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Math

Math Snap

PROBLEM

A contractor records the areas, in square feet, of a small sample of houses in a neighborhood to determine data about the neighborhood. They are:
2,400;1,750;1,900;2,500;2,250;2,1002,400 ; 1,750 ; 1,900 ; 2,500 ; 2,250 ; 2,100 Which of the following represents the numerator in the calculation of variance and standard deviation?
(225)2+(−425)2+(−275)2+(325)2+(75)2+(−75)2=423,750(225)^{2}+(-425)^{2}+(-275)^{2}+(325)^{2}+(75)^{2}+(-75)^{2}=423,750
(650)2+(−150)2+(−600)2+(250)2+(150)2+(−300)2=980,000(650)^{2}+(-150)^{2}+(-600)^{2}+(250)^{2}+(150)^{2}+(-300)^{2}=980,000
(250)2+(−400)2+(−250)2+(350)2+(100)2+(−50)2=420,000(250)^{2}+(-400)^{2}+(-250)^{2}+(350)^{2}+(100)^{2}+(-50)^{2}=420,000
DONE

STEP 1

1. The variance and standard deviation are calculated using the formula for a sample.
2. The numerator in the calculation of variance is the sum of squared deviations from the mean.

STEP 2

1. Calculate the mean of the data set.
2. Determine the deviations from the mean for each data point.
3. Square each deviation.
4. Sum the squared deviations to find the numerator for variance.

STEP 3

Calculate the mean of the data set:
Mean=2400+1750+1900+2500+2250+21006 \text{Mean} = \frac{2400 + 1750 + 1900 + 2500 + 2250 + 2100}{6} Mean=12,9006=2,150 \text{Mean} = \frac{12,900}{6} = 2,150

STEP 4

Determine the deviations from the mean for each data point:
2400−2150=250 2400 - 2150 = 250 1750−2150=−400 1750 - 2150 = -400 1900−2150=−250 1900 - 2150 = -250 2500−2150=350 2500 - 2150 = 350 2250−2150=100 2250 - 2150 = 100 2100−2150=−50 2100 - 2150 = -50

STEP 5

Square each deviation:
(250)2=62,500 (250)^2 = 62,500 (−400)2=160,000 (-400)^2 = 160,000 (−250)2=62,500 (-250)^2 = 62,500 (350)2=122,500 (350)^2 = 122,500 (100)2=10,000 (100)^2 = 10,000 (−50)2=2,500 (-50)^2 = 2,500

SOLUTION

Sum the squared deviations:
62,500+160,000+62,500+122,500+10,000+2,500=420,000 62,500 + 160,000 + 62,500 + 122,500 + 10,000 + 2,500 = 420,000 The numerator in the calculation of variance and standard deviation is:
420,000 \boxed{420,000}

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