Answer:
a) left handed people
b) Binomial probability distribution with pdf
[tex]P(X=x)=15Cx0.1^{x} 0.9^{15-x}[/tex]
where x=0,1,2,...,15.
c) Histogram is attached
d) The shape of histogram depicts that distribution is rightly skewed.
e) 1.5
f) 1.35
g) 1.16
Step-by-step explanation:
a)
The random variable in the given scenario is " left handed people"
b)
The scenario represents the binomial probability distribution as the outcome is divided into one of two categories and experiment is repeated fixed number of times i.e. 15 and trails are independent. The pdf of binomial distribution is
[tex]P(X=x)=nCxp^{x} q^{n-x}[/tex]
Here n=15, p=0.1 and q=1-p=0.9.
So, the pdf would be
[tex]P(X=x)=15Cx0.1^{x} 0.9^{n-x}[/tex]
where x=0,1,2,...,15.
c)
Histogram is constructed by first computing probabilities on all x points i.e. x=0, x=1 , .... ,x=15 and then plotting all probabilities with respective x values. Histogram is in attached image.
d)
The tail of histogram is to the right side and thus the histogram depicts that given probability distribution is rightly skewed.
e)
The mean of binomial probability distribution is computed by multiplying number of trails and probability of success.
mean=np=15*0.1=1.5
f)
The variance of binomial probability distribution is computed by multiplying number of trails and probability of success and probability of failure.
variance=npq=15*0.1*0.9=1.35
g)
The standard deviation can be calculated by simply taking square root of variance
S.D=√npq=√1.35=1.16
The proportion of left-handed people follows a binomial distribution
The given parameters are:
[tex]\mathbf{n = 15}[/tex] -- the sample size
[tex]\mathbf{p = 10\%}[/tex] --- the proportion of left-handed people
(a) The random variable
The distribution is about left-handed people.
Hence, the random variable is left-handed people
(b) The probability distribution
If the proportion of left-handed people is 10%, then the proportion of right-handed people is 90%.
So, the probability distribution function is:
[tex]\mathbf{P(x) = ^nC_x p^x (1 - p)^{n -x}}[/tex]
This gives
[tex]\mathbf{P(x) = ^nC_x (10\%)^x (1 - 10\%)^{n -x}}[/tex]
[tex]\mathbf{P(x) = ^nC_x 0.1^x 0.9^{n -x}}[/tex]
Hence, the probability distribution function is [tex]\mathbf{P(x) = ^nC_x 0.1^x 0.9^{n -x}}[/tex]
(c) The histogram
To do this, we calculate P(x) for x = 0 to 15
[tex]\mathbf{P(0) = ^{15}C_0 \times 0.1^0 \times 0.9^{15 -0} = 0.206}[/tex]
[tex]\mathbf{P(1) = ^{15}C_1 \times 0.1^1 \times 0.9^{15 -1} = 0.343}[/tex]
.....
..
[tex]\mathbf{P(15) = ^{15}C_{15} \times 0.1^{15} \times 0.9^{15 -15} = 10^{-15}}[/tex]
See attachment for the histogram
(d) The mean
This is calculated as:
[tex]\mathbf{\bar x = np}[/tex]
So, we have:
[tex]\mathbf{\bar x = 15 \times 10\% }[/tex]
[tex]\mathbf{\bar x= 1.5}[/tex]
Hence, the mean is 1.5
(e) The variance
This is calculated as:
[tex]\mathbf{Var = np(1 - p)}[/tex]
So, we have:
[tex]\mathbf{Var = 15 \times 10\% \times (1 - 10\%)}[/tex]
[tex]\mathbf{Var = 1.35}[/tex]
Hence, the variance is 1.35
(f) The standard deviation
This is calculated as:
[tex]\mathbf{\sigma = \sqrt{Var}}[/tex]
So, we have:
[tex]\mathbf{\sigma = \sqrt{1.35}}[/tex]
[tex]\mathbf{\sigma =1.16}[/tex]
Hence, the standard deviation is 1.16
Read more about distributions at:
https://brainly.com/question/16355734
