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 = □ (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 \( x_{i} \) are the data points, \( \bar{x} \) is the mean, and \( n \) is the number of data points. Here, \( n = 7 \) and \( \bar{x}=9 \).

Step2: Calculate \( (x_{i}-\bar{x})^{2} \) for each data point

For the set \( \{8,8,9,9,9,10,10\} \):

• For \( x_{i}=8 \): \( (8 - 9)^{2}=(- 1)^{2}=1 \). There are 2 data points with value 8, so total for 8s: \( 2\times1 = 2 \).
• For \( x_{i}=9 \): \( (9 - 9)^{2}=0^{2}=0 \). There are 3 data points with value 9, so total for 9s: \( 3\times0 = 0 \).
• For \( x_{i}=10 \): \( (10 - 9)^{2}=1^{2}=1 \). There are 2 data points with value 10, so total for 10s: \( 2\times1=2 \).

Step3: Sum the squared deviations

\( \sum_{i = 1}^{7}(x_{i}-\bar{x})^{2}=2 + 0+2=4 \).

Step4: Apply the standard deviation formula

Substitute into the formula: \( s=\sqrt{\frac{4}{7 - 1}}=\sqrt{\frac{4}{6}}=\sqrt{\frac{2}{3}}\approx0.8 \) (rounded to the nearest tenth).

Answer:

\( 0.8 \)