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? (250)² + (-400)² + (-250)² + (350)² + (100)² + (-50)² = 420,0 (650)² + (-150)² + (-600)² + (250)² + (150)² + (-300)² = 980, (225)² + (-425)² + (-275)² + (325)² + (75)² + (-75)² = 423,750 what is the variance? what is the standard deviation, rounded to the nearest whole number?

Explanation:

Step1: Find the mean of the data set

The data set is \(2400, 1750, 1900, 2500, 2250, 2100\). The number of data points \(n = 6\).
The mean \(\bar{x}=\frac{2400 + 1750+1900 + 2500+2250+2100}{6}\)
\(=\frac{2400+1750 = 4150; 4150+1900 = 6050; 6050+2500 = 8550; 8550+2250 = 10800; 10800+2100 = 12900}{6}=\frac{12900}{6}=2150\)

Step2: Calculate the deviations from the mean

For each data point \(x_i\), calculate \(x_i-\bar{x}\):

• \(2400 - 2150=250\)
• \(1750 - 2150=- 400\)
• \(1900 - 2150=-250\)
• \(2500 - 2150 = 350\)
• \(2250 - 2150=100\)
• \(2100 - 2150=-50\)

Step3: Square the deviations

Square each of the deviations:

• \((250)^2\)
• \((-400)^2\)
• \((-250)^2\)
• \((350)^2\)
• \((100)^2\)
• \((-50)^2\)

Step4: Sum the squared deviations (this is the numerator for sample variance)

The sum of squared deviations \(=\ (250)^2+(-400)^2+(-250)^2+(350)^2+(100)^2+(-50)^2\)
We know from the first option (the checked one) that this sum is \(420000\) (assuming the typo in the image, it should be \(420000\) instead of \(420,0\)).

Step5: Calculate the sample variance

Sample variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}\)
We have \(\sum_{i=1}^{n}(x_i - \bar{x})^2=420000\) and \(n=6\), so \(n - 1=5\)
\(s^{2}=\frac{420000}{5}=84000\)

Step6: Calculate the standard deviation

Standard deviation \(s=\sqrt{s^{2}}=\sqrt{84000}\approx289.83\approx290\)

Answer:

Variance: \(\boldsymbol{84000}\)
Standard Deviation: \(\boldsymbol{290}\)