Question

In the following probability distribution, the random variable x represents the number of activities a parent of a K to 5 th-grade student is involved in. Complete parts (a) through (f) below.

x	0	1	2	3	4
P(x)	0.167	0.194	0.344	0.178	0.117

(a) Verify that this is a discrete probability distribution.

This is a discrete probability distribution because the sum of the probabilities is _______  and each probability is _______ 

(b) Graph the discrete probability distribution. Choose the correct graph below.

(c) Compute and interpret the mean of the random variable x .

The mean is _______  activities.

(Type an integer or a decimal. Do not round.)

Which of the following interpretations of the mean is correct?

A. As the number of experiments decreases, the mean of the observations will approach the mean of the random variable.

B. The observed value of an experiment will be less than the mean of the random variable in most experiments.

C. As the number of experiments increases, the mean of the observations will approach the mean of the random variable.

D. The observed value of an experiment will be equal to the mean of the random variable in most experiments.

(d) Compute the standard deviation of the random variable x.

The standard deviation is _______  activities.

(Round to one decimal place as needed.)

(e) What is the probability that a randomly selected student has a parent involved in three activities?

The probability is

(Type an integer or a decimal. Do not round.)

(f) What is the probability that a randomly selected student has a parent involved in three or four activities?

The probability is

(Type an integer or a decimal. Do not round.)

Answer 1

a) This is a discrete probability distribution because the sum of the probabilities is 1 and each probability is occurred against a discrete random variable.[here, sum of the probabilities is = 0.167+0.194+0.344+0.178+0.117=1]

b) the graph of the distribution is --

0.35 0.349 0.37 0.251 0.167 0,199 0.178 0.15 0.051 0 1 2 3 4

the correct graph is (c)...[ans]

c) the mean of the random variable X is -->

E(X)=

Στίp()

=0*0.167+1*0.194+2*0.344+3*0.178+4*0.117=1.884

the correct interpretation is , as the no of experiments increases , the mean of the observations approaches to the mean of the random variable . because , for n tends to infinite , the all observations tends to takes the value of mean and as a result it the correct interpretation will be (c).....[ans]

d) The formula of v(X), where X is the r.v.

V(X)=E(X²) -E²(X)

now,

E(X²)= 0² *0.167+1² *0.194+2² *0.344+3² *0.178+4² *0.117=5.044=5.0

and E²(X)=(1.884)² =3.549456

then, V(X)=5.044 -3.549456=1.494544

so, standard deviation =SD=sqrt(1.494544)=1.22251544 =1.2...[Ans]

e) The probability that randomly selected student have a parent involved in 3 activities

=p(the student's parent involved in exactly 3 activities)

=p(X=3)=0.178 .... [Ans]

f) The probability that randomly selected student have a parent involved in 3 or 4 activities

=p( parent involved in 3 activities or involved in 4 activities)

=p(parent involved in 3 activities) + p(parent involved in 4 activities)

=p(X=3)+p(X=4)

=0.178+0.117=0.295....[Ans]