Exercise 2 Last year, ballots in Champaign-Urbana contained the following questi

Question

Exercise 2 Last year, ballots in Champaign-Urbana contained the following question to assess public opinion on an issue: "Should the State of Illinois legalize and regulate the sale and use of marijuana in a similar fashion as the State of Colorado?" Suppose that we would like to understand Champaign-Urbana's 2017 opinion on marijuana legalization. To satisfy our curiosity, we obtain a random sample of 120 Champaign-Urbanians and find that 87 support marijuana legalization (a) Construct a 99% confidence interval for p, the true proportion of Champaign-Urbanians that support marijuana legalization (b) Suppose that a pollster wants to estimate the true proportion of Champaign-Urbanians that support marijuana legalization to within 0.04, with 95% confidence. How many Champaign-Urbanians should this pollster poll? Assume the pollster has no prior knowledge about the proportion (c) Now assume the pollster believes that support for legalization is somewhere between 65% and 85% and they would like to estimate the true proportion of Champaign-Urbanians that support marijuana legalization to within 0.04, with 90% confidence. How many Champaign-Urbanians should this pollster poll?

Explanation / Answer

Proportion support marijuana i.e. p : 87/120 = 0.725

(a) 99% Confidence interval for p : p+-z(p(1-p)/n)0.5

= 0.725 +- 2.5758((0.725*0.275)/120)0.5

= (0.620, 0.830)

(b) Margin of error , z0.025 (p(1-p)/n)0.5 = 0.04

1.96((0.725)(0.275)/n)0.5 = 0.04

n = 479 (approx)

(c) Support is between 65% and 85%

so using 65% : 1.6449((0.65(0.35))/n)0.5 = 0.04 ; n = 385

for p = 85% : 1.6449((0.85)(0.15)/n)0.5 = 0.04 ; n = 216

so, 216<n<385

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