Question

another measure of dispersion is standard deviation, which describes the typical difference (or deviation) between a data value and the mean. the standard deviation \\(\sigma\\) (read as \sigma\) of \\(x_1,\dots,x_n\\) is: \\(\sigma = \sqrt{\frac{(x - \bar{x})^2}{n}}\\) \\(\sigma = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \dots + (x_n - \bar{x})^2}{n}}\\) (dont worry, we will use the calculator!) 6. what is the standard deviation of the waiting times in each data set in #3? office a = \underline{\qquad\qquad\qquad\qquad} office b = \underline{\qquad\qquad\qquad\qquad} which off shows more variability in waiting times? \underline{\qquad\qquad\qquad\qquad} 7. what is the standard deviation of the times in each data set in #2? race a = \underline{\qquad\qquad\qquad\qquad} race b = \underline{\qquad\qquad\qquad\qquad}

Explanation:

To solve for the standard deviation, we need the data sets from #3 (for Office A and Office B waiting times) and #2 (for Race A and Race B times). Since the data sets are not provided here, we assume we can use a calculator or software (as hinted in the problem) to compute the standard deviation using the formula $\sigma = \sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}}$, where $\bar{x}$ is the mean of the data set.

Step 1: Recall the Standard Deviation Formula

The population standard deviation formula is $\sigma=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}}$, where $x_{i}$ are the data points, $\bar{x}$ is the mean, and $n$ is the number of data points.

Step 2: Obtain the Data Sets

We need the data from problem #3 (waiting times for Office A and Office B) and problem #2 (times for Race A and Race B). Let's assume the data for #3 (Office A and Office B) and #2 (Race A and Race B) are as follows (example data, since actual data is missing; in a real scenario, use the given data):

For Problem 6 (Waiting Times - Office A and Office B)

• Office A Data (example): Let's say data points are \(1, 2, 3, 4, 5\).
• Mean \(\bar{x}=\frac{1 + 2+3 + 4+5}{5}=\frac{15}{5} = 3\).
• Sum of squared deviations: \((1 - 3)^{2}+(2 - 3)^{2}+(3 - 3)^{2}+(4 - 3)^{2}+(5 - 3)^{2}=4 + 1+0 + 1+4 = 10\).
• Variance \(=\frac{10}{5}=2\).
• Standard deviation \(\sigma=\sqrt{2}\approx1.414\).

• Office B Data (example): Let's say data points are \(1, 3, 5, 7, 9\).
• Mean \(\bar{x}=\frac{1+3 + 5+7 + 9}{5}=\frac{25}{5}=5\).
• Sum of squared deviations: \((1 - 5)^{2}+(3 - 5)^{2}+(5 - 5)^{2}+(7 - 5)^{2}+(9 - 5)^{2}=16 + 4+0 + 4+16 = 40\).
• Variance \(=\frac{40}{5}=8\).
• Standard deviation \(\sigma=\sqrt{8}\approx2.828\).

• To find which office has more variability, we compare the standard deviations. Since \(2.828>1.414\), Office B has more variability.

For Problem 7 (Times - Race A and Race B)

• Race A Data (example): Let's say data points are \(10, 20, 30, 40, 50\).
• Mean \(\bar{x}=\frac{10 + 20+30 + 40+50}{5}=\frac{150}{5}=30\).
• Sum of squared deviations: \((10 - 30)^{2}+(20 - 30)^{2}+(30 - 30)^{2}+(40 - 30)^{2}+(50 - 30)^{2}=400 + 100+0 + 100+400 = 1000\).
• Variance \(=\frac{1000}{5}=200\).
• Standard deviation \(\sigma=\sqrt{200}\approx14.14\).

• Race B Data (example): Let's say data points are \(15, 25, 35, 45, 55\).
• Mean \(\bar{x}=\frac{15 + 25+35 + 45+55}{5}=\frac{175}{5}=35\).
• Sum of squared deviations: \((15 - 35)^{2}+(25 - 35)^{2}+(35 - 35)^{2}+(45 - 35)^{2}+(55 - 35)^{2}=400 + 100+0 + 100+400 = 1000\).
• Variance \(=\frac{1000}{5}=200\).
• Standard deviation \(\sigma=\sqrt{200}\approx14.14\).

(Note: In a real problem, substitute the actual data from #2 and #3. The above is a demonstration of the calculation process.)

Answer:

Problem 6:

• Office A (standard deviation) \(=\boldsymbol{\approx1.414}\) (example value, use actual data)
• Office B (standard deviation) \(=\boldsymbol{\approx2.828}\) (example value, use actual data)
• More variability: \(\boldsymbol{Office\ B}\) (example, based on data)

Problem 7:

• Race A (standard deviation) \(=\boldsymbol{\approx14.14}\) (example value, use actual data)
• Race B (standard deviation) \(=\boldsymbol{\approx14.14}\) (example value, use actual data)

(Remember to replace example values with the actual data from problems #2 and #3, and use a calculator for precise computation.)