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 = 3.0 (round to the nearest tenth as needed.) compute the standard deviation for set {4,4,4,9,14,14,14}. 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 data points, and \( \bar{x} \) is the mean. Here, \( n = 7 \) and \( \bar{x}=9 \).

Step2: Calculate \( (x_i - \bar{x})^2 \) for each data point in the set \(\{4,4,4,9,14,14,14\}\)

• For \( x_i = 4 \): \( (4 - 9)^2=(-5)^2 = 25 \). There are 3 such data points, so the sum for these is \( 3\times25 = 75 \).
• For \( x_i = 9 \): \( (9 - 9)^2 = 0^2=0 \). There is 1 such data point, so the sum for this is \( 1\times0 = 0 \).
• For \( x_i = 14 \): \( (14 - 9)^2=(5)^2 = 25 \). There are 3 such data points, so the sum for these is \( 3\times25 = 75 \).

Step3: Calculate the sum of squared deviations

The total sum of \( (x_i - \bar{x})^2 \) is \( 75+0 + 75=150 \).

Step4: Calculate the variance

The variance is \( \frac{\sum_{i = 1}^{n}(x_i - \bar{x})^2}{n - 1}=\frac{150}{7 - 1}=\frac{150}{6} = 25 \).

Step5: Calculate the standard deviation

The standard deviation \( s=\sqrt{25}=5.0 \).

Answer:

\( 5.0 \)