In the next presidential election of year 2020, five candidates running for the

Question

In the next presidential election of year 2020, five candidates running for the election are Bernie Sanders, Elizabeth Warren, Donald Trump, Ted Cruz, and John Kasich. It is desired to estimate the true proportion (pi) favoring one of the five candidates. a) Based on Sample size 1000 with 96% confidence, how close (plus or minus) should we expect our estimate to be to the true value? b) If we wanted to be 99% sure that our estimate was within 0.05 of the true value, how large the sample size should be? c) If 460 in the sample of 1000 were found to prefer Donald Trump, what is the lower limit of the two-sided 99% confidence interval for the true proportion (pi)? d) Based on the information in part (c), at the 0.025 level of significance test the hypothesis H_0: pi greaterthanorequalto 0.5 vs. H_1: pi

Explanation / Answer

a)for parameter p not known estimated p=0.5

here std error of mean =(p(1-p)/n)1/2 =0.0158

for 96% CI; z=2.0537

therefore margin of error or closeness from mean =z*std error =0.0325

b)here p=0.5 ; margin of error E =0.05

and for 99% CI ; z=2.5758

therefore smple size n=p(1-p)*(z/E)2 =664

c)here estimated p =460/1000=0.46

std error of mean =(p(1-p)/n)1/2 =0.0158

for 99% CI ; z=2.5758

therefore lower limit =p-z*std error =0.46-2.5758*0.0158=0.4194

d)here p=0.5 ; therefore std error of mean =(p(1-p)/n)1/2 =0.0158

phat=0.46

hence test stat z=(phat-p)/std error=(0.46-0.5)/0.0158=-2.8284

for 0.025 level critical value of z=-1.96

as test stat is lower then critical value we reject null hypothesis and conclude that proportion is less then 0.5 at 0.025 level of signficance.

e)p value for above test stat =0.0023

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