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 487 eggs in group I boxes, of which a field count showed about 270 hatched. Group II nesting boxes were placed in highly visible locations and grouped closely together. There were a total of 808 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 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

Does the interval indicate that the proportion of eggs hatched from group I nest boxes is higher than, lower than, or equal to the proportion of eggs hatched from group II nest boxes?

Because the interval contains only negative numbers, we can say that a higher proportion of eggs hatched in highly visible, closely grouped nesting boxes.We can not make any conclusions using this confidence interval.     Because the interval contains only positive numbers, we can say that a higher proportion of eggs hatched in well-separated and well-hidden nesting boxes.Because the interval contains both positive and negative numbers, we can not say that a higher proportion of eggs hatched in well-separated and well-hidden nesting boxes.

(d) What conclusions about placement of nest boxes can be drawn? In the article discussed in the Focus Problem, additional concerns are raised about the higher cost of placing and maintaining group I nest box placements. Also at issue is the cost efficiency per successful wood duck hatch.

A greater proportion of wood duck eggs hatch if the eggs are laid in highly visible, closely grouped nesting boxes.No conclusion can be made.     A greater proportion of wood duck eggs hatch if the eggs are laid in well-separated, well-hidden nesting boxes.The eggs hatch equally well in both conditions.

Answer 1

Part a)
p̂ = X / n = 270/487 = 0.554

p̂ ± Z(α/2) √( (p * q) / n)
0.5544 ± Z(0.05/2) √( (0.5544 * 0.4456) / 487)
Z(α/2) = Z(0.05/2) = 1.96
Lower Limit = 0.554 - Z(0.05) √( (0.5544 * 0.4456) / 487) = 0.51
upper Limit = 0.554 + Z(0.05) √( (0.5544 * 0.4456) / 487) = 0.599
95% Confidence interval is ( 0.51 , 0.599 )
( 0.510 < P < 0.599 )

Part b)
p̂ = X / n = 262/808 = 0.324

p̂ ± Z(α/2) √( (p * q) / n)
0.3243 ± Z(0.05/2) √( (0.3243 * 0.6757) / 808)
Z(α/2) = Z(0.05/2) = 1.96
Lower Limit = 0.324 - Z(0.05) √( (0.3243 * 0.6757) / 808) = 0.292
upper Limit = 0.324 + Z(0.05) √( (0.3243 * 0.6757) / 808) = 0.357
95% Confidence interval is ( 0.292 , 0.357 )
( 0.292 < P < 0.357 )

Part c)
(p̂1 - p̂2) ± Z(α/2) * √( ((p̂1 * q̂1)/ n1) + ((p̂2 * q̂2)/ n2) )
Z(α/2) = Z(0.05 /2) = 1.96
Lower Limit = ( 0.5544 - 0.3243 )- Z(0.05/2) * √(((0.5544 * 0.4456 )/ 487 ) + ((0.3243 * 0.6757 )/ 808 ) = 0.175
upper Limit = ( 0.5544 - 0.3243 )+ Z(0.05/2) * √(((0.5544 * 0.4456 )/ 487 ) + ((0.3243 * 0.6757 )/ 808 )) = 0.285
95% Confidence interval is ( 0.175 , 0.285 )
( 0.175 < ( P1 - P2 ) < 0.285 )

Because the interval contains only positive numbers, we can say that a higher proportion of eggs hatched in well-separated and well-hidden nesting boxes.

Part d)

A greater proportion of wood duck eggs hatch if the eggs are laid in well-separated, well-hidden nesting boxes.