Question

The default rate on government-guaranteed student loans at a certain public four-year institution is 8 percent and 1,000 student loans are made at the institution. If we assume that the defaults are statistically independent across loans, the number of total defaults at the institution is a binomial random variable. However, we will approximate the binomial distribution by the standard normal distribution using the central limit theorem. Calculate the probability of 65 or fewer defaults using the central limit theorem. a. 4.019% b. 12. 188% C. 39.743% d. 28.001% e. 0.987% QUESTION 4 A bottling company uses a machine to fill bottles with olive oil. The bottles are designed to contain 475 mililitters. In fact, the contents vary according to a normal distribution with a mean of 477 mililitters and standard deviation of 3 mililitters. Nine bottles are randomly chosen and the total amount of olive oil in the nine bottles is measured, which is supposed to be 4275 mililitters if there is no error. What is the probability that the total amount of olive oil is less than 4275 mililitters? a. 3.593% b. 11.507% C. 2.275% d.6.681% e. 15.866%

Answer 1

3)

here mean of distribution=μ=np=		80
and standard deviation σ=sqrt(np(1-p))=		8.5790
for normal distribution z score =(X-μ)/σx
therefore from normal approximation of binomial distribution and continuity correction:

probability =

P(X<65)

=

P(Z<-1.75)=

0.0402~ 4.019 %

option A is correct

4)

mean μ=	477.000
standard deviation σ=	3.000
sample size       =n=	9
std error=σx̅=σ/√n=	1.0000

probability =

P(X<4275/9 =475)

=

P(Z<-2)=

0.0228 ~ 2.275 %

option C is correct