Question

In the Focus Problem at the beginning of this chapter, a study was described comparing the hatch ratios of wood duck nesting boxes. Group I nesting boxes were well separated from each other and well hidden by available brush. There were a total of 495 eggs in group I boxes, of which a field count showed about 268 hatched. Group II nesting boxes were placed in highly visible locations and grouped closely together. There were a total of 784 eggs in group II boxes, of which a field count showed about 272 hatched.

(a) Find a point estimate p̂₁ for p₁, the proportion of eggs that hatch in group I nest box placements. (Round your answer to three decimal places.)
p̂₁ =  

Find a 95% confidence interval for p₁. (Round your answers to three decimal places.)

lower limit
upper limit

(b) Find a point estimate p̂₂ for p₂, the proportion of eggs that hatch in group II nest box placements. (Round your answer to three decimal places.)
p̂₂ =  

Find a 95% confidence interval for p₂. (Round your answers to three decimal places.)

lower limit
upper limit

(c) Find a 95% confidence interval for p₁ − p₂. (Round your answers to three decimal places.)

lower limit
upper limit

Answer 1

a)

p1cap = 268/495 = 0.541

sample proportion, = 0.541
sample size, n = 495
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.541 * (1 - 0.541)/495) = 0.0224

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

Margin of Error, ME = zc * SE
ME = 1.96 * 0.0224
ME = 0.0439

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.541 - 1.96 * 0.0224 , 0.541 + 1.96 * 0.0224)
CI = (0.497 , 0.585)

lower limit = 0.497
Upper limit = 0.585

b)

p2cap= 272/784 = 0.347

sample proportion, = 0.347
sample size, n = 784
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.347 * (1 - 0.347)/784) = 0.017

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

Margin of Error, ME = zc * SE
ME = 1.96 * 0.017
ME = 0.0333

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.347 - 1.96 * 0.017 , 0.347 + 1.96 * 0.017)
CI = (0.314 , 0.38)

lower limit = 0.314
Upper limit = 0.380

c)

Here, , n1 = 495 , n2 = 784
p1cap = 0.541 , p2cap = 0.347

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.541 * (1-0.541)/495 + 0.347*(1-0.347)/784)
SE = 0.0281

For 0.95 CI, z-value = 1.96
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.541 - 0.347 - 1.96*0.0281, 0.541 - 0.347 + 1.96*0.0281)
CI = (0.139 , 0.249)

lower limit = 0.139
Upper limit = 0.249