Answer:
b. P(X≥40)= 0.30
c. P(X<35)= 0.55
d. E(X)= 33.75
Standard deviation 8.5
Step-by-step explanation:
Hello!
The variable of interest is
X: Number of new clients seeking tax advice.
a. Is this a valid probability distribution? Explain.
The conditions that distribution should be met to be considered a distribution of probability are two:
-All probabilities should be between 0 and 1
-The sum of all probabilities should be 1
Looking at the given data, the first condition checks.
∑f(x)= 0.05+0.20+0.30+0.15+0.10+0.10+0.10= 1 ⇒ The second condition checks.
This is a valid probability distribution.
b. What is the probability that Backens and Hayes LLC will obtain 40 or more new clients?
You can symbolize this as:
P(X≥40)
"40 or more new clients" means that they can obtain 40 or they can obtain 50 new clients, then the probability is
P(X≥40)= f(40) + f(45) + f(50)= 0.10 + 0.10 + 0.10= 0.30
c. What is the probability that Backens and Hayes LLC will obtain fewer than 35 new clients?
Symbolically:
P(X<35)
In this expresion there are included the situations: "30 new clients", "25 new clients" and "20 new clients", if you add the point probabilites of each situeation you'll get the asked one:
P(X<35)= f(30)+f(25)+f(20)= 0.30+0.20+0.05= 0.55
d. Compute the expected value, variance, and standard deviation of x.
To calculate the mean and standard deviation of a probability distribution you have to use the folowwing formulas:
Mean: E(X)= ∑xfi= 20*0.05 + 25*0.20 + 30*0.30 + 35*0.15 + 40*0.10 + 45*0.10 + 50*0.10= 33.75 new customers
Variance V(X)=∑x²fi - (∑xfi)²=
∑x²fi= 20²*0.05 + 25²*0.20 + 30²*0.30 + 35²*0.15 + 40²*0.10 + 45²*0.10 + 50²*0.10= 1211.25
V(X)= ∑x²fi - (∑xfi)²= 1211.25 - (33.75)²= 72.1875
Stander deviation √V(X)= √72.1875= 8.496 ≅ 8.50 new customers
I hope this helps!
