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Summary:
A 95% confidence interval for the true proportion of people who own tablets is 0.693 to 0.767.
This means that we are 95% confident that the true proportion of people who own tablets is between 69.3% and 76.7%.
Step-by-step explanation:
A confidence interval is a range of values that is likely to contain the true population proportion. The confidence interval is constructed by taking the sample proportion and adding and subtracting a margin of error. The margin of error is calculated using the sample proportion, the sample size, and the z-score for the desired confidence level.
In this problem, we are given a sample proportion of 189 / 270 = 0.70, a sample size of 270, and a z-score for a 95% confidence interval of 1.96.
The first step is to find the margin of error. The margin of error is calculated using the following formula:
Code snippet
ME = z * √p(1 - p) / n
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where z is the z-score, p is the sample proportion, and n is the sample size.
Plugging in the values, we get:
Code snippet
ME = 1.96 * √0.70(1 - 0.70) / 270
= 0.037
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The second step is to construct the confidence interval. The confidence interval is calculated using the following formula:
Code snippet
CI = p ± ME
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where p is the sample proportion and ME is the margin of error.
Plugging in the values, we get:
Code snippet
CI = 0.70 ± 0.037
= 0.663 to 0.737
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Therefore, a 95% confidence interval for the true proportion of people who own tablets is 66.3% to 73.7%.
Interpretation:
We are 95% confident that the true proportion of people who own tablets is between 66.3% and 73.7%. This means that if we were to take repeated samples of 270 people, 95% of the confidence intervals would contain the true proportion of people who own tablets.
It is important to note that the confidence interval does not tell us the exact value of the true proportion of people who own tablets. It only tells us a range of values that is likely to contain the true proportion of people who own tablets.