Question

the following four sets of 7 numbers all have a mean of 9. complete parts (a) through (d) below. {9,9,9,9,9,9,9} , {8,8,9,9,9,10,10} , {6,6,6,9,12,12,12} , {4,4,4,9,14,14,14} c. compute the standard deviation for each set. compute the standard deviation for set {9,9,9,9,9,9,9}. s = 0.0 (round to the nearest tenth as needed.) compute the standard deviation for set {8,8,9,9,9,10,10}. s = 0.8 (round to the nearest tenth as needed.) compute the standard deviation for set {6,6,6,9,12,12,12}. s = \square (round to the nearest tenth as needed.)

Explanation:

Step1: Recall the formula for sample standard deviation

The formula for the sample standard deviation \( s \) is:

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

where \( n \) is the number of data points, \( x_i \) are the individual data points, and \( \bar{x} \) is the mean. We know the mean \( \bar{x}=9 \) and \( n = 7 \).

Step2: Calculate \( (x_i - \bar{x})^2 \) for each data point in the set \( \{6,6,6,9,12,12,12\} \)

• For \( x_i = 6 \): \( (6 - 9)^2=(- 3)^2 = 9 \). There are 3 data points with value 6, so the sum of their squared deviations is \( 3\times9 = 27 \).
• For \( x_i = 9 \): \( (9 - 9)^2=0^2 = 0 \). There is 1 data point with value 9, so the sum of its squared deviation is \( 1\times0=0 \).
• For \( x_i = 12 \): \( (12 - 9)^2=(3)^2 = 9 \). There are 3 data points with value 12, so the sum of their squared deviations is \( 3\times9 = 27 \).

Step3: Calculate the sum of squared deviations

The total sum of squared deviations \( \sum_{i = 1}^{n}(x_i - \bar{x})^2=27 + 0+27=54 \).

Step4: Calculate the variance and then the standard deviation

First, calculate the variance \( s^2=\frac{\sum_{i = 1}^{n}(x_i - \bar{x})^2}{n - 1}=\frac{54}{7 - 1}=\frac{54}{6} = 9 \).
Then, the standard deviation \( s=\sqrt{9}=3.0 \).

Answer:

\( 3.0 \)