Independent solution

How to solve this Uniform Distribution question

Answer in brief

This is a uniform-distribution probability centered at the mean. One standard deviation spans a total interval of length 30/sqrt(3), so the requested probability is 1/sqrt(3)=0.577350, which rounds to choice C.

Setup

Setup

For a uniform variable on an interval with endpoints a and b, write its mean and standard deviation.

μ=a+b2=80\mu=\frac{a+b}{2}=80
σ=b−a12=3012=53\sigma=\frac{b-a}{\sqrt{12}}=\frac{30}{\sqrt{12}}=5\sqrt{3}

Model

Model

The interval one standard deviation from the mean remains completely inside the distribution's support. Uniform probability is interval length divided by total support length.

Pr⁡(∣X−μ∣≤σ)=(μ+σ)−(μ−σ)b−a\Pr(|X-\mu|\le \sigma)=\frac{(\mu+\sigma)-(\mu-\sigma)}{b-a}
Pr⁡(∣X−μ∣≤σ)=2σ30\Pr(|X-\mu|\le \sigma)=\frac{2\sigma}{30}

Compute

Compute

Substitute the uniform standard deviation and simplify.

2(30/12)30=212=13\frac{2(30/\sqrt{12})}{30}=\frac{2}{\sqrt{12}}=\frac{1}{\sqrt{3}}
13=0.5773502692…\frac{1}{\sqrt{3}}=0.5773502692\ldots

Answer

Answer

The probability rounds to 0.58 at the precision of the listed answers.

Pr⁡(∣X−μ∣≤σ)≈0.58(C)\boxed{\Pr(|X-\mu|\le \sigma)\approx 0.58\quad\text{(C)}}