Certainly! Let's go through the solution step-by-step to complete the given table and find the standard deviation, [tex]\(\sigma\)[/tex].
Given values:
[tex]\[
\mu = 3.08
\][/tex]
[tex]\[
\text{Observations} = [(1, 0.5), (2, 0.1), (3, 0.1), (6, 0.1), (12, 0.1)]
\][/tex]
### Step 1: Compute [tex]\((x - \mu)^2\)[/tex] rounded to the nearest hundredth
1. [tex]\(x = 1\)[/tex]
[tex]\[
(x - \mu)^2 = (1 - 3.08)^2 = (-2.08)^2 = 4.33
\][/tex]
2. [tex]\(x = 2\)[/tex]
[tex]\[
(x - \mu)^2 = (2 - 3.08)^2 = (-1.08)^2 = 1.17
\][/tex]
3. [tex]\(x = 3\)[/tex]
[tex]\[
(x - \mu)^2 = (3 - 3.08)^2 = (-0.08)^2 = 0.01
\][/tex]
4. [tex]\(x = 6\)[/tex]
[tex]\[
(x - \mu)^2 = (6 - 3.08)^2 = 2.92^2 = 8.53
\][/tex]
5. [tex]\(x = 12\)[/tex]
[tex]\[
(x - \mu)^2 = (12 - 3.08)^2 = 8.92^2 = 79.57
\][/tex]
### Step 2: Compute [tex]\((x - \mu)^2 \cdot P(x)\)[/tex] rounded to 3 decimal places
1. [tex]\(x = 1\)[/tex], [tex]\(P(x) = 0.5\)[/tex]
[tex]\[
(x - \mu)^2 \cdot P(x) = 4.33 \cdot 0.5 = 2.165
\][/tex]
2. [tex]\(x = 2\)[/tex], [tex]\(P(x) = 0.1\)[/tex]
[tex]\[
(x - \mu)^2 \cdot P(x) = 1.17 \cdot 0.1 = 0.117
\][/tex]
3. [tex]\(x = 3\)[/tex], [tex]\(P(x) = 0.1\)[/tex]
[tex]\[
(x - \mu)^2 \cdot P(x) = 0.01 \cdot 0.1 = 0.001
\][/tex]
4. [tex]\(x = 6\)[/tex], [tex]\(P(x) = 0.1\)[/tex]
[tex]\[
(x - \mu)^2 \cdot P(x) = 8.53 \cdot 0.1 = 0.853
\][/tex]
5. [tex]\(x = 12\)[/tex], [tex]\(P(x) = 0.1\)[/tex]
[tex]\[
(x - \mu)^2 \cdot P(x) = 79.57 \cdot 0.1 = 7.957
\][/tex]
### Step 3: Summarize the Table
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
\text{Number of Bagels} & (X - \mu)^2 \text{ (rounded to nearest hundredth)} & P(x) & (x - \mu)^2 \cdot P(x) \text{ (rounded to 3 decimal places)} \\
\hline
1 & 4.33 & 0.5 & 2.165 \\
\hline
2 & 1.17 & 0.1 & 0.117 \\
\hline
3 & 0.01 & 0.1 & 0.001 \\
\hline
6 & 8.53 & 0.1 & 0.853 \\
\hline
12 & 79.57 & 0.1 & 7.957 \\
\hline
\end{array}
\][/tex]
### Step 4: Calculate the Variance and Standard Deviation
1. Calculate Variance ([tex]\(\sigma^2\)[/tex]):
[tex]\[
\sigma^2 = \sum (x - \mu)^2 \cdot P(x) = 2.165 + 0.117 + 0.001 + 0.853 + 7.957 = 11.093
\][/tex]
2. Calculate Standard Deviation ([tex]\(\sigma\)[/tex]):
[tex]\[
\sigma = \sqrt{\sigma^2} = \sqrt{11.093} \approx 3.33
\][/tex]
Thus, the final conclusion is:
- The variance [tex]\(\sigma^2\)[/tex] is 11.093.
- The standard deviation [tex]\(\sigma\)[/tex] is approximately 3.33.
