Chapter 3, Section 4, Exercise 115 The following gives information about the pro

Question

Chapter 3, Section 4, Exercise 115 The following gives information about the proportion of a sample that agree with a certain statement. Use StatKey or other technology to find a confidence interval at the given confidence level for the proportion of the population to agree, using percentiles from a bootstrap distribution. Sta Key tip Use CI for Single Proportion" and then "Edit Data" to enter the sample information. Find a 99% confidence interval if, in a random sample of 1000 people, 382 agree. 578 disagree, and 40 cant decide. Click here to access Statkey. Round your answers to three decimal places. The 99% confidence interval is to

Explanation / Answer

exercise 115.
TRADITIONAL METHOD
given that,
possibile chances (x)=382
sample size(n)=1000
success rate ( p )= x/n = 0.382
I.
sample proportion = 0.382
standard error = Sqrt ( (0.382*0.618) /1000) )
= 0.015
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 * 0.015
= 0.04
III.
CI = [ p ± margin of error ]
confidence interval = [0.382 ± 0.04]
= [ 0.342 , 0.422]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
possibile chances (x)=382
sample size(n)=1000
success rate ( p )= x/n = 0.382
CI = confidence interval
confidence interval = [ 0.382 ± 2.576 * Sqrt ( (0.382*0.618) /1000) ) ]
= [0.382 - 2.576 * Sqrt ( (0.382*0.618) /1000) , 0.382 + 2.576 * Sqrt ( (0.382*0.618) /1000) ]
= [0.342 , 0.422]
-----------------------------------------------------------------------------------------------
interpretations:
1. We are 99% sure that the interval [ 0.342 , 0.422] 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

exercise114.

TRADITIONAL METHOD
given that,
possibile chances (x)=112
sample size(n)=400
success rate ( p )= x/n = 0.28
I.
sample proportion = 0.28
standard error = Sqrt ( (0.28*0.72) /400) )
= 0.022
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 * 0.022
= 0.037
III.
CI = [ p ± margin of error ]
confidence interval = [0.28 ± 0.037]
= [ 0.243 , 0.317]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
possibile chances (x)=112
sample size(n)=400
success rate ( p )= x/n = 0.28
CI = confidence interval
confidence interval = [ 0.28 ± 1.645 * Sqrt ( (0.28*0.72) /400) ) ]
= [0.28 - 1.645 * Sqrt ( (0.28*0.72) /400) , 0.28 + 1.645 * Sqrt ( (0.28*0.72) /400) ]
= [0.243 , 0.317]
-----------------------------------------------------------------------------------------------
interpretations:
1. We are 90% sure that the interval [ 0.243 , 0.317] 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

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