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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]$
2400, 1750, 1900, 2500, 2250, 2100
$[/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]



Answer :

GinnyAnswer
Let's go through the steps to calculate the variance and standard deviation of the given data set:

### 1. Calculate the Mean of the Areas
Given areas:
[tex]\[ 2400, 1750, 1900, 2500, 2250, 2100 \][/tex]

First, we calculate the mean (average) of these areas:
[tex]\[ \text{Mean} = \frac{2400 + 1750 + 1900 + 2500 + 2250 + 2100}{6} = \frac{12900}{6} = 2150 \][/tex]

### 2. Calculate the Deviations from the Mean
Next, we find the deviation of each area from the mean:
[tex]\[ 2400 - 2150 = 250 \\ 1750 - 2150 = -400 \\ 1900 - 2150 = -250 \\ 2500 - 2150 = 350 \\ 2250 - 2150 = 100 \\ 2100 - 2150 = -50 \][/tex]

These deviations are: [tex]\(250, -400, -250, 350, 100, -50\)[/tex].

### 3. Square Each Deviation
Now, we square each deviation:
[tex]\[ 250^2 = 62500 \\ (-400)^2 = 160000 \\ (-250)^2 = 62500 \\ 350^2 = 122500 \\ 100^2 = 10000 \\ (-50)^2 = 2500 \][/tex]

### 4. Sum of the Squared Deviations
The sum of the squared deviations (numerator for variance) is:
[tex]\[ 62500 + 160000 + 62500 + 122500 + 10000 + 2500 = 420000 \][/tex]

So, the correct expression representing the numerator for the variance calculation is:
[tex]\((250)^2 + (-400)^2 + (-250)^2 + (350)^2 + (100)^2 + (-50)^2 = 420,000\)[/tex]
Thus, the correct answer to the numerator is:
[tex]\[ \boxed{(250)^2 + (-400)^2 + (-250)^2 + (350)^2 + (100)^2 + (-50)^2 = 420,000} \][/tex]

### 5. Calculate the Variance
Divide the numerator by the number of data points to find the variance:
[tex]\[ \text{Variance} = \frac{420000}{6} = 70000 \][/tex]

So, the variance is:
[tex]\[ \boxed{70000} \][/tex]

### 6. Calculate the 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, we get:
[tex]\[ \boxed{265} \][/tex]

Conclusively, the answers are:
- Numerator for the variance: [tex]\((250)^2 + (-400)^2 + (-250)^2 + (350)^2 + (100)^2 + (-50)^2 = 420,000\)[/tex]
- Variance: [tex]\(\boxed{70000}\)[/tex]
- Standard Deviation (rounded to the nearest whole number): [tex]\(\boxed{265}\)[/tex]

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