Question

1. A simple random sample was conducted of 1405 Canadian adults. They were asked whether they were in favor of tighter enforcement of tobacco laws. Of the 1405 adults, 1060 responded yes. Obtain a 95% confidence interval for the proportion of Canadian adults who are in favor of tighter enforcement of tobacco laws.
2. Of 15, 344 Virginians tested for COVID-19 in March of 2020, 1,484 had positive or presumptive positive results for the virus. Assuming the other 49 states had results similar to Virginia, obtain a 90% confidence level for the proportion of Americans tested that would receive a positive result for COVID-19.
3. Of the 157 graduates of Page County High School in 2015, 68% claimed they had tried smoking weed at some point in their high school years. A 95% confidence level for this claim was .234 to.405. Find the point estimate  (p hat) and the margin of error E for this data.

Answer 1

1)
sample proportion, = 0.7544
sample size, n = 1405
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.7544 * (1 - 0.7544)/1405) = 0.011

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

Margin of Error, ME = zc * SE
ME = 1.96 * 0.011
ME = 0.0216

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.7544 - 1.96 * 0.011 , 0.7544 + 1.96 * 0.011)
CI = (0.733 , 0.776)

2)

sample proportion, = 0.0967
sample size, n = 15344
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.0967 * (1 - 0.0967)/15344) = 0.002

Given CI level is 90%, hence α = 1 - 0.9 = 0.1
α/2 = 0.1/2 = 0.05, Zc = Z(α/2) = 1.64

Margin of Error, ME = zc * SE
ME = 1.64 * 0.002
ME = 0.0033

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.0967 - 1.64 * 0.002 , 0.0967 + 1.64 * 0.002)
CI = (0.093 , 0.1)

c)

phat = ( upper bound+ lower bound)/2
= ( 0.234 + 0.405)/2
= 0.320

MArgin of error = upper bound - phat
= 0.405 - 0.320
= 0.085