# Consider a binomial experiment with n = 20\ and\ p

Consider a binomial experiment with .
1. Compute f (12).
2. Compute f (16).
3. Compute $P\left(x\ge 16\right)$.
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Heather Fulton

Step 1
Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for you. To get remaining sub-part solved please repost the complete question and mention the sub-parts to be solved.”
(1) Compute f (12).
The value of the function f (12) is obtained below:
Let X denotes the random variable which follows binomial distribution with the probability of success 0.70 with the number trails 20.
That is, $$\displaystyle{n}={12},{p}={0.70},{q}={0.30}{\left(={1}-{0.70}\right)}$$
The probability distribution is given by,
$P(X=x)=(\begin{array}{c}n\\ x\end{array})p^{x}(1-p)^{n-x};\ here\ x=0,1,2,...,n\ for\ 0 \le p \le 1$
Where n is the number of trials and p is the probability of success for each trial.
The required probability is,
$$\displaystyle{f{{\left({12}\right)}}}={P}{\left({X}\le{12}\right)}$$
Use Excel to obtain the probability value for x equals 12.
Follow the instruction to obtain the P-value:
1. Open EXCEL
2. Go to Formula bar.
3. In formula bar enter the function as“=BINOMDIST”
4. Enter the number of success as 12.
5. Enter the Trails as 20
6. Enter the probability as 0.70
7. Enter the cumulative as True.
8. Click enter
EXCEL output:
From the Excel output, the P-value is 0.2277
Thus, the value of the function f (12) is 0.2277.
Step 2
2. Compute f (16).
The value of the function f (16) is obtained below:
The required probability is,
$$\displaystyle{f{{\left({16}\right)}}}={P}{\left({X}\le{16}\right)}$$
Use Excel to obtain the probability value for x equals 16.
Follow the instruction to obtain the P-value:
1. Open EXCEL
2. Go to Formula bar.
3. In formula bar enter the function as“=BINOMDIST”
4. Enter the number of success as 16.
5. Enter the Trails as 20
6. Enter the probability as 0.70
7. Enter the cumulative as True.
8. Click enter
EXCEL output:
From the Excel output, the P-value is 0.8929
Thus, the value of the function f (16) is 0.8929.
Step 3
(3) Compute the probability $$\displaystyle{P}{\left({x}\geq{16}\right)}$$.
The probability $$\displaystyle{P}{\left({x}\geq{16}\right)}$$ is obtained below as follows:
The required probability is,
$$\displaystyle{P}{\left({x}\geq{16}\right)}={1}-{P}{\left({x}{ < }{16}\right)}$$
$$\displaystyle={1}-{P}{\left({x}\le{15}\right)}$$
Use Excel to obtain the probability value for x equals 15.
Follow the instruction to obtain the P-value:
1.Open EXCEL
2. Go to Formula bar.
3. In formula bar enter the function as“=BINOMDIST”
4. Enter the number of success as 15.
5. Enter the Trails as 20
6. Enter the probability as 0.70
7. Enter the cumulative as True.
8. Click enter
EXCEL output:
From the Excel output, the P-value is 0.7625.
$$\displaystyle{P}{\left({x}\geq{16}\right)}={1}-{P}{\left({x}{ < }{16}\right)}$$
$$\displaystyle={1}-{P}{\left({x}\le{15}\right)}$$
$$\displaystyle={1}-{0.7625}$$
$$\displaystyle={0.2375}$$
The probability $$\displaystyle{P}{\left({x}\geq{16}\right)}\ {i}{s}\ {0.2375}$$.

###### Not exactly what you’re looking for?
Neil Dismukes
Step 1
Given:
$p=0.70$
$n=20$
Formula binomial probability:
$f\left(k\right)=\left(\begin{array}{c}n\\ k\end{array}\right)×{p}^{k}×\left(1-p{\right)}^{n-k}$
a) $f\left(12\right)=\left(\begin{array}{c}20\\ 12\end{array}\right)×{0.70}^{12}×\left(1-0.70{\right)}^{20-12}=0.114397$
b) $f\left(16\right)=\left(\begin{array}{c}20\\ 16\end{array}\right)×{0.70}^{16}×\left(1-0.70{\right)}^{20-16}=0.130421$
$P\left(X\ge 16\right)=f\left(16\right)+f\left(17\right)+f\left(18\right)+f\left(19\right)+f\left(20\right)=0.2375$
d) Use the complement rule for probabilities:
$P\left(X\le 15\right)=1-P\left(X\ge 16\right)=1-0.2375=0.7625$
Step 2
e) The mean of a binomial distribution is the sample size n and the probability p:
$\mu =np=20×0.70=14$
f) The standard deviation of a binomial distribution is the square root of the product of the sample size n and the probabilities p and q. The variance is the square of the standard deviation.
${\sigma }^{2}=npq=np\left(1-p\right)=20\left(0.70\right)\left(1-0.70\right)=4.2$
$\sigma =\sqrt{npq}=\sqrt{np\left(1-p\right)}=\sqrt{20\left(0.70\right)\left(1-0.70\right)}\approx 2..0494$
###### Not exactly what you’re looking for?
Jeffrey Jordon

Step 1
Given that

a) $F\left(x\right)\therefore P\left(x\right){=}^{n}{C}_{x}{P}^{x}\left(1-P{\right)}^{n-x}$
$=\frac{n!}{\left(n-x\right)!x!}{P}^{x}\left(1-P{\right)}^{n-x}$
$\therefore F\left(13\right){=}^{20}{C}_{13}\left(0.7{\right)}^{13}\left(1-0.7{\right)}^{20-13}$
$=\frac{20!}{7!13!}\left(0.7{\right)}^{13}\left(1-0.7{\right)}^{7}$
$F\left(13\right)=0.1642$
b) $F\left(16\right)=P\left(x=16\right){=}^{20}{C}_{16}\left(0.7{\right)}^{16}\left(1-0.7{\right)}^{20-16}$
$=\frac{20!}{16!4!}\left(0.7{\right)}^{16}\left(1-0.7{\right)}^{4}$
$F\left(16\right)=0.1304$
c) $P\left(X\ge 16\right)=1-P\left(x<16\right)$
$=1-P\left(X\le 15\right)$
=1-0.7643(Using Binomial Table)
$P\left(X\ge 16\right)=0.2357$
d) $P\left(X\le 15\right)=P\left(x=0\right)+P\left(x=1\right)+P\left(x=2\right)+P\left(x=3\right)$
$+P\left(x=4\right)+P\left(x=5\right)+P\left(x=6\right)+P\left(x=7\right)+P\left(x=8\right)$
$+P\left(x=9\right)+P\left(x=10\right)+P\left(x=11\right)+P\left(x=12\right)$
$+P\left(x=13\right)+P\left(x=14\right)+P\left(x=15\right)$
$=0+0+0+0$
$+0+0+0.0002+0.0010+0.0039$
$+0.120+0.308+0.0554+0.1144$
$+0.1643+0.1916+0.1789$
$P\left(X\le 15\right)=0.7643$
e) $F\left(x\right)=np$
$\because X\sim B\left(np\right)$
$=20×0.7$
$F\left(x\right)=14$
f) $var\left(x\right)=npq$
$\because q=1-p$
$=20×0.7\left(1-0.7\right)$
$var\left(x\right)4.2$
$\sigma =\sqrt{var\left(x\right)}=2.04$