1. Assume that the per capita income of residents in a country is normally distr

Question

1. Assume that the per capita income of residents in a country is normally distributed with mean $1000 and variance $10,000 ($ squared).
a. What is the probability that the per capita income lies between $800 and
$1200?
b. What is the probability that it exceeds $1200?
c. What is the probability that it is less than $800?
d. Is it true that the probability of per capita income exceeding $5000 is practically zero?

Continuing with problem 1, based on a random sample of 1000 members,
suppose that you find the sample mean income, , to be $900.
a. Given that , what is the probability of obtaining such a sample
mean value?
b. Based on the sample mean, establish a 95% confidence interval for and
find out if this confidence interval includes . If it does not, what
conclusions would you draw?
c. Using the test of significance approach, decide whether you want to accept
or reject the hypothesis that . Which test did you use and why?

Explanation / Answer

1.a. Assume that the per capita income of residents in a country is normally distributed with mean $1000 and variance $10,000

For x=800, z1=(x-)/=(800-1000)/100=-2

For x=1200, z2=(1200-1000)/100=2

Thus, P(800)<x<1200)=P(x<1200)-P(x<800)

=P(z2<2)-P(z1<-2)=0.9772-0.0228=0.9544

b.P(x>1200)=P(z>2)=1-P(z<2)=1-0.9772=0.0228

c.P(x<800)=P(z<-2)=0.0288

d.For x=5000, z=(5000-1000)/100=40

Thus, the probability of per capita income exceeding $5000 is practically zero.

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