Question

Given two independent random samples with the following results:

n1=154pˆ1=0.27n1=154p^1=0.27   n2=351pˆ2=0.43n2=351p^2=0.43

Use this data to find the 98%98%confidence interval for the true difference between the population proportions.

Copy Data

Step 1 of 3 :  

Find the critical value that should be used in constructing the confidence interval.

Step 2: Find the value of the standard error. Round your answer to three decimal places

Step 3: Construct the 98% confidence interval. Round your answers to three decimal places

Answer 1

Solution

Back-up Theory

100(1 - α) % Confidence Interval for (p₁ - p₂) is:

[(p₁cap – p₂capx/n) ± MoE, …………………………………………………….. (1)

where

MoE = (Z_α/2)√[p_cap(1 - p_cap){(1/n₁) + (1/n₂)}] ………………………..………….. (2)

with

p₁cap and p₂cap being the corresponding sample proportions,

p_cap = (np₁cap + mp₂cap)/(n + m),

Z_α/2 is the upper (α /2)% point of N(0, 1),

n₁ and n₂ being the two sample sizes.

Now, to work out the solution,

Step 1 of 3 : Critical value that should be used in constructing the confidence interval = Z_0.01 = 2.3263 [Using Excel Function: Statistical NORMSINV]

Step 2: Standard error = √[p_cap(1 - p_cap){(1/n₁) + (1/n₂)}] = 0.0469

= 0.047

Step 3: 98% confidence interval = [- 0.269 < (p1 – p2) < - 0.051] Answer

Details of calculations

n1	154	p1cap	0.27
n2	351	p2cap	0.43
pcap	0.38120792	1 - pcap	0.618792

{(1/n1)+(1/n2)}	0.00934251	SE	0.046945
p1cap - p2cap	-0.16	α	0.02
Zα/2	2.32634787	1 - (α/2)	0.99

MoE	0.10920933
LB	-0.2692093
UB	-0.0507906

DONE