Question

c) Repeat part () using a confidence level of 80% d) Compare the contdence intervals from parts (b) and (c) and identify the interval that is wider. Why is it wider? parameter increases parameter increases

Answer 1

The 99% confidence interval

p̂ = 0.495, 1 - p̂ = 0.505, n = 568, α = 0.01

The Zcritical (2 tail) for α = 0.01, is 2.576

The Confidence Interval is given by p̂ ± ME, where

$ME = Zcritical * \sqrt{\frac{\hat{p}(1-\hat{p})}{n} }= 2.576 * \sqrt{\frac{0.495*0.505}{568}} = 0.054$

The Lower Limit = 0.495 - 0.054 = 0.441

The Upper Limit = 0.495 + 0.054 =0.549

The 99% Confidence Interval is 0.441 < p < 0.549

The 80% confidence interval

p̂ = 0.495, 1 - p̂ = 0.505, n = 568, α = 0.20

The Zcritical (2 tail) for α = 0.20, is 1.282

The Confidence Interval is given by p̂ ± ME, where

$ME = Zcritical * \sqrt{\frac{\hat{p}(1-\hat{p})}{n} }= 1.282 * \sqrt{\frac{0.495*0.505}{568}} = 0.027$

The Lower Limit = 0.495 - 0.027 = 0.468

The Upper Limit = 0.495 + 0.027 =0.522

The 80% Confidence Interval is 0.468 < p < 0.522

The Width of the 99% CI is = 0.549 - 0.441 = 0.108

The Width of the 80% CI is = 0.522- 0.468 = 0.054

Therefore for question (d) OPTION (C) is the correct answer. The 99% confidence interval is wider than the 80% confidence interval. As the confidence interval widens, the probability the probability that the confidence interval actually does contain the population parameter increases.