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# A 10 points for a poll of 300 voters what is the

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a. (10 points) For a poll of 300 voters, what is the expected proportion of Democtratic voters favoring Obama? What is the variance of this proportion? Expected proportion=true proportion=.46 or 46%. Variance of the proportion is p*(1-p)/n=.46*.54/300=.00083. b. (10 points) What is the probability that the result of the poll in part a has Obama receiving less than 43% of the Democratic vote? We will apply the normal approximation to a proportional random variable. That is, approximately, the sample proportion ˆ p is distributed normally with mean .46 and variance .00083. Every value of ˆ p thus has a corresponding standard normal z-value, found via the transformation: ˆ ˆ ˆ .46 .46 .029 ˆ ˆ *(1 ) .00083 p p p p z p p n - - - = = = - . We’re interested in the value ˆ p =.43. This corresponds to the z-value z= -1.03. From the standard normal table, P( z<-1.03)=1-P(z<1.03)=1-.8485=.1515=P( ˆ p <43%). There’s a 15.15% chance that the sample proportion of democrats favoring Obama is less than 43%.

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Economics 390, Midterm #2 Page 5 6. Each spring break thousands of university students descend on Cancun, Mexico. Suppose the following table presents the joint PDF of two variables for each student: the number of trips they make to the beach, and the number of times they get a sunburn. 0 Sunburns 1 Sunburn 1 Trips to Beach .2 .3 2 Trips to Beach .1 X a. (10 points) What is the value of X? The sum of terms in the cells must equal 1, so X must be .4. b. (10 points) Find the marginal PDF for the number of sunburns. This is found by adding for each column (the sunburns) the joint PDF values. P(Sunburns=0)= .3, P(Sunburns=1)=.7 c. (10 points) Amongst the students who take 2 trips to the beach, what is the expected number of sunburns? .5 of all students take 2 trips to the beach. .1 of all students get 0 sunburns. Thus, .2 of students who take 2 trips to the beach get 0 sunburns. Or in our standard notation, P(S=0|B=2)=.2 By the complement rule, .8 of the students who take 2 trips of the beach get 1 sunburn. P(S=1|B=2)=.8. The expected number of sunburns for those who take two trips to the beach is 0*.2+1*.8=.8. Or, E[S|B=2]=.8 d. (10 points*) Calculate the covariance between the number of trips to the beach and the number of sunburns. It is fastest to use the formula Cov(B,S)=E[BS]-µ B µ S .
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