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 491 eggs in group I boxes, of which a field count showed about 260 hatched. Group II nesting boxes were placed in highly visible locations and grouped closely together. There were a total of 816 eggs in group II boxes, of which a field count showed about 268 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 99% 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 99% confidence interval for p₂. (Round your answers to three decimal places.)

lower limit
upper limit

(c) Find a 99% confidence interval for p₁ − p₂. (Round your answers to three decimal places.)

lower limit
upper limit

Answer 1

Solution :

Given that n1 = 491 , x1 = 260 , n2 = 816 , x2 = 268

(a)
=> Proportion p1 = x1/n1

= 260/491

= 0.5295

= 0.530 (rounded)

=> q1 = 1 - p1 = 0.470

=> For 99% confidence interval , Z = 2.58

=> The 99% confidence interval for p1 is

=> p1 +/- Z*sqrt(p1*q1/n1)

=> 0.530 +/- 2.58*sqrt(0.530*0.470/491)

=> (0.4719 , 0.5881)

=> (0.472 , 0.588) (rounded)

=> Lower limit = 0.472

=> Upper limit = 0.588

(b)
=> Proportion p2 = x2/n2

= 268/816

= 0.3284

= 0.328 (rounded)

=> q2 = 1 - p2 = 0.672

=> For 99% confidence interval , Z = 2.58

=> The 99% confidence interval for p1 is

=> p2 +/- Z*sqrt(p2*q2/n2)

=> 0.328 +/- 2.58*sqrt(0.328*0.672/816)

=> (0.2856 , 0.3704)

=> (0.286 , 0.370) (rounded)

=> Lower limit = 0.286

=> Upper limit = 0.370

(c)
=> The 99% confidence interval for p1 − p2 is

=> (p1 - p2) +/- Z*sqrt((p1*q1/n1) + (p2*q2/n2))

=> (0.530 - 0.328) +/- 2.58*sqrt((0.530*0.470/491) + (0.328*0.672/816))

=> 0.202 +/- 2.58*sqrt(0.0005 + 0.0003)

=> 0.202 +/- 0.073

=> (0.129 , 0.275)

=> Lower limit = 0.129

=> Upper limit = 0.275