Question

find the variance for the given sample data. round your answer to one more decimal place than the original data. 7 7 2 5 1 a. 7.7 b. 11.8 c. 7.8 d. 6.2

Explanation:

Step1: Identify the data set

The data set is \( 7, 7, 2, 5, 1 \).

Step2: Calculate the mean (\(\bar{x}\))

The formula for the mean is \(\bar{x} = \frac{\sum_{i = 1}^{n} x_i}{n}\), where \(n\) is the number of data points.
\(\sum_{i = 1}^{5} x_i = 7 + 7 + 2 + 5 + 1 = 22\)
\(n = 5\)
So, \(\bar{x} = \frac{22}{5} = 4.4\)

Step3: Calculate the squared differences from the mean

For each data point \(x_i\), calculate \((x_i - \bar{x})^2\):

• For \(x_1 = 7\): \((7 - 4.4)^2 = (2.6)^2 = 6.76\)
• For \(x_2 = 7\): \((7 - 4.4)^2 = (2.6)^2 = 6.76\)
• For \(x_3 = 2\): \((2 - 4.4)^2 = (-2.4)^2 = 5.76\)
• For \(x_4 = 5\): \((5 - 4.4)^2 = (0.6)^2 = 0.36\)
• For \(x_5 = 1\): \((1 - 4.4)^2 = (-3.4)^2 = 11.56\)

Step4: Calculate the sum of squared differences

\(\sum_{i = 1}^{5} (x_i - \bar{x})^2 = 6.76 + 6.76 + 5.76 + 0.36 + 11.56 = 31.2\)

Step5: Calculate the sample variance (\(s^2\))

The formula for sample variance is \(s^2 = \frac{\sum_{i = 1}^{n} (x_i - \bar{x})^2}{n - 1}\)
Here, \(n - 1 = 5 - 1 = 4\)
So, \(s^2 = \frac{31.2}{4} = 7.8\)

Answer:

C. 7.8