Question

1. in a certain town, 70% of adults have a college degree. the accompanying table describes the probability distribution for the number of adults (among 4 randomly selected adults) who have a college degree. find the standard deviation for the probability distribution.

x	p(x)
0	0.0081
1	0.0756
2	0.2646
3	0.4116
4	0.2401

a) σ = 0.92
b) σ = 0.84
c) σ = 1.06
d) σ = 2.95

Explanation:

Step1: Calculate the mean $\mu$

$\mu=\sum_{x = 0}^{4}x\cdot P(x)=0\times0.0081 + 1\times0.0756+2\times0.2646 + 3\times0.4116+4\times0.2401=2.8$

Step2: Calculate $\sum_{x = 0}^{4}(x - \mu)^2\cdot P(x)$

$(0 - 2.8)^2\times0.0081+(1 - 2.8)^2\times0.0756+(2 - 2.8)^2\times0.2646+(3 - 2.8)^2\times0.4116+(4 - 2.8)^2\times0.2401$
$=(- 2.8)^2\times0.0081+(-1.8)^2\times0.0756+(-0.8)^2\times0.2646+(0.2)^2\times0.4116+(1.2)^2\times0.2401$
$=7.84\times0.0081 + 3.24\times0.0756+0.64\times0.2646+0.04\times0.4116+1.44\times0.2401$
$=0.063504+0.244944+0.169344+0.016464+0.345744 = 0.84$

Step3: Calculate the standard - deviation $\sigma$

The formula for the standard deviation of a probability distribution is $\sigma=\sqrt{\sum_{x}(x - \mu)^2\cdot P(x)}$. Since $\sum_{x}(x - \mu)^2\cdot P(x)=0.84$, then $\sigma=\sqrt{0.84}\approx0.92$

Answer:

A. $\sigma = 0.92$