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Sagot :
Answer:
[tex](0.17 - 0.27) \pm 1.65\sqrt{\frac{0.17*0.83 + 0.27*0.73}{200}}[/tex], that is, option C
Step-by-step explanation:
From a random sample of 200 people in City C, 34 were found to subscribe to the streaming service. From a random sample of 200 people in City K, 54 were found to subscribe to the streaming service.
This means that the proportions are:
[tex]p_C = \frac{34}{200} = 0.17[/tex]
[tex]p_K = \frac{54}{200} = 0.27[/tex]
Subtraction of proportions:
In the confidence interval, we subtract the proportions. So:
[tex]p = p_C - p_K = 0.17 - 0.27[/tex]
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].
Standard error:
For a subtraction, as the standard deviation of the distribution is the square root of the sum of the variances, we have that:
[tex]\sqrt{\frac{\pi(1-\pi)}{n}} = \sqrt{\frac{0.17*0.83 + 0.27*0.73}{200}}[/tex]
90% confidence level
So [tex]\alpha = 0.1[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.1}{2} = 0.95[/tex], so [tex]Z = 1.645[/tex].
So the confidence interval is:
[tex](0.17 - 0.27) \pm 1.65\sqrt{\frac{0.17*0.83 + 0.27*0.73}{200}}[/tex], that is, option C
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