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 60 75 90 160 195 200 265 375 400 1750 2750 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 (round to one decimal place as needed.)

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 (in dollars)

Step2: Calculate the mean ($\bar{x}$)

First, find the sum of the data:
$38 + 45 + 50 + 60 + 60 + 75 + 90 + 160 + 195 + 200 + 265 + 375 + 400 + 1750 + 2750$
$= 38+45 = 83; 83+50 = 133; 133+60 = 193; 193+60 = 253; 253+75 = 328; 328+90 = 418; 418+160 = 578; 578+195 = 773; 773+200 = 973; 973+265 = 1238; 1238+375 = 1613; 1613+400 = 2013; 2013+1750 = 3763; 3763+2750 = 6513$
Number of data points ($n$) = 15
Mean $\bar{x} = \frac{6513}{15} = 434.2$

Step3: Calculate the variance ($s^2$)

Variance formula for sample: $s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$
Calculate each $(x_i - \bar{x})^2$:

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

Sum of these squared differences:
$157074.44 + 151476.64 = 308551.08; +147609.64 = 456160.72; +140025.64 = 596186.36; +140025.64 = 736212; +129024.64 = 865236.64; +118473.64 = 983710.28; +75185.64 = 1058895.92; +57216.64 = 1116112.56; +54849.64 = 1170962.2; +28628.64 = 1199590.84; +3504.64 = 1203095.48; +1169.64 = 1204265.12; +1731330.64 = 2935595.76; +5362930.64 = 8298526.4$

Variance $s^2 = \frac{8298526.4}{15 - 1} = \frac{8298526.4}{14} \approx 592751.9$ (dollars²)

Step4: Calculate the standard deviation ($s$)

Standard deviation is the square root of variance:
$s = \sqrt{592751.9} \approx 770.0$ (dollars)

Answer:

The standard deviation of the sample data is approximately $\boldsymbol{770.0}$ dollars.