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? the range of the sample data is 2712 dollars. (type an integer or a decimal. do not round.) the standard deviation of the sample data is 770 dollars. (round to one decimal place as needed.) the variance of the sample data is (round to one decimal place as needed.) data: 38, 45, 50, 60, 60, 75, 90, 160, 195, 200, 265, 375, 400, 1750, 2750

Explanation:

Step1: Identify the data set

The data set is: 38, 45, 50, 60, 60, 75, 90, 160, 195, 200, 265, 375, 400, 1750, 2750.

Step2: Find the range

Range = Maximum value - Minimum value
Maximum value = 2750, Minimum value = 38
Range = 2750 - 38 = 2712 (dollars)

Step3: Find the mean ($\bar{x}$)

First, sum all the values:
38 + 45 + 50 + 60 + 60 + 75 + 90 + 160 + 195 + 200 + 265 + 375 + 400 + 1750 + 2750
= 38 + 45 = 83; 83 + 50 = 133; 133 + 60 = 193; 193 + 60 = 253; 253 + 75 = 328; 328 + 90 = 418; 418 + 160 = 578; 578 + 195 = 773; 773 + 200 = 973; 973 + 265 = 1238; 1238 + 375 = 1613; 1613 + 400 = 2013; 2013 + 1750 = 3763; 3763 + 2750 = 6513.

Number of data points ($n$) = 15.

Mean $\bar{x} = \frac{6513}{15} = 434.2$.

Step4: Calculate the variance ($s^2$)

Variance formula for sample data: $s^2 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n - 1}$

Calculate each $(x_i - \bar{x})^2$:

• $(38 - 434.2)^2 = (-396.2)^2 = 157074.44$
• $(45 - 434.2)^2 = (-389.2)^2 = 151476.64$
• $(50 - 434.2)^2 = (-384.2)^2 = 147609.64$
• $(60 - 434.2)^2 = (-374.2)^2 = 140025.64$
• $(60 - 434.2)^2 = (-374.2)^2 = 140025.64$
• $(75 - 434.2)^2 = (-359.2)^2 = 129024.64$
• $(90 - 434.2)^2 = (-344.2)^2 = 118473.64$
• $(160 - 434.2)^2 = (-274.2)^2 = 75185.64$
• $(195 - 434.2)^2 = (-239.2)^2 = 57216.64$
• $(200 - 434.2)^2 = (-234.2)^2 = 54849.64$
• $(265 - 434.2)^2 = (-169.2)^2 = 28628.64$
• $(375 - 434.2)^2 = (-59.2)^2 = 3504.64$
• $(400 - 434.2)^2 = (-34.2)^2 = 1169.64$
• $(1750 - 434.2)^2 = (1315.8)^2 = 1731330.64$
• $(2750 - 434.2)^2 = (2315.8)^2 = 5362930.64$

Now sum all these squared differences:

157074.44 + 151476.64 = 308551.08; +147609.64 = 456160.72; +140025.64 = 596186.36; +140025.64 = 736212; +129024.64 = 865236.64; +118473.64 = 983710.28; +75185.64 = 1,058,895.92; +57216.64 = 1,116,112.56; +54849.64 = 1,170,962.2; +28628.64 = 1,199,590.84; +3504.64 = 1,203,095.48; +1169.64 = 1,204,265.12; +1,731,330.64 = 2,935,595.76; +5,362,930.64 = 8,298,526.4.

Now divide by $n - 1 = 14$:

Variance $s^2 = \frac{8298526.4}{14} \approx 592751.9$ (dollars²)

Step5: Calculate the standard deviation ($s$)

Standard deviation is the square root of variance:

$s = \sqrt{592751.9} \approx 770.0$ (dollars)

Answer:

• Range: 2712 dollars
• Standard Deviation: 770.0 dollars
• Variance: 592751.9 dollars²