Use the information in problem 3 to construct a 99% confidence interval for the

Question

Use the information in problem 3 to construct a 99% confidence interval for the difference in the proportion of New Hampshire residents and Maine residents who enjoy mountain biking. Interpret your interval (i.e. what, exactly, does it mean in this case). Round to 4 (FOUR) decimal places.

INFORMATION FROM PROBLEM 3- - - > A biking industry analyst believes that the proportion of New Hampshire residents who enjoy mountain biking is greater than the proportion of Maine residents who enjoy mountain biking. She randomly samples 400 New Hampshire residents and 60 of those sampled indicate an affinity for mountain biking. She randomly samples 500 Maine residents, 40 of whom indicate they enjoy mountain biking. At the .01 level of significance, is there enough information to conclude that the proportion of New Hampshire residents who enjoy mountain biking is greater than the proportion of Maine residents who enjoy mountain biking? List and clearly label all eight steps. Round to 4 (FOUR) decimal places.

Explanation / Answer

PART A.
Given that,
sample one, x1 =60, n1 =400, p1= x1/n1=0.15
sample two, x2 =40, n2 =500, p2= x2/n2=0.08
null, Ho: p1 = p2
alternate, H1: p1 > p2
level of significance, = 0.01
from standard normal table,right tailed z /2 =2.326
since our test is right-tailed
reject Ho, if zo > 2.326
we use test statistic (z) = (p1-p2)/(p^q^(1/n1+1/n2))
zo =(0.15-0.08)/sqrt((0.111*0.889(1/400+1/500))
zo =3.32
| zo | =3.32
critical value
the value of |z | at los 0.01% is 2.326
we got |zo| =3.32 & | z | =2.326
make decision
hence value of | zo | > | z | and here we reject Ho
p-value: right tail - Ha : ( p > 3.3204 ) = 0.00045
hence value of p0.01 > 0.00045,here we reject Ho
ANSWERS
---------------
null, Ho: p1 = p2
alternate, H1: p1 > p2
test statistic: 3.32
critical value: 2.326
decision: reject Ho
p-value: 0.00045

the proportion of New Hampshire residents who enjoy mountain biking is greater than the proportion of Maine residents who enjoy mountain biking

PART B.
TRADITIONAL METHOD
given that,
sample one, x1 =60, n1 =400, p1= x1/n1=0.15
sample two, x2 =40, n2 =500, p2= x2/n2=0.08
I.
standard error = sqrt( p1 * (1-p1)/n1 + p2 * (1-p2)/n2 )
where
p1, p2 = proportion of both sample observation
n1, n2 = sample size
standard error = sqrt( (0.15*0.85/400) +(0.08 * 0.92/500))
=0.0216
II.
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, = 0.01
from standard normal table, two tailed z /2 =2.58
margin of error = 2.58 * 0.0216
=0.0557
III.
CI = (p1-p2) ± margin of error
confidence interval = [ (0.15-0.08) ±0.0557]
= [ 0.0143 , 0.1257]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
sample one, x1 =60, n1 =400, p1= x1/n1=0.15
sample two, x2 =40, n2 =500, p2= x2/n2=0.08
CI = (p1-p2) ± sqrt( p1 * (1-p1)/n1 + p2 * (1-p2)/n2 )
where,
p1, p2 = proportion of both sample observation
n1,n2 = size of both group
a = 1 - (confidence Level/100)
Za/2 = Z-table value
CI = confidence interval
CI = [ (0.15-0.08) ± 2.58 * 0.0216]
= [ 0.0143 , 0.1257 ]
-----------------------------------------------------------------------------------------------
interpretations:
1) we are 99% sure that the interval [ 0.0143 , 0.1257] contains the difference between
true population proportion P1-P2
2) if a large number of samples are collected, and a confidence interval is created
for each sample, 99% of these intervals will contains the difference between
true population mean P1-P2

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