Question

Independent random samples of

n₁ = 800

and

n₂ = 670

observations were selected from binomial populations 1 and 2, and

x₁ = 336

and

x₂ = 378

successes were observed.

(a) Find a 90% confidence interval for the difference (p₁ − p₂) in the two population proportions. (Round your answers to three decimal places.)
to

(b) What assumptions must you make for the confidence interval to be valid? (Select all that apply.)nq̂ > 5 for samples from both populationssymmetrical distributions for both populationsnp̂ > 5 for samples from both populationsindependent random samples

n₁ + n₂ > 1,000

(c) Can you conclude that there is a difference in the population proportions based on the confidence interval found in part (a)?

No. Since zero is not contained in the confidence interval, the two population proportions are likely to be equal.Yes. Since zero is contained in the confidence interval, the two population proportions are likely to be different.    No. Since zero is contained in the confidence interval, the two population proportions are likely to be equal.Nothing can be determined about the difference between the two population proportions.Yes. Since zero is not contained in the confidence interval, the two population proportions are likely to be different.

Answer 1

a)

Here, , n1 = 800 , n2 = 670
p1cap = 0.42 , p2cap = 0.5642

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.42 * (1-0.42)/800 + 0.5642*(1-0.5642)/670)
SE = 0.0259

For 0.9 CI, z-value = 1.64
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.42 - 0.5642 - 1.64*0.0259, 0.42 - 0.5642 + 1.64*0.0259)
CI = (-0.187 , -0.102)

b)

nq̂ > 5 for samples from both populationssymmetrical distributions for both populationsnp̂ > 5 for samples from both populationsindependent random samples

c)

.Yes. Since zero is not contained in the confidence interval, the two population proportions are likely to be different.