Question

Suppose you and three friends, Allan, Bella, and Chris, are designing a study to measure
the approval rating of the current president. For each case, construct the desired confidence
interval and briefly explain why the results are or are not valid.
(a) You poll 120 randomly selected people and find that 30 of them “approve”. You cnstruct
a 95% confidence interval for the proportion.
(b) Allan polls 200 people who he knows on Facebook and only 30 of them “approve”. He
constructs a 90% confidence interval for the proportion.
(c) Bella polls 50 randomly selected people and finds that 7 of them “approve”. She constructs a 99% confidence interval for the proportion.
(d) Chris polls 400 randomly selected people and finds that 21 of them “approve”. He
constructs a 99% confidence interval for the proportion

(e) If the true approval rating of the president was 11%, which of the above confidence intervals does it lie in?

Answer 1

a)
sample proportion, = 0.25
sample size, n = 120
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.25 * (1 - 0.25)/120) = 0.0395

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 * 0.0395
ME = 0.0774

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.25 - 1.96 * 0.0395 , 0.25 + 1.96 * 0.0395)
CI = (0.1726 , 0.3274)

b)

sample proportion, = 0.15
sample size, n = 200
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.15 * (1 - 0.15)/200) = 0.0252

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 * 0.0252
ME = 0.0413

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.15 - 1.64 * 0.0252 , 0.15 + 1.64 * 0.0252)
CI = (0.1087 , 0.1913)

c)

sample proportion, = 0.14
sample size, n = 50
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.14 * (1 - 0.14)/50) = 0.0491

Given CI level is 99%, hence α = 1 - 0.99 = 0.01
α/2 = 0.01/2 = 0.005, Zc = Z(α/2) = 2.58

Margin of Error, ME = zc * SE
ME = 2.58 * 0.0491
ME = 0.1267

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.14 - 2.58 * 0.0491 , 0.14 + 2.58 * 0.0491)
CI = (0.0133 , 0.2667)

d)

sample proportion, = 0.0525
sample size, n = 400
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.0525 * (1 - 0.0525)/400) = 0.0112

Given CI level is 99%, hence α = 1 - 0.99 = 0.01
α/2 = 0.01/2 = 0.005, Zc = Z(α/2) = 2.58

Margin of Error, ME = zc * SE
ME = 2.58 * 0.0112
ME = 0.0289

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.0525 - 2.58 * 0.0112 , 0.0525 + 2.58 * 0.0112)
CI = (0.0236 , 0.0814)

e)

it lies in the option b) and option c) 90% and 99% interval