This Question: 6 pts poll found that 382 of 2,351 adults aged 18 or older have a

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This Question: 6 pts poll found that 382 of 2,351 adults aged 18 or older have at least one tattoo. (a) Obtain a point estimate for the proportion of interval for the adults who have at least one tattoo. of adults with at least one tattoo. confidence interval for the proportion of aduts with at least one tattoo 99% the level of confidence on the width of the interval? @p=(Round to three docinal places as needed.) Construct the 90% contidence interval. Select the correct choice below and, if necessary i compliete your choice OA. Lower bound: Upper bound: (Round to three deçimal places as needed.) O B. The (c) Construct the 99% for constructing a confidence interval are not satisfied. confidence interval. Select the correct choice below and, E necessary, ti in the answer boxes to Lower bound: Upper bound: (Round to three decimal places as needed.) O B. The requirements for are not satisfied OB. widens the interval. Increasing the level of confidence has no effect on tell the effect of increasing the O D. It is not possible to for constructing a interval in parts (D) and (c) were not met

Explanation / Answer

a.
Given that,
possibile chances (x)=382
sample size(n)=2351
success rate ( p )= x/n = 0.1625
success probability,( po )=0.5
failure probability,( qo) = 0.5
null, Ho:p=0.5  
alternate, H1: p!=0.5
level of significance, = 0.05
from standard normal table, two tailed z /2 =1.96
since our test is two-tailed
reject Ho, if zo < -1.96 OR if zo > 1.96
we use test statistic z proportion = p-po/sqrt(poqo/n)
zo=0.16248-0.5/(sqrt(0.25)/2351)
zo =-32.7303
| zo | =32.7303
critical value
the value of |z | at los 0.05% is 1.96
we got |zo| =32.73 & | z | =1.96
make decision
hence value of | zo | > | z | and here we reject Ho
p-value: two tailed ( double the one tail ) - Ha : ( p != -32.73035 ) = 0
hence value of p0.05 > 0,here we reject Ho
ANSWERS
---------------
null, Ho:p=0.5
alternate, H1: p!=0.5
test statistic: -32.7303
critical value: -1.96 , 1.96
decision: reject Ho
p-value: 0

b.
TRADITIONAL METHOD
given that,
possibile chances (x)=382
sample size(n)=2351
success rate ( p )= x/n = 0.162
I.
sample proportion = 0.162
standard error = Sqrt ( (0.162*0.838) /2351) )
= 0.008
II.
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, = 0.1
from standard normal table, two tailed z /2 =1.645
margin of error = 1.645 * 0.008
= 0.013
III.
CI = [ p ± margin of error ]
confidence interval = [0.162 ± 0.013]
= [ 0.15 , 0.175]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
possibile chances (x)=382
sample size(n)=2351
success rate ( p )= x/n = 0.162
CI = confidence interval
confidence interval = [ 0.162 ± 1.645 * Sqrt ( (0.162*0.838) /2351) ) ]
= [0.162 - 1.645 * Sqrt ( (0.162*0.838) /2351) , 0.162 + 1.645 * Sqrt ( (0.162*0.838) /2351) ]
= [0.15 , 0.175]
-----------------------------------------------------------------------------------------------
interpretations:
1. We are 90% sure that the interval [ 0.15 , 0.175] contains the true population proportion
2. If a large number of samples are collected, and a confidence interval is created
for each sample, 90% of these intervals will contains the true population proportion

c.
TRADITIONAL METHOD
given that,
possibile chances (x)=382
sample size(n)=2351
success rate ( p )= x/n = 0.162
I.
sample proportion = 0.162
standard error = Sqrt ( (0.162*0.838) /2351) )
= 0.008
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 =2.576
margin of error = 2.576 * 0.008
= 0.02
III.
CI = [ p ± margin of error ]
confidence interval = [0.162 ± 0.02]
= [ 0.143 , 0.182]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
possibile chances (x)=382
sample size(n)=2351
success rate ( p )= x/n = 0.162
CI = confidence interval
confidence interval = [ 0.162 ± 2.576 * Sqrt ( (0.162*0.838) /2351) ) ]
= [0.162 - 2.576 * Sqrt ( (0.162*0.838) /2351) , 0.162 + 2.576 * Sqrt ( (0.162*0.838) /2351) ]
= [0.143 , 0.182]
-----------------------------------------------------------------------------------------------
interpretations:
1. We are 99% sure that the interval [ 0.143 , 0.182] contains the true population proportion
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 true population proportion

d.
increasing the confidence interval narrowing the interval

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