Question

IVILVIU Cancy OX. Healthcare A Gallup poll conducted in November 2010 found that 493 of 1050 adult Americans believe it is the responsibility of the federal government to make sure all Americans have healthcare coverage. (a) Obtain a point estimate for the proportion of adult Americans who believe it is the responsibility of the federal government to make sure all Americans have healthcare coverage. (b) Verify that the requirements for constructing a confidence interval for p are satisfied. (c) Construct a 95% confidence interval for the proportion of adult Americans who believe it is the responsibility of the federal government to make sure all Americans have healthcare coverage (d) Construct a 99% confidence interval for the proportion of adult Americans who believe it is the responsibility of the federal government to make sure all Americans have healthcare coverage. (e) What is the effect of increasing the level of confidence on the width of the interval? ... . .. . ..antin

Answer 1

a) $\widehat p$ = 493/1050 = 0.4695

b) n$\widehat p$(1 - $\widehat p$ ) = 1050 * 0.4695 * (1 - 0.4695) = 261.52

Since n$\widehat p$(1 - $\widehat p$ ) > 10 and n < 0.05N, so assume that the population is normally distributed.

c) At 95% confidence level, the critical value is z_0.025 = 1.96

The 95% confidence interval for population proportion is

$\widehat p$ +/- z_0.025 * sqrt($\widehat p$(1 - $\widehat p$ )/n)

= 0.4695 +/- 1.96 * sqrt(0.4695 * (1 - 0.4695)/1050)

= 0.4695 +/- 0.0302

= 0.4393, 0.4997

d) At 99% confidence level, the critical value is z_0.005 = 2.58

The 99% confidence interval for population proportion is

$\widehat p$ +/- z_0.005 * sqrt($\widehat p$(1 - $\widehat p$ )/n)

= 0.4695 +/- 2.58 * sqrt(0.4695 * (1 - 0.4695)/1050)

= 0.4695 +/- 0.0397

= 0.4298, 0.5092

e) As the confidence level increases, the width of the confidence interval also increases.