Suppose that 50% of all adult Americans believe that federal budget deficits at

Question

Suppose that 50% of all adult Americans believe that federal budget deficits at recent levels cause long-term harm to the nation’s economy.

12. (10 points) What is the probability that more than 58% of a random sample of 250 adult Americans would hold this belief? Show your calculation

13. (5 points) If the sample of 250 was made up of adults randomly selected from each of the eight regions of the U.S. (i.e. New England, Midwest, Great Lakes, Plains, Southeast, Southwest, Rocky Mountains and Far West) which sampling design was used?

Explanation / Answer

let X be the random variable denoting the number of adult americans out of 250 americans believe that federal budget deficits at recent levels cause long-term harm to the nation’s economy.

Suppose that 50% of all adult Americans believe that federal budget deficits at recent levels cause long-term harm to the nation’s economy.

hence X~Bin(250,0.5)

but here n=250 is very large.

hence X can be approximated by a Normal distribution with mean E[X]=250*0.5=125

and variance=V[X]=250*0.5*(1-0.5)=62.5

12. now 58% of 250 people means 250*58%=145 people

so the probability that more than 58% of a random sample of 250 adult Americans would hold this belief is

P[X>145]=P[(X-125)/sqrt(62.5)>(145-125)/sqrt(62.5)]=P[Z>2.5298] where Z~N(0,1)

             =1-P[Z<2.5298]=1-0.994294=0.005706 [answer]

13. If the sample of 250 was made up of adults randomly selected from each of the eight regions of the U.S. (i.e. New England, Midwest, Great Lakes, Plains, Southeast, Southwest, Rocky Mountains and Far West) then stratified sampling was used where the 8 regions were the eight strata and from each strata adult americans are randomly selected such that the total number of selected americans were 250.

here basically the whole population was divided into 8 regions and then the sample was collected randomly from the regions individually such that total sample size is 250 [answer]

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