We appreciate your visit to Part 1 Solve all problems in this part Question 1 A 2020 survey of 1 250 American adults indicates that 75 of cell phone owners. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
a) With 95% confidence, we need a sample size of 1,086 to estimate the true proportion within 2%.
b) The point estimate is 75%, the margin of error is 2.2%, and the confidence interval is 72.8% to 77.2%.
c) We can be 95% confident that between 72.8% and 77.2% of Americans access the internet on their cell phones.
a) To estimate the true proportion of Americans who access the internet on their cell phone within 2% with 95% confidence, we need to determine the required sample size. We can use the formula:
[tex]n = (Z^2 * p * (1 - p)) / E^2[/tex]
where n is the required sample size, Z is the Z-score corresponding to the desired level of confidence (in this case, 95% confidence corresponds to a Z-score of approximately 1.96)
p is the estimated proportion from the pilot survey (in this case, 75%), and E is the desired margin of error (in this case, 2%).
Substituting the values into the formula:
[tex]n = (1.96^2 * 0.75 * (1 - 0.75)) / (0.02^2)[/tex]
n ≈ 1,085.33
We round up to the nearest whole number, so the required sample size is 1,086.
b) To calculate a 95% confidence interval for the true proportion of Americans who access the internet on their phones, we can use the formula:
(i) Point estimate: The point estimate is the proportion observed in the sample. In this case, the point estimate is 75%.
(ii) Margin of error: The margin of error can be calculated using the formula:
[tex]ME = Z * \sqrt{(p * (1 - p)) / n)}[/tex]
where ME is the margin of error, Z is the Z-score corresponding to the desired level of confidence (in this case, approximately 1.96), p is the estimated proportion, and n is the sample size.
Substituting the values into the formula:
[tex]ME = 1.96 * \sqrt{\sqrt{x} (0.75 * (1 - 0.75)) / 1,250)}[/tex]
ME ≈ 0.022
(iii) Confidence Interval: The confidence interval can be calculated by subtracting the margin of error from the point estimate and adding it to the point estimate.
Confidence Interval = Point estimate ± Margin of error
Confidence Interval = 75% ± 2.2%
c) The 95% confidence interval for the true proportion of Americans who access the internet on their phones is approximately 72.8% to 77.2%.
This means that we can be 95% confident that the true proportion of Americans who access the internet on their phones falls within this interval.
In other words, based on the sample data, we estimate that between 72.8% and 77.2% of Americans access the internet on their cell phones.
For more question on proportion visit:
https://brainly.com/question/1496357
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