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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 :

GinnyAnswer
Let's solve the problem step-by-step.

### Step 1: Calculate the mean of the data
Given data: [tex]\(2400, 1750, 1900, 2500, 2250, 2100\)[/tex]

The mean ([tex]\(\mu\)[/tex]) of the data is calculated as:
[tex]\[ \mu = \frac{2400 + 1750 + 1900 + 2500 + 2250 + 2100}{6} = 2150 \][/tex]

### Step 2: Calculate the squared deviations from the mean
For each value, we subtract the mean and then square the result:
[tex]\[ (2400 - 2150)^2 = 250^2 = 62500 \][/tex]
[tex]\[ (1750 - 2150)^2 = (-400)^2 = 160000 \][/tex]
[tex]\[ (1900 - 2150)^2 = (-250)^2 = 62500 \][/tex]
[tex]\[ (2500 - 2150)^2 = 350^2 = 122500 \][/tex]
[tex]\[ (2250 - 2150)^2 = 100^2 = 10000 \][/tex]
[tex]\[ (2100 - 2150)^2 = (-50)^2 = 2500 \][/tex]

### Step 3: Sum the squared deviations (Numerator of the variance)
[tex]\[ 62500 + 160000 + 62500 + 122500 + 10000 + 2500 = 420000 \][/tex]

The numerator in the calculation of variance is thus [tex]\(420000\)[/tex].

### Step 4: Calculate the variance
Variance ([tex]\(\sigma^2\)[/tex]) is the average of these squared deviations:
[tex]\[ \sigma^2 = \frac{420000}{6} = 70000 \][/tex]

### Step 5: Calculate the standard deviation
The standard deviation ([tex]\(\sigma\)[/tex]) is the square root of the variance:
[tex]\[ \sigma = \sqrt{70000} \approx 264.575 \ (\text{rounded to the nearest whole number: } 265) \][/tex]

### Summary of Results:
Numerator for the calculation of variance: [tex]\(420000\)[/tex]

Variance: [tex]\(70000\)[/tex]

Standard Deviation (rounded to the nearest whole number): [tex]\(265\)[/tex]

Therefore, to fill in the given blanks:
- What is the variance? [tex]\( \boxed{70000} \)[/tex]
- What is the standard deviation, rounded to the nearest whole number? [tex]\( \boxed{265} \)[/tex]
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