Respuesta :
Answer:
a) 0.9
b) Mean = 1.58
Standard Deviation = 0.89
Step-by-step explanation:
We are given the following in the question:
A marketing firm is considering making up to three new hires.
Let X be the variable describing the number of hiring in the company.
Thus, x can take values 0,1 ,2 and 3.
[tex]P(x\geq 2) = 50\%= 0.5\\P(x = 0) = 10\% = 0.1\\P(x = 3) = 18\% = 0.18[/tex]
a) P(firm will make at least one hire)
[tex]P(x\geq 2) = P(x=2) + P(x=3)\\0.5 = P(x=2) + 0.18\\ P(x=2) = 0.32[/tex]
Also,
[tex]P(x= 0) +P(x= 1) + P(x= 2) + P(x= 3) = 1\\ 0.1 + P(x= 1) + 0.32 + 0.18 = 1\\ P(x= 1) = 1- (0.1+0.32+0.18) = 0.4[/tex]
[tex]\text{P(firm will make at least one hire)}\\= P(x\geq 1)\\=P(x=1) + P(x=2) + P(x=3)\\ = 0.4 + 0.32 + 0.18 = 0.9[/tex]
b) expected value and the standard deviation of the number of hires.
[tex]E(X) = \displaystyle\sum x_iP(x_i)\\=0(0.1) + 1(0.4) + 2(0.32)+3(0.18) = 1.58[/tex]
[tex]E(x^2) = \displaystyle\sum x_i^2P(x_i)\\=0(0.1) + 1(0.4) + 4(0.32) +9(0.18) = 3.3\\V(x) = E(x^2)-[E(x)]^2 = 3.3-(1.58)^2 = 0.80\\\text{Standard Deviation} = \sqrt{V(x)} = \sqrt{0.8036} = 0.89[/tex]
From the probability distribution, we have that:
a) 0.9 = 90% probability that the firm will make at least one hire.
b)
i. The expected value is of 1.58.
ii. The standard deviation is of 0.9.
--------------------
10% chance that it will not make any hires.
This means that [tex]P(X = 0) = 0.1[/tex]
18% chance that it will make all three hires.
This means that [tex]P(X = 3) = 0.18[/tex]
50% chance of hiring at least two candidates.
This means that:
[tex]P(X = 2) + P(X = 3) = 0.5[/tex]
[tex]P(X = 2) + 0.18 = 0.5[/tex]
[tex]P(X = 2) = 0.32[/tex]
Item a:
- We need to find P(X = 1) = x.
- The sum of all probabilities is 100% = 1, thus:
[tex]0.1 + x + 0.32 + 0.18 = 1[/tex]
[tex]0.6 + x = 1[/tex]
[tex]x = 0.4[/tex]
The probability of at least one is:
[tex]P(X \geq 1) = P(X = 1) + P(X = 2) + P(X = 3) = 0.4 + 0.32 + 0.18 = 0.9[/tex]
0.9 = 90% probability that the firm will make at least one hire.
Item b:
- The expected value is the sum of the multiplication of each outcome by it's probability. Then:
[tex]E(X) = 0(0.1) + 1(0.4) + 2(0.32) + 3(0.18) = 1.58[/tex]
The expected value is of 1.58.
- The standard deviation is the square root of the sum of the difference squared between each outcome and the mean, multiplied by it's probability. Then:
[tex]\sqrt{V(x)} = \sqrt{0.1(0 - 1.58)^2 + 0.4(1 - 1.58)^2 + 0.32(2 - 1.58)^2 + 0.18(3 - 1.58)^2} = 0.9[/tex]
The standard deviation is of 0.9.
A similar problem is given at https://brainly.com/question/14464022
