.)

Explanation:

Step1: Calculate the mean

First, find the sum of the data values: \(36 + 45+50 + 60+70+75+85+105+195+215+250+375+450+1750+2500=6131\).
The number of data points \(n = 15\).
The mean \(\bar{x}=\frac{6131}{15}\approx408.7\).

Step2: Calculate the variance

Use the formula \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\).
\((36-408.7)^{2}=(- 372.7)^{2}=138805.29\)
\((45 - 408.7)^{2}=(-363.7)^{2}=132277.69\)
\((50-408.7)^{2}=(-358.7)^{2}=128665.69\)
\((60 - 408.7)^{2}=(-348.7)^{2}=121591.69\)
\((70-408.7)^{2}=(-338.7)^{2}=114617.69\)
\((75-408.7)^{2}=(-333.7)^{2}=111355.69\)
\((85-408.7)^{2}=(-323.7)^{2}=104781.69\)
\((105 - 408.7)^{2}=(-303.7)^{2}=92233.69\)
\((195-408.7)^{2}=(-213.7)^{2}=45667.69\)
\((215-408.7)^{2}=(-193.7)^{2}=37519.69\)
\((250-408.7)^{2}=(-158.7)^{2}=25185.69\)
\((375-408.7)^{2}=(-33.7)^{2}=1135.69\)
\((450-408.7)^{2}=(41.3)^{2}=1705.69\)
\((1750-408.7)^{2}=(1341.3)^{2}=1799085.69\)
\((2500-408.7)^{2}=(2091.3)^{2}=4373515.69\)

\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=138805.29+132277.69+128665.69+121591.69+114617.69+111355.69+104781.69+92233.69+45667.69+37519.69+25185.69+1135.69+1705.69+1799085.69+4373515.69 = 8437966.85\)

\(s^{2}=\frac{8437966.85}{14}\approx602711.9\) (dollars²)

Step3: Calculate the standard deviation

Use the formula \(s=\sqrt{s^{2}}\).
\(s=\sqrt{602711.9}\approx776.4\) (dollars)

Answer:

The variance of the sample data is \(602711.9\) dollars².
The standard deviation of the sample data is \(776.4\) dollars.

Yes, there are outliers (\(1750\) and \(2500\)). Outliers can have a large effect on the measures of variation. The variance and standard - deviation are non - resistant measures, so the presence of large outliers will increase the values of variance and standard deviation.