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Construct a confidence interval for p1​−p2​ at the given level of confidence. x1​=367,n1​=535,x2​=436,n2​=562,90% confidence The researchers are \% confident the difference between the two population proportions, p1​−p2​, is between and (Use ascending order. Type an integer or decimal rounded to three decimal places as needed.)

User Bitbyter
by
8.0k points

1 Answer

5 votes

Given:

1

=

367

,

1

=

535

,

2

=

436

,

2

=

562

,

confidence level

=

90

%

x

1

=367,n

1

=535,x

2

=436,n

2

=562,confidence level=90%

First, calculate the sample proportions:

^

1

=

1

1

=

367

535

p

^

1

=

n

1

x

1

=

535

367

^

2

=

2

2

=

436

562

p

^

2

=

n

2

x

2

=

562

436

Next, calculate the standard error:

SE

=

^

1

(

1

^

1

)

1

+

^

2

(

1

^

2

)

2

SE=

n

1

p

^

1

(1−

p

^

1

)

+

n

2

p

^

2

(1−

p

^

2

)

Then, find the critical value for a 90% confidence level. Since the confidence level is given as 90%, the corresponding two-tailed critical value is

=

1.645

z=1.645 (obtained from a standard normal distribution table).

Finally, plug the values into the formula to calculate the confidence interval:

Confidence Interval

=

(

(

^

1

^

2

)

±

SE

)

Confidence Interval=((

p

^

1

p

^

2

)±z⋅SE)

Let's calculate it step by step:

^

1

=

367

535

0.686

p

^

1

=

535

367

≈0.686

^

2

=

436

562

0.775

p

^

2

=

562

436

≈0.775

SE

=

0.686

(

1

0.686

)

535

+

0.775

(

1

0.775

)

562

0.034

SE=

535

0.686(1−0.686)

+

562

0.775(1−0.775)

≈0.034

Confidence Interval

=

(

(

0.686

0.775

)

±

1.645

0.034

)

Confidence Interval=((0.686−0.775)±1.645⋅0.034)

Now, calculate the upper and lower bounds of the confidence interval:

Lower bound

=

(

0.686

0.775

)

1.645

0.034

Lower bound=(0.686−0.775)−1.645⋅0.034

Upper bound

=

(

0.686

0.775

)

+

1.645

0.034

Upper bound=(0.686−0.775)+1.645⋅0.034

Rounding the values to three decimal places, the confidence interval is approximately:

Confidence Interval

=

(

0.102

,

0.065

)

Confidence Interval=(−0.102,−0.065)

Therefore, the researchers are 90% confident that the difference between the two population proportions,

1

2

p

1

−p

2

, is between -0.102 and -0.065 (in ascending order).

User Anton N
by
8.2k points