Beer: The following table presents the number of active breweries for samples of states located east and west of the Mississippi River.

To compute the sample standard deviation for both the East and West data sets, we can use the following steps:

1. Calculate the mean (average) of the given data points.
2. Subtract the mean from each data point and square the result.
3. Compute the variance (the average of these squared differences).
4. Take the square root of the variance to get the standard deviation.

(a) Calculating for the East:

Step 1: Calculate the Mean

• East breweries: 19, 20, 18, 10, 47, 22, 52, 13, 38, 23
• Number of states (n): 10
• Mean \( \mu \): \[ \mu = \frac{19 + 20 + 18 + 10 + 47 + 22 + 52 + 13 + 38 + 23}{10} = \frac{ 19 + 20 + 18 + 10 + 47 + 22 + 52 + 13 + 38 + 23}{10} = \frac{ 172 }{10} = 17.2 \]

Step 2: Calculate Squared Deviations

• Deviations and their squares: \[ \begin{align*} (19 - 27.2)^2 & = ( -8.2 )^2 = 67.24 \ (20 - 27.2)^2 & = (-7.2)^2 = 51.84 \ (18 - 27.2)^2 & = (-9.2)^2 = 84.64 \ (10 - 27.2)^2 & = (-17.2)^2 = 295.84 \ (47 - 27.2)^2 & = (19.8)^2 = 392.04 \ (22 - 27.2)^2 & = ( -5.2 )^2 = 27.04 \ (52 - 27.2)^2 & = (24.8)^2 = 615.04 \ (13 - 27.2)^2 & = ( -14.2 )^2 = 201.64 \ (38 - 27.2)^2 & = (10.8)^2 = 116.64 \ (23 - 27.2)^2 & = (-4.2)^2 = 17.64 \ \end{align*} \]

Step 3: Calculate Variance \[ \text{Variance} = \frac{\text{Sum of squared deviations}}{n - 1} \] \[ = \frac{67.24 + 51.84 + 84.64 + 295.84 + 392.04 + 27.04 + 615.04 + 201.64 + 116.64 + 17.64}{10 - 1} \] \[ = \frac{ 2,844.6 }{9} \approx 315.5 \]

Step 4: Standard Deviation \[ \sigma \approx \sqrt{315.5} \approx 17.7 \]

So the sample standard deviation for the number of breweries east of the Mississippi River is approximately 17.7.

(b) Calculating for the West:

Step 1: Calculate the Mean

• West breweries: 305, 6, 41, 5, 37, 15, 17
• Number of states (n): 7
• Mean \( \mu \): \[ \mu = \frac{305 + 6 + 41 + 5 + 37 + 15 + 17}{7} = \frac{ 426 }{7} \approx 60.857 \]

Step 2: Calculate Squared Deviations

• Deviations and their squares: \[ \begin{align*} (305 - 60.857)^2 & \approx (244.143)^2 \approx 59639.027 \ (6 - 60.857)^2 & \approx (-54.857)^2 \approx 3015.029 \ (41 - 60.857)^2 & \approx (-19.857)^2 \approx 394.308 \ (5 - 60.857)^2 & \approx (-55.857)^2 \approx 3125.098 \ (37 - 60.857)^2 & \approx (-23.857)^2 \approx 569.616 \ (15 - 60.857)^2 & \approx (-45.857)^2 \approx 2107.693 \ (17 - 60.857)^2 & \approx (-43.857)^2 \approx 1924.399 \ \end{align*} \]

Step 3: Calculate Variance \[ \text{Variance} = \frac{\text{Sum of squared deviations}}{n - 1} \] \[ = \frac{ 59639.027 + 3015.029 + 394.308 + 3125.098 + 569.616 + 2107.693 + 1924.399 }{6} \] \[ \approx \frac{ 65195.171 }{6} \approx 10865.862. \]

Step 4: Standard Deviation \[ \sigma \approx \sqrt{10865.862} \approx 104.28 \]

So the sample standard deviation for the number of breweries west of the Mississippi River is approximately 104.3.

Thus, the final answers are:

• (a) Approximately 17.7
• (b) Approximately 104.3