Question

A random sample of 100 people from Country S had 15 people with blue eyes. A separate random sample of 100 people from Country B had 25 people with blue eyes.
Assuming all conditions are met, which of the following is a 95 percent confidence interval to estimate the difference in population proportions of people with blue eyes (Country S - Country B)?

A) (-0.01, 0.21)
B) (-0.15, -0.05)
C) (-0.19, -0.01)
D (-0.21, 0.01)
E) (-0.24, 0.01)

Answer 1

Given Information:

Sample size of country S = ns = 100

Sample size of country B = nb = 100

Number of people with blue eyes in country S = 15

Number of people with blue eyes in country B = 25

Confidence level = 95%

Required Information:

Difference in population proportion = ?

Answer:

A (-0.01, 0.21)

Step-by-step explanation:

The difference in mean is

μ = ps - pb

μ = 15/100 - 25/100

μ = 0.15 - 0.25

μ = 0.10

The difference in standard error is

ο = z*√(ps(1 - ps)/ns + pb(1 - pb)/nb)

The z-score corresponding to 95% confidence interval is 1.96

ο = 1.96*√(0.15(1 - 0.15)/100 + 0.25(1 - 0.25)/100)

ο = 1.96*√(0.15(0.85)/100 + 0.25(0.75)/100)

ο = 0.11

Therefore, the difference in population proportion with blue eyes is

μ ± ο

μ + ο, μ - ο

0.10 + 0.11, 0.10 - 0.11

0.21, -0.01

(-0.01, 0.21)

Therefore, the correct option is A.

Answer 2

Answer:

The correct answer is option (D)  (-0.21, 0.01)

Step-by-step explanation:

Solution:

Data Given;

Total number of people in country S = 100

people with blue eyes in country  S = 15

Total number of people in country B = 100

People with blue eyes in country  B = 25

Let the Y represent number of people with blue eyes in both countries and let Z represent total number of people in both country.

Therefore,

Y₁= 15

Y₂= 25

Z₁ = 100

Z₂ = 100

Sample proportion of country (S₁)  = Y1/Z1 = 15/100 = 0.15

Sample proportion of B country  (S₂) =  Y2/Z2 = 25/100 = 0.25

To calculate the 95% confidence interval, we use the formula;

Confidence interval (CI) =

(S₁ -S₂)  ± Zₐ₋₂ * √[S₁( 1-S₁)/z₁ + S₂(1-S₂)/z₂]

                                                                                                     

At 95% confidence interval, the z-score from standard normal table is  1.96.

Substituting into the formula, we have;

CI = (0.15-0.25) ± 1.96*√[0.15( 1-0.15/100) + 0.25(1-0.25)/100]

   =  -0.1 ± 1.96*√[(0.15*0.85)/100 + (0.25*0.75)/100]

   = -0.1 ± 1.96*√[0.1275/100+ 0.1875/100]

   = -0.1 ± 1.96*√[0.001275 + 0.001875]

   = -0.1 ± 1.96* √0.00315

  = -0.1 ± 1.96*0.0561

  = -0.1 +0.11

  = -0.1 -0.11 , -0.1 + 0.11

 CI  = (-0.21, 0.01)