Chapter 6, Section 1-CI, Exercise 020 Have You Made a Friend Online? A survey 1

Question

Chapter 6, Section 1-CI, Exercise 020

Have You Made a Friend Online?

A survey1 of 1060 randomly selected US teens ages 13 to 17 found that 605 of them say they have made a new friend online.

1Lenhart A, “Teens, Technology, and Friendships,” Pew Research Center, pewresearch.org, August 6, 2015.

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Chapter 6, Section 1-CI, Exercise 020

Have You Made a Friend Online?

A survey1 of 1060 randomly selected US teens ages 13 to 17 found that 605 of them say they have made a new friend online.

1Lenhart A, “Teens, Technology, and Friendships,” Pew Research Center, pewresearch.org, August 6, 2015.

(a) Find a 90% confidence interval for the proportion, p, of all US teens who have made a new friend online.

Round your answers to three decimal places.

The 90% confidence interval is to .

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(b) Give the best estimate for p and give the margin of error for the estimate.

Round your answers to three decimal places.

p^=

margin of error=

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(c) Use the interval to determine whether we can be 90% confident that more than half of US teens have made a new friend online.

Yes No

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Explanation / Answer

TRADITIONAL METHOD
given that,
possibile chances (x)=605
sample size(n)=1060
success rate ( p )= x/n = 0.5708
I.
sample proportion = 0.5708
standard error = Sqrt ( (0.5708*0.4292) /1060) )
= 0.0152
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 * 0.0152
= 0.025
III.
CI = [ p ± margin of error ]
confidence interval = [0.5708 ± 0.025]
= [ 0.5457 , 0.5958]
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DIRECT METHOD
given that,
possibile chances (x)=605
sample size(n)=1060
success rate ( p )= x/n = 0.5708
CI = confidence interval
confidence interval = [ 0.5708 ± 1.645 * Sqrt ( (0.5708*0.4292) /1060) ) ]
= [0.5708 - 1.645 * Sqrt ( (0.5708*0.4292) /1060) , 0.5708 + 1.645 * Sqrt ( (0.5708*0.4292) /1060) ]
= [0.5457 , 0.5958]
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interpretations:
1. We are 90% sure that the interval [ 0.5457 , 0.5958] 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

[ANSWERS]
a. [0.546 , 0.596]
b. ( p )= x/n = 0.5708 , ME = 0.025
c. Yes

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