Question

In each of the following cases, compute 95 percent, 98 percent, and 99 percent confidence intervals for the population proportion p.

(a) pˆp^ = .6 and n = 100 (Round your answers to 3 decimal places.)

95 percent confidence intervals is [, ]

98 percent confidence intervals is [, ]

99 percent confidence intervals is [, ]

(b) pˆp^ = .5 and n = 299. (Round your answers to 3 decimal places.)

95 percent confidence intervals is [, ]

98 percent confidence intervals is [, ]

99 percent confidence intervals is [, ]

(c) pˆp^ = .7 and n = 121. (Round your answers to 3 decimal places.)

95 percent confidence intervals is [, ]

98 percent confidence intervals is [, ]

99 percent confidence intervals is [, ]

(d) pˆp^ = .8 and n = 56. (Round your answers to 3 decimal places.)

95 percent confidence intervals is [, ]

98 percent confidence intervals is [, ]

99 percent confidence intervals is [, ]

Answer 1

Part a)

i)

p̂ = 0.6
p̂ = 1 - p̂ = 0.4
n = 100
p̂ ± Z(α/2) √( (p * q) / n)
0.6 ± Z(0.05/2) √( (0.6 * 0.4) / 100)
Z(α/2) = Z(0.05/2) = 1.96
Lower Limit = 0.6 - Z(0.05) √( (0.6 * 0.4) / 100) = 0.504
upper Limit = 0.6 + Z(0.05) √( (0.6 * 0.4) / 100) = 0.696
95% Confidence interval is ( 0.504 , 0.696 )
( 0.504 < P < 0.696 )

ii)

p̂ = 0.6
p̂ = 1 - p̂ = 0.4
n = 100
p̂ ± Z(α/2) √( (p * q) / n)
0.6 ± Z(0.02/2) √( (0.6 * 0.4) / 100)
Z(α/2) = Z(0.02/2) = 2.326
Lower Limit = 0.6 - Z(0.02) √( (0.6 * 0.4) / 100) = 0.486
upper Limit = 0.6 + Z(0.02) √( (0.6 * 0.4) / 100) = 0.714
98% Confidence interval is ( 0.486 , 0.714 )
( 0.486 < P < 0.714 )

iii)

p̂ = 0.6
p̂ = 1 - p̂ = 0.4
n = 100
p̂ ± Z(α/2) √( (p * q) / n)
0.6 ± Z(0.01/2) √( (0.6 * 0.4) / 100)
Z(α/2) = Z(0.01/2) = 2.576
Lower Limit = 0.6 - Z(0.01) √( (0.6 * 0.4) / 100) = 0.4738
upper Limit = 0.6 + Z(0.01) √( (0.6 * 0.4) / 100) = 0.7262
99% Confidence interval is ( 0.474 , 0.726 )
( 0.474 < P < 0.726 )

Part b)

i)

p̂ = 0.5
p̂ = 1 - p̂ = 0.5
n = 299
p̂ ± Z(α/2) √( (p * q) / n)
0.5 ± Z(0.05/2) √( (0.5 * 0.5) / 299)
Z(α/2) = Z(0.05/2) = 1.96
Lower Limit = 0.5 - Z(0.05) √( (0.5 * 0.5) / 299) = 0.4433
upper Limit = 0.5 + Z(0.05) √( (0.5 * 0.5) / 299) = 0.5567
95% Confidence interval is ( 0.443 , 0.557 )
( 0.443 < P < 0.557 )

ii)
p̂ = 0.5
p̂ = 1 - p̂ = 0.5
n = 299
p̂ ± Z(α/2) √( (p * q) / n)
0.5 ± Z(0.02/2) √( (0.5 * 0.5) / 299)
Z(α/2) = Z(0.02/2) = 2.326
Lower Limit = 0.5 - Z(0.02) √( (0.5 * 0.5) / 299) = 0.4327
upper Limit = 0.5 + Z(0.02) √( (0.5 * 0.5) / 299) = 0.5673
98% Confidence interval is ( 0.433 , 0.567 )
( 0.433 < P < 0.567 )

iii)
p̂ = 0.5
p̂ = 1 - p̂ = 0.5
n = 299
p̂ ± Z(α/2) √( (p * q) / n)
0.5 ± Z(0.01/2) √( (0.5 * 0.5) / 299)
Z(α/2) = Z(0.01/2) = 2.576
Lower Limit = 0.5 - Z(0.01) √( (0.5 * 0.5) / 299) = 0.4255
upper Limit = 0.5 + Z(0.01) √( (0.5 * 0.5) / 299) = 0.5745
99% Confidence interval is ( 0.426 , 0.574 )
( 0.426 < P < 0.574 )

Part c)

i)

p̂ = 0.7
p̂ = 1 - p̂ = 0.3
n = 121
p̂ ± Z(α/2) √( (p * q) / n)
0.7 ± Z(0.05/2) √( (0.7 * 0.3) / 121)
Z(α/2) = Z(0.05/2) = 1.96
Lower Limit = 0.7 - Z(0.05) √( (0.7 * 0.3) / 121) = 0.6183
upper Limit = 0.7 + Z(0.05) √( (0.7 * 0.3) / 121) = 0.7817
95% Confidence interval is ( 0.618 , 0.782 )
( 0.618 < P < 0.782 )

ii)

p̂ = 0.7
p̂ = 1 - p̂ = 0.3
n = 121
p̂ ± Z(α/2) √( (p * q) / n)
0.7 ± Z(0.02/2) √( (0.7 * 0.3) / 121)
Z(α/2) = Z(0.02/2) = 2.326
Lower Limit = 0.7 - Z(0.02) √( (0.7 * 0.3) / 121) = 0.6031
upper Limit = 0.7 + Z(0.02) √( (0.7 * 0.3) / 121) = 0.7969
98% Confidence interval is ( 0.603 , 0.797 )
( 0.603 < P < 0.797 )

iii)

p̂ = 0.7
p̂ = 1 - p̂ = 0.3
n = 121
p̂ ± Z(α/2) √( (p * q) / n)
0.7 ± Z(0.01/2) √( (0.7 * 0.3) / 121)
Z(α/2) = Z(0.01/2) = 2.576
Lower Limit = 0.7 - Z(0.01) √( (0.7 * 0.3) / 121) = 0.5927
upper Limit = 0.7 + Z(0.01) √( (0.7 * 0.3) / 121) = 0.8073
99% Confidence interval is ( 0.593 , 0.807 )
( 0.593 < P < 0.807 )

Part d)

i)
p̂ = 0.8
p̂ = 1 - p̂ = 0.2
n = 56
p̂ ± Z(α/2) √( (p * q) / n)
0.8 ± Z(0.05/2) √( (0.8 * 0.2) / 56)
Z(α/2) = Z(0.05/2) = 1.96
Lower Limit = 0.8 - Z(0.05) √( (0.8 * 0.2) / 56) = 0.6952
upper Limit = 0.8 + Z(0.05) √( (0.8 * 0.2) / 56) = 0.9048
95% Confidence interval is ( 0.695 , 0.905 )
( 0.695 < P < 0.905 )

ii)
p̂ = 0.8
p̂ = 1 - p̂ = 0.2
n = 56
p̂ ± Z(α/2) √( (p * q) / n)
0.8 ± Z(0.02/2) √( (0.8 * 0.2) / 56)
Z(α/2) = Z(0.02/2) = 2.326
Lower Limit = 0.8 - Z(0.02) √( (0.8 * 0.2) / 56) = 0.6757
upper Limit = 0.8 + Z(0.02) √( (0.8 * 0.2) / 56) = 0.9243
98% Confidence interval is ( 0.676 , 0.924 )
( 0.676 < P < 0.924 )

iii)
p̂ = 0.8
p̂ = 1 - p̂ = 0.2
n = 56
p̂ ± Z(α/2) √( (p * q) / n)
0.8 ± Z(0.01/2) √( (0.8 * 0.2) / 56)
Z(α/2) = Z(0.01/2) = 2.576
Lower Limit = 0.8 - Z(0.01) √( (0.8 * 0.2) / 56) = 0.6623
upper Limit = 0.8 + Z(0.01) √( (0.8 * 0.2) / 56) = 0.9377
99% Confidence interval is ( 0.662 , 0.938 )
( 0.662 < P < 0.938 )