Step-by-Step Solution:

(a) Compute $s$ for the original data set $\{3, 13, 17, 19, 9\}$:

The formula for the standard deviation of a sample is:

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

Where:

• $n$ is the number of data points.
• $x_i$ is each data point.
• $\bar{x}$ is the mean of the data.

1. Compute the mean $\bar{x}$: $\bar{x} = \frac{\text{sum of data points}}{\text{number of data points}} = \frac{3 + 13 + 17 + 19 + 9}{5} = \frac{61}{5} = 12.2$

2. Compute $(x_i - \bar{x})^2$ for each data point:

   $(3 - 12.2)^2 = (-9.2)^2 = 84.64$ $(13 - 12.2)^2 = (0.8)^2 = 0.64$ $(17 - 12.2)^2 = (4.8)^2 = 23.04$ $(19 - 12.2)^2 = (6.8)^2 = 46.24$ $(9 - 12.2)^2 = (-3.2)^2 = 10.24$

3. Sum the squared deviations: $\sum (x_i - \bar{x})^2 = 84.64 + 0.64 + 23.04 + 46.24 + 10.24 = 160.8$

4. Compute $s$: $s = \sqrt{\frac{1}{n-1} \sum (x_i - \bar{x})^2} = \sqrt{\frac{1}{5-1} \cdot 160.8} = \sqrt{\frac{160.8}{4}} = \sqrt{40.2} \approx 6.3411$

Thus, the standard deviation for the original data set is: $s \approx 6.3411$

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(b) Add 6 to each data value:

The new data set is $\{9, 19, 23, 25, 15\}$.

1. Compute the mean $\bar{x}$ of the new data: Adding a constant to each data value shifts the mean by the same constant: $\text{New } \bar{x} = \bar{x}_{\text{old}} + 6 = 12.2 + 6 = 18.2$

2. Compute $(x_i - \bar{x})^2$ for each new data point: Since the spread of the data (distance from the mean) doesn’t change when a constant is added, the deviations $(x_i - \bar{x})^2$ will be identical to the original data set.

3. Reuse the previous sum of squared deviations: $\sum (x_i - \bar{x})^2 = 160.8$

4. Compute $s$: $s = \sqrt{\frac{1}{n-1} \sum (x_i - \bar{x})^2} = \sqrt{\frac{160.8}{4}} = \sqrt{40.2} \approx 6.3411$

Thus, the standard deviation for the new data set is: $s \approx 6.3411$

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Key Conclusion:

Adding a constant to each data value in a data set does not change the standard deviation, as it only shifts the data but does not alter the spread.

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Final Answers:

(a) $s \approx 6.3411$
(b) $s \approx 6.3411$

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Do you have any questions or need further clarification? Here are some related questions to consider:

1. What happens to the variance when a constant is added to all data values?
2. How does multiplying each data value by a constant affect the standard deviation?
3. Why is the standard deviation a measure of spread rather than location?
4. Can the standard deviation ever be negative? Why or why not?
5. How is the population standard deviation different from the sample standard deviation?

Tip: Practice calculating both variance and standard deviation to better understand their relationship and significance in statistics.