Questions: Consider the following data: x 2 3 4 5 6 P(X=x) 0.3 0.2 0.1 0.1 0.3 Step 3 of 5: Find the standard deviation. Round your answer to one decimal place.

Solution Steps

The mean \( \mu \) of the distribution is calculated as follows:

\[ \mu = E(X) = \sum_{x} x \cdot P(X=x) = 2 \times 0.3 + 3 \times 0.2 + 4 \times 0.1 + 5 \times 0.1 + 6 \times 0.3 = 3.9 \]

The variance \( \sigma^2 \) is computed using the formula:

\[ \sigma^2 = E[(X - \mu)^2] = \sum_{x} (x - \mu)^2 \cdot P(X=x) \]

Substituting the values:

\[ \sigma^2 = (2 - 3.9)^2 \times 0.3 + (3 - 3.9)^2 \times 0.2 + (4 - 3.9)^2 \times 0.1 + (5 - 3.9)^2 \times 0.1 + (6 - 3.9)^2 \times 0.3 = 2.7 \]

The standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{\sigma^2} = \sqrt{2.7} \approx 1.6 \]

Final Answer