Respuesta :
Answer:
Step-by-step explanation:
Confidence interval for the difference in the two proportions is written as
Difference in sample proportions ± margin of error
Sample proportion, p= x/n
Where x = number of success
n = number of samples
For district C,
x = 22
n1 = 100
p1 = 22/100 = 0.22
For district M,
x = 26
n2 = 100
p2 = 26/100 = 0.26
Margin of error = z√[p1(1 - p1)/n1 + p2(1 - p2)/n2]
To determine the z score, we subtract the confidence level from 100% to get α
α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025
This is the area in each tail. Since we want the area in the middle, it becomes
1 - 0.025 = 0.975
The z score corresponding to the area on the z table is 1.96. Thus, the z score for the confidence level of 95% is 1.96
Margin of error = 1.96 × √[0.22(1 - 0.22)/100 + 0.26(1 - 0.26)/100]
= 1.96 × √0.00364
= 0.12
Confidence interval = (0.22 - 0.26) ± 0.12
= - 0.04 ± 0.12
The required 95 percent confidence interval for the difference is ( 0.8, 0.16).
Given that,
A random sample of 100 voters in District C, 22 percent responded yes to The question "Are you in favor of an increase in state spending on the arts?"
An independent random sample of 100 voters in District M resulted in 26 percent responding yes to the question.
We have to determine,
A 95 percent confidence interval for the difference.
According to the question,
A random sample of 100 voters in District C, 22 percent responded yes to The question "Are you in favor of an increase in state spending on the arts?"
An independent random sample of 100 voters in District M resulted in 26 percent responding yes to the question.
Confidence interval for the difference in the two proportions is written as,
Difference in sample proportions ± margin of error
Sample proportion, p= x/n
Where x = number of success
n = number of samples
From a random sample of 100 voters in District C,
[tex]x = 22\\\\n_1= 100\\\\p_1 = \dfrac{22}{100} = 0.22[/tex]
From a random sample of 100 voters in District M,
[tex]x = 26\\\\n_1= 100\\\\p_1 = \dfrac{26}{100} = 0.26[/tex]
Therefore,
[tex]Margin \ of \ error = \sqrt{\dfrac{p_1(1-p_1}{n_1} + \dfrac{p_2(1-p_2)}{n_2}}[/tex]
To determine the z score, subtract the confidence level from 100% to get b
α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025
This is the area in each tail. Since, the area in the middle, it becomes
1 - 0.025 = 0.975
The z-score corresponding to the area on the z table is 1.96. Thus, the z score for the confidence level of 95% is 1.96,
[tex]Margin \ of \ error = \sqrt{\dfrac{0.22(1-0.22)}{100} + \dfrac{0.26(1-0.26)}{100}}[/tex]
[tex]= \sqrt{\dfrac{0.22(0.78)}{100} +\dfrac{0.26(0.76)}{100}}\\\\= \sqrt{\dfrac{0.17}{100}+ \dfrac{0.19}{100}}\\\\= \sqrt{\dfrac{0.36}{100}}\\\\= \sqrt{0.0036}\\\\= 0.06[/tex]
Then, Confidence interval is given as;
[tex](0.22 - 0.26) \pm 0.12\\\\(-0.4) \pm 0.12\\\\(-0.4+0.12 , -0.4-0.12)\\\\(0.8, 0.16)[/tex]
Hence, The required 95 percent confidence interval for the difference is( 0.8, 0.16)
To know more about Confidence interval click the link given below.
https://brainly.com/question/15349022
