Let's solve the problem step by step: --- ## **Step 1: Write Down the Data** | x | f(x) | |---|------| | 1 | .25 | | 2 | .15 | | 3 | .60 | --- ## **Step 2: Find the Mean (\(\mu\))** The mean (expected value) is found by: \[ \mu = \sum [x \cdot f(x)] \] Calculate each term: - For \( x = 1 \): \( 1 \times .25 = .25 \) - For \( x = 2 \): \( 2 \times .15 = .30 \) - For \( x = 3 \): \( 3 \times .60 = 1.80 \) Sum them: \[ \mu = .25 + .30 + 1.80 = 2.35 \] **Explanation:** The mean is the weighted average of x, using their probabilities. --- ## **Step 3: Find the Variance (\(\sigma^2\))** The variance formula is: \[ \sigma^2 = \sum [ (x - \mu)^2 \cdot f(x) ] \] Calculate each squared difference and multiply by f(x): ### **For \(x = 1\):** \[ (x - \mu)^2 = (1 - 2.35)^2 = (-1.35)^2 = 1.8225 \] \[ 1.8225 \times .25 = .455625 \] ### **For \(x = 2\):** \[ (x - \mu)^2 = (2 - 2.35)^2 = (-.35)^2 = .1225 \] \[ .1225 \times .15 = .018375 \] ### **For \(x = 3\):** \[ (x - \mu)^2 = (3 - 2.35)^2 = (.65)^2 = .4225 \] \[ .4225 \times .60 = .2535 \] --- ## **Step 4: Add the Results** \[ \sigma^2 = .455625 + .018375 + .2535 = .7275 \] --- ## **Step 5: Round to 2 Decimal Places** \[ \sigma^2 \approx .73 \] --- ## **Explanation and Answer** - The mean is calculated to find the "center" of the data. - Variance measures how spread out the values are around the mean. **None of the provided options (1.54, 2.37, 5.52, 2.35) match the calculated variance of .73.** Double check the calculations and the table: > If you meant to use the variance formula with the expected value of \(x^2\), let's check that: ### Calculate \(E[x^2]\): - \(1^2 \times .25 = .25\) - \(2^2 \times .15 = 4 \times .15 = .6\) - \(3^2 \times .6 = 9 \times .6 = 5.4\) - Sum: \(.25 + .6 + 5.4 = 6.25\) ### Variance: \[ \sigma^2 = E[x^2] - (\mu)^2 = 6.25 - (2.35)^2 = 6.25 - 5.5225 = .7275 \] Still, \(\boxed{.73}\). **Final Answer: \(\sigma^2 = .73\) (rounded to 2 decimal places), which is not in the choices.** --- ### **If you need to match an answer choice, double-check the problem statement or data. But mathematically, the variance is \(.73\).**