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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:
[tex]\[2,400 ; 1,750 ; 1,900 ; 2,500 ; 2,250 ; 2,100\][/tex]

Which of the following represents the numerator in the calculation of variance and standard deviation?

A. [tex]\[(225)^2+(-425)^2+(-275)^2+(325)^2+(75)^2+(-75)^2=423,750\][/tex]

B. [tex]\[(650)^2+(-150)^2+(-600)^2+(250)^2+(150)^2+(-300)^2=980,000\][/tex]

C. [tex]\[(250)^2+(-400)^2+(-250)^2+(350)^2+(100)^2+(-50)^2=420,000\][/tex]

What is the variance? [tex]\[\square\][/tex]

What is the standard deviation, rounded to the nearest whole number? [tex]\[\square\][/tex]

Sagot :

GinnyAnsweridn
To solve the problem, let's go through it step-by-step.

### Step-by-Step Solution

1. List of Areas:
We have the areas in square feet: [tex]\( 2400, 1750, 1900, 2500, 2250, 2100 \)[/tex].

2. Calculate the Mean:
First, calculate the mean of these areas.
[tex]\[ \text{Mean} = \frac{2400 + 1750 + 1900 + 2500 + 2250 + 2100}{6} = \frac{12900}{6} = 2150 \, \text{square feet} \][/tex]

3. Calculating Squared Deviations from the Mean:
Next, calculate how much each value deviates from the mean, square these deviations, and sum them up.
- Deviation for 2400: [tex]\( (2400 - 2150) = 250 \)[/tex]
[tex]\[ (250)^2 = 62500 \][/tex]
- Deviation for 1750: [tex]\( (1750 - 2150) = -400 \)[/tex]
[tex]\[ (-400)^2 = 160000 \][/tex]
- Deviation for 1900: [tex]\( (1900 - 2150) = -250 \)[/tex]
[tex]\[ (-250)^2 = 62500 \][/tex]
- Deviation for 2500: [tex]\( (2500 - 2150) = 350 \)[/tex]
[tex]\[ (350)^2 = 122500 \][/tex]
- Deviation for 2250: [tex]\( (2250 - 2150) = 100 \)[/tex]
[tex]\[ (100)^2 = 10000 \][/tex]
- Deviation for 2100: [tex]\( (2100 - 2150) = -50 \)[/tex]
[tex]\[ (-50)^2 = 2500 \][/tex]

Adding up these squared deviations:
[tex]\[ 62500 + 160000 + 62500 + 122500 + 10000 + 2500 = 420000 \][/tex]

Therefore, the correct numerator in the calculation of variance and standard deviation is [tex]\( 420000 \)[/tex].

4. Calculating Variance:
Variance is calculated by dividing the sum of squared deviations by the number of observations.
[tex]\[ \text{Variance} = \frac{420000}{6} = 70000 \, \text{square feet}^2 \][/tex]

5. Calculating Standard Deviation:
The standard deviation is the square root of the variance.
[tex]\[ \text{Standard Deviation} = \sqrt{70000} \approx 264.575 \][/tex]

Rounding to the nearest whole number, the standard deviation is [tex]\( 265 \)[/tex].

### Summary
- The numerator in the calculation of variance and standard deviation is [tex]\( 420000 \)[/tex].
- The variance is [tex]\( 70000 \, \text{square feet}^2 \)[/tex].
- The standard deviation, rounded to the nearest whole number, is [tex]\( 265 \)[/tex].
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