To calculate the sample standard deviation, we first need to compute the mean of the given data set, and then find the squared differences from the mean, sum them up, divide by \(n-1\) (where \(n\) is the sample size), and finally take the square root.
Here are the steps in detail:
Given Data:
East: 14, 19, 20, 18, 10, 47, 22, 52, 13, 38
Step 1: Calculate the Mean
\[ \text{Mean} (\bar{x}) = \frac{\text{sum of values}}{n} \]
Sum of values: \[ 14 + 19 + 20 + 18 + 10 + 47 + 22 + 52 + 13 + 38 = 14 + 19 + 20 + 18 + 10 + 47 + 22 + 52 + 13 + 38 = 252 \]
Number of values \(n = 10\).
Mean: \[ \bar{x} = \frac{252}{10} = 25.2 \]
Step 2: Calculate the Squared Differences
Now we calculate each squared difference from the mean: \[ (14 - 25.2)^2 = (-11.2)^2 = 125.44 \] \[ (19 - 25.2)^2 = (-6.2)^2 = 38.44 \] \[ (20 - 25.2)^2 = (-5.2)^2 = 27.04 \] \[ (18 - 25.2)^2 = (-7.2)^2 = 51.84 \] \[ (10 - 25.2)^2 = (-15.2)^2 = 231.04 \] \[ (47 - 25.2)^2 = (21.8)^2 = 475.24 \] \[ (22 - 25.2)^2 = (-3.2)^2 = 10.24 \] \[ (52 - 25.2)^2 = (26.8)^2 = 718.24 \] \[ (13 - 25.2)^2 = (-12.2)^2 = 148.84 \] \[ (38 - 25.2)^2 = (12.8)^2 = 163.84 \]
Step 3: Sum of Squared Differences
Now we add these squared differences: \[ 125.44 + 38.44 + 27.04 + 51.84 + 231.04 + 475.24 + 10.24 + 718.24 + 148.84 + 163.84 = 1817.6 \]
Step 4: Calculate Sample Variance
The sample variance \(s^2\) is given by: \[ s^2 = \frac{\text{sum of squared differences}}{n - 1} \] \[ s^2 = \frac{1817.6}{10 - 1} = \frac{1817.6}{9} = 202.0 \]
Step 5: Calculate Sample Standard Deviation
Finally, take the square root to find the standard deviation \(s\): \[ s = \sqrt{202.0} \approx 14.2 \]
Final Answer:
The sample standard deviation for the number of breweries east of the Mississippi River is approximately 14.2.
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