Questions: 36% 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=0.36. x P(x) 0 1 2 3 4 5 6 7 8 (Type integers or decimals rounded to four decimal places as needed.)

36% 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=0.36.

x P(x)
0
1
2
3
4
5
6
7
8

(Type integers or decimals rounded to four decimal places as needed.)

Transcript text: $36\%$ 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 $\mathrm{n}=8$ and $\mathrm{p}=0.36$. \begin{tabular}{|c|c|} \hline $\mathbf{x}$ & $\mathbf{P}(\mathbf{x})$ \\ \hline 0 & $\square$ \\ \hline 1 & $\square$ \\ \hline 2 & $\square$ \\ \hline 3 & $\square$ \\ \hline 4 & $\square$ \\ \hline 5 & $\square$ \\ \hline 6 & $\square$ \\ \hline 7 & $\square$ \\ \hline 8 & $\square$ \\ \hline \end{tabular} (Type integers or decimals rounded to four decimal places as needed.)

Solution

Solution Steps

Step 1: Define the Problem

We are tasked with constructing a binomial distribution for the random variable $ X $, which represents the number of employees who judge their peers by the cleanliness of their workspaces. Given parameters are $ n = 8 $ (the number of trials) and $ p = 0.36 $ (the probability of success).

Step 2: Calculate the Probabilities

Using the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

where $ q = 1 - p = 0.64 $, we calculate the probabilities for $ x = 0, 1, 2, \ldots, 8 $:

$ P(0) = 0.0281 $
$ P(1) = 0.1267 $
$ P(2) = 0.2494 $
$ P(3) = 0.2805 $
$ P(4) = 0.1973 $
$ P(5) = 0.0888 $
$ P(6) = 0.0250 $
$ P(7) = 0.0040 $
$ P(8) = 0.0003 $

Step 3: Present the Binomial Distribution Table

The probabilities calculated can be summarized in the following binomial distribution table:

\[ \begin{array}{|c|c|} \hline x & P(x) \\ \hline 0 & 0.0281 \\ 1 & 0.1267 \\ 2 & 0.2494 \\ 3 & 0.2805 \\ 4 & 0.1973 \\ 5 & 0.0888 \\ 6 & 0.0250 \\ 7 & 0.0040 \\ 8 & 0.0003 \\ \hline \end{array} \]

Final Answer

The probabilities for the binomial distribution are summarized in the table above. The final answer for the probabilities of each number of employees judging their peers by the cleanliness of their workspaces is:

\[ \boxed{ \begin{array}{|c|c|} \hline x & P(x) \\ \hline 0 & 0.0281 \\ 1 & 0.1267 \\ 2 & 0.2494 \\ 3 & 0.2805 \\ 4 & 0.1973 \\ 5 & 0.0888 \\ 6 & 0.0250 \\ 7 & 0.0040 \\ 8 & 0.0003 \\ \hline \end{array} } \]

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