Question

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, 2,100

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\\)
\\((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\\)

what is the variance?

what is the standard deviation, rounded to the nearest whole number?

Explanation:

Calculate the sample mean

The given data set represents a small sample of house areas in square feet:

$$2400, 1750, 1900, 2500, 2250, 2100$$

First, we find the sample size \(n\):

$$n = 6$$

Next, we calculate the sample mean \(\bar{x}\):

$$\bar{x} = \frac{2400 + 1750 + 1900 + 2500 + 2250 + 2100}{6}$$

$$\bar{x} = \frac{12900}{6} = 2150$$

Determine the sum of squared deviations

Using the Sample Variance Formula, the numerator is the sum of squared deviations from the mean, \(\sum (x_i - \bar{x})^2\):

$$(2400 - 2150)^2 = (250)^2$$

$$(1750 - 2150)^2 = (-400)^2$$

$$(1900 - 2150)^2 = (-250)^2$$

$$(2500 - 2150)^2 = (350)^2$$

$$(2250 - 2150)^2 = (100)^2$$

$$(2100 - 2150)^2 = (-50)^2$$

Summing these squared values:

$$(250)^2 + (-400)^2 + (-250)^2 + (350)^2 + (100)^2 + (-50)^2 = 420,000$$

This matches the third multiple-choice option, which is already selected in the image.

Calculate the sample variance

Since the data is a "small sample of houses", we use the Sample Variance Formula with Bessel's correction, dividing by \(n - 1\):

$$s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$$

$$s^2 = \frac{420,000}{6 - 1} = \frac{420,000}{5} = 84,000$$

Calculate the sample standard deviation

Using the Sample Standard Deviation Formula, we take the square root of the sample variance:

$$s = \sqrt{84,000} \approx 289.8275$$

Rounding to the nearest whole number:

$$s \approx 290$$

Answer:

Question 1

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

• (A) \((225)^2 + (-425)^2 + (-275)^2 + (325)^2 + (75)^2 + (-75)^2 = 423,750\)
• (B) \((650)^2 + (-150)^2 + (-600)^2 + (250)^2 + (150)^2 + (-300)^2 = 980,000\)
• (C) \((250)^2 + (-400)^2 + (-250)^2 + (350)^2 + (100)^2 + (-50)^2 = 420,000\) (Correct answer)

Question 2

What is the variance?
<blank>84000</blank>

Question 3

What is the standard deviation, rounded to the nearest whole number?
<blank>290</blank>