Answer the following questions showing all work. Full credit will not be given t

Question

Answer the following questions showing all work. Full credit will not be given to answers without work shown. If you use Minitab Express include the appropriate output (copy + paste) along with an explanation. Output without explanation will not receive full credit. Round all answers to 3 decimal places. If you have any questions, post them to the course discussion board.

1. In Summer 2016 a sample of 440 Penn State World Campus STAT 200 students was asked how many languages they can speak. Below is the probability distribution of their responses. (30 points)

X = Languages

1

2

3

4            

5

Probability

.7000

.2318

.0068

.0023

A. Is number of languages spoken a discrete or continuous variable? Explain why.

B. What proportion of students in this sample can speak exactly 3 languages?

C. Compute the mean of X [i.e., E(X)].

D. Compute the standard deviation of X.

E. Complete the cumulative probability row of the following table:

X = Languages

1

2

3

4            

5

Probability

.7000

.2318

See part B

.0068

.0023

Cumulative Probability

F. If you were to randomly select one student from this sample, what is the probability that he or she would speak more than two languages?

X = Languages

1

2

3

4            

5

Probability

.7000

.2318

.0068

.0023

Explanation / Answer

Ans:

a)Number of languages spoken is a discrete random variable.

b)P(1)+P(2)+P(3)+P(4)+P(5)=1

0.7+0.2318+P(3)+0.0068+0.0023=1

P(3)=1-0.9409

P(3)=0.0591

c) Mean=1*0.7+2*0.23+3*0.0591+4*0.0068+5*0.0023=1.3796=1.38

d)Variance=(1.38-1)^2*0.7+(1.38-2)^2*0.2318+(1.38-3)^2*0.0591+(4-1.38)^2*0.0068+(5-1.38)^2*0.0023=0.422

standard deviation=sqrt(0.422)=0.65

e)

Probability of speaking of more than 2 languages=0.0591+0.0068+0.0023=0.0682

or

P(x>=2)=1-P(x<=2)=1-0.9318=0.0682

prob. Cummulative prob. 1 0.7 0.7 2 0.2318 0.9318 3 0.0591 0.9909 4 0.0068 0.9977 5 0.0023 1

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