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For the newborn-crying example, a child psychologist finds μ = 2.1 times per week that crying wakes the mother, with (x-μ)^2*P(x) values of 0.1764, 0.2662, 0.0046, 0.1458, 0.2888, and 0.1682 across the six outcomes. What is the standard deviation, σ, of X?
Aσ ≈ 2.1000
Bσ ≈ 1.0247
Cσ ≈ 1.0500
Dσ ≈ 0.5124
Answer & Solution
Correct answer: B. σ ≈ 1.0247
1. Standard deviation of a discrete distribution is the square root of the sum of (x-μ)^2*P(x) across all outcomes.
2. Add the six given products: 0.1764 + 0.2662 + 0.0046 + 0.1458 + 0.2888 + 0.1682 = 1.05.
3. Take the square root of 1.05: sqrt(1.05) ≈ 1.0247.
4. Option C gives the sum itself before the square root, and option B mistakes σ for μ, so neither is the standard deviation.
_Source: OpenStax Introductory Statistics (CC BY 4.0), Ch 4 "Discrete Random Variables", section 4.2 Mean or Expected Value and Standard Deviation_
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