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? 38 45 50 60 70 75 85 150 185 215 250 375 450 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 717.3 dollars (round to one decimal place as needed.) the variance of the sample data is (round to one decimal place as needed)

Explanation:

Step1: Recall the formula for variance

The variance \(s^{2}\) of a sample is given by \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\), where \(n\) is the sample size, \(x_{i}\) are the data points, and \(\bar{x}\) is the sample mean. Another formula \(s^{2}=\frac{n\sum_{i=1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}{n(n - 1)}\) can also be used.

First, find \(n\). Counting the data points: \(n = 18\)
Find \(\sum_{i=1}^{n}x_{i}=38 + 45+50+60+70+75+85+150+185+215+250+375+450+1750+2500\)
\(\sum_{i=1}^{n}x_{i}=6493\)
Find \(\sum_{i=1}^{n}x_{i}^{2}=38^{2}+45^{2}+50^{2}+60^{2}+70^{2}+75^{2}+85^{2}+150^{2}+185^{2}+215^{2}+250^{2}+375^{2}+450^{2}+1750^{2}+2500^{2}\)
\(\sum_{i=1}^{n}x_{i}^{2}=38^{2}+45^{2}+50^{2}+60^{2}+70^{2}+75^{2}+85^{2}+150^{2}+185^{2}+215^{2}+250^{2}+375^{2}+450^{2}+1750^{2}+2500^{2}=1444+2025+2500+3600+4900+5625+7225+22500+34225+46225+62500+140625+202500+3062500+6250000 = 9834373\)

Step2: Calculate the variance

Using the formula \(s^{2}=\frac{n\sum_{i=1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}{n(n - 1)}\)
Substitute \(n = 18\), \(\sum_{i=1}^{n}x_{i}=6493\), \(\sum_{i=1}^{n}x_{i}^{2}=9834373\)
\(s^{2}=\frac{18\times9834373-(6493)^{2}}{18\times(18 - 1)}\)
First, calculate \(18\times9834373=177018714\)
\((6493)^{2}=6493\times6493 = 42159049\)
\(18\times9834373-(6493)^{2}=177018714 - 42159049=134859665\)
\(18\times(18 - 1)=18\times17 = 306\)
\(s^{2}=\frac{134859665}{306}\approx440718.0\)

Answer:

The variance of the sample data is \(440718.0\) dollars².