Question

On March 30, 2015, President Obama had an approval rating of 47% (according to a Gallup Poll). Suppose a sample of 100 Americans is taken that same week. What is the probability that the sample proportion has between a 40% and a 50% approval rating? (Use 4 or more decimal places in your computations!)

Answer 1

Prob = 0.6449

Explanation:

Given that on March 30, 2015, President Obama had an approval rating of 47% (according to a Gallup Poll).

i.e. population proportion P = 0.47

Sampling proportion p follows a normal distribution with mean = p = 0.47 and

std deviation =
`\sqrt{(pq)/(n) } =\sqrt{(0.47(1-0.47))/(100) }\\=0.0499`

Probability that the sample proportion has between a 40% and a 50% approval rating

= P(0.4<p<0.5)

Convert to z score

Z score for 0.4 =
`(0.4-0.47)/(0.0499) =-1.40`

Z score for 0.5 =
`(0.5-0.47)/(0.0499) =0.601`

P(0.4<p<0.5) = P(-1.40<z<0.60)

= 0.4192+0.2257

=0.6449