Question

35% of employees judge their peers by the cleanliness of their workspaces. you randomly select 8 employees and ask them whether they judge their peers by the cleanliness of their workspaces. the random variable represents the number of employees who judge their peers by the cleanliness of their workspaces. complete parts (a) through (c) below. (a) construct a binomial distribution using n = 8 and p = 0.35.

x

p(x)

|0|
|1|
|2|
|3|
|4|
|5|
|6|
|7|
|8|
(type integers or decimals rounded to four decimal places as needed.)

Explanation:

Step1: Recall binomial - probability formula

The binomial - probability formula is $P(x)=C(n,x)\times p^{x}\times(1 - p)^{n - x}$, where $C(n,x)=\frac{n!}{x!(n - x)!}$, $n = 8$, $p=0.35$, and $1-p = 0.65$.

Step2: Calculate for $x = 0$

$C(8,0)=\frac{8!}{0!(8 - 0)!}=1$, $P(0)=1\times0.35^{0}\times0.65^{8}=0.0498$.

Step3: Calculate for $x = 1$

$C(8,1)=\frac{8!}{1!(8 - 1)!}=8$, $P(1)=8\times0.35^{1}\times0.65^{7}=0.1977$.

Step4: Calculate for $x = 2$

$C(8,2)=\frac{8!}{2!(8 - 2)!}=28$, $P(2)=28\times0.35^{2}\times0.65^{6}=0.3053$.

Step5: Calculate for $x = 3$

$C(8,3)=\frac{8!}{3!(8 - 3)!}=56$, $P(3)=56\times0.35^{3}\times0.65^{5}=0.2587$.

Step6: Calculate for $x = 4$

$C(8,4)=\frac{8!}{4!(8 - 4)!}=70$, $P(4)=70\times0.35^{4}\times0.65^{4}=0.1373$.

Step7: Calculate for $x = 5$

$C(8,5)=\frac{8!}{5!(8 - 5)!}=56$, $P(5)=56\times0.35^{5}\times0.65^{3}=0.0467$.

Step8: Calculate for $x = 6$

$C(8,6)=\frac{8!}{6!(8 - 6)!}=28$, $P(6)=28\times0.35^{6}\times0.65^{2}=0.0108$.

Step9: Calculate for $x = 7$

$C(8,7)=\frac{8!}{7!(8 - 7)!}=8$, $P(7)=8\times0.35^{7}\times0.65^{1}=0.0013$.

Step10: Calculate for $x = 8$

$C(8,8)=\frac{8!}{8!(8 - 8)!}=1$, $P(8)=1\times0.35^{8}\times0.65^{0}=0.0001$.

Answer:

x	P(x)
1	0.1977
2	0.3053
3	0.2587
4	0.1373
5	0.0467
6	0.0108
7	0.0013
8	0.0001