Question

1. Suppose the random variables ?? and ?? have the joint probability density function,

??(??,??) =(4?? + 2??) / 75 ?????? 0 < ?? < 3 ?????? 0 < ?? < ?? + 1
a) Determine the marginal probability density function of ??.
  

b) Determine the conditional probability of ?? given ?? = 2.

2. To estimate the average monthly rent for 1 bedroom apartments, 13 complexes were randomly selected in Orlando. The mean cost is $970 with a standard deviation of $72.

a) Find and interpret a 90% confidence interval for the mean monthly rent of all 1 bedroom apartments.


   b) If the confidence level was increased to 95%, what would happen to the width of the interval? Hint: Do not calculate a second interval; just think about the relationship between the    confidence level and the interval width.

Answer 1

As per company policy, I am answering question 1 Only.

Q2)

A)

Mean= 970.0000
sd= 72.00
u= 24.00
n= 13.00
alpha= 10%

T(a/2,n-1)
t(0.1/2,13-1)
1.782

CI = mean +-t(a/2,n-1)*(sd/sqrt(n))
lower = 970-1.782*(72/sqrt(13))= 934.41
upper = 970 + 1.782*(72/sqrt(13))= 1005.59

B)

Width of the confidence interval = margin of error, ME = t(a/2,n-1)*(sd/sqrt(n))

With the increase in confidence level, the value of t(a/2,n-1) increases. And hence the width of the confidence interval also increases.