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 489 eggs in group I boxes, of which a field count showed about 266 hatched. Group II nesting boxes were placed in highly visible locations and grouped closely together. There were a total of 798 eggs in group II boxes, of which a field count showed about 262 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 90% 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 90% confidence interval for p2. (Round your answers to three decimal places.)

lower limit

upper limit

(c) Find a 90% confidence interval for p1 − p2. (Round your answers to three decimal places.)

lower limit

upper limit

Answer 1

a)

sample proportion, = 0.544
sample size, n = 489
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.544 * (1 - 0.544)/489) = 0.0225

Given CI level is 90%, hence α = 1 - 0.9 = 0.1
α/2 = 0.1/2 = 0.05, Zc = Z(α/2) = 1.64

Margin of Error, ME = zc * SE
ME = 1.64 * 0.0225
ME = 0.0369

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.544 - 1.64 * 0.0225 , 0.544 + 1.64 * 0.0225)
CI = (0.507 , 0.581)
Lower limit = 0.507
Upper limit = 0.581

b)

sample proportion, = 0.328
sample size, n = 798
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.328 * (1 - 0.328)/798) = 0.0166

Given CI level is 90%, hence α = 1 - 0.9 = 0.1
α/2 = 0.1/2 = 0.05, Zc = Z(α/2) = 1.64

Margin of Error, ME = zc * SE
ME = 1.64 * 0.0166
ME = 0.0272

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.328 - 1.64 * 0.0166 , 0.328 + 1.64 * 0.0166)
CI = (0.301 , 0.355)
Lower limit = 0.301
upper limit = 0.355

c)

Here, , n1 = 489 , n2 = 798
p1cap = 0.544 , p2cap = 0.328

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.544 * (1-0.544)/489 + 0.328*(1-0.328)/798)
SE = 0.028

For 0.9 CI, z-value = 1.64
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.544 - 0.328 - 1.64*0.028, 0.544 - 0.328 + 1.64*0.028)
CI = (0.17 , 0.262)

Lower limit = 0.170
Upper imit = 0.262