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

Irvin Dukes 2021-12-16 Answered
Consider a binomial experiment with n=20 and p=.70.
1. Compute f (12).
2. Compute f (16).
3. Compute P(x16).
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Expert Answer

Heather Fulton
Answered 2021-12-17 Author has 31 answers

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 \(\displaystyle{P}{\left({x}\geq{16}\right)}\ {i}{s}\ {0.2375}\).

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Neil Dismukes
Answered 2021-12-18 Author has 37 answers
Step 1
Given:
p=0.70
n=20
Formula binomial probability:
f(k)=(nk)×pk×(1p)nk
a) f(12)=(2012)×0.7012×(10.70)2012=0.114397
b) f(16)=(2016)×0.7016×(10.70)2016=0.130421
c) Add the corresponding probabilities:
P(X16)=f(16)+f(17)+f(18)+f(19)+f(20)=0.2375
d) Use the complement rule for probabilities:
P(X15)=1P(X16)=10.2375=0.7625
Step 2
e) The mean of a binomial distribution is the sample size n and the probability p:
μ=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.
σ2=npq=np(1p)=20(0.70)(10.70)=4.2
σ=npq=np(1p)=20(0.70)(10.70)2..0494
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Jeffrey Jordon
Answered 2021-12-27 Author has 2027 answers

Step 1
Given that
XBinomial(n=20, p=0.7)
a) F(x)P(x)=nCxPx(1P)nx
=n!(nx)!x!Px(1P)nx
F(13)=20C13(0.7)13(10.7)2013
=20!7!13!(0.7)13(10.7)7
F(13)=0.1642
b) F(16)=P(x=16)=20C16(0.7)16(10.7)2016
=20!16!4!(0.7)16(10.7)4
F(16)=0.1304
c) P(X16)=1P(x<16)
=1P(X15)
=1-0.7643(Using Binomial Table)
P(X16)=0.2357
d) P(X15)=P(x=0)+P(x=1)+P(x=2)+P(x=3)
+P(x=4)+P(x=5)+P(x=6)+P(x=7)+P(x=8)
+P(x=9)+P(x=10)+P(x=11)+P(x=12)
+P(x=13)+P(x=14)+P(x=15)
=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(X15)=0.7643
e) F(x)=np
XB(np)
=20×0.7
F(x)=14
f) var(x)=npq
q=1p
=20×0.7(10.7)
var(x)4.2
σ=var(x)=2.04

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