Final answer:
To compute a 90% confidence interval for the proportion of letters mailed in the United States that were delivered the next day, we calculate the sample proportion, standard error, margin of error, and then determine the lower and upper limits of the confidence interval. The result is that we are 90% confident that the true proportion falls between 0.699 and 0.779 after rounding to two decimal places.
Step-by-step explanation:
To compute a 90% confidence interval for the proportion of all letters mailed in the United States that were delivered the day after they were mailed, based on the given sample where 277 out of 375 letters were delivered the next day, we use the following steps:
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- Calculate the sample proportion (p-hat), which is the number of successes (letters delivered the next day) divided by the sample size: p-hat = 277 / 375.
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- Determine the Z-score associated with a 90% confidence level. For a 90% confidence interval, the Z-score is approximately 1.645 because we exclude 5% in each tail of the normal distribution.
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- Calculate the standard error (SE) using the formula SE = sqrt(p-hat(1 - p-hat) / n), where n is the sample size.
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- Calculate the margin of error (ME) by multiplying the Z-score by the standard error: ME = Z * SE.
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- Determine the lower and upper limits of the confidence interval using the sample proportion and the margin of error: Lower limit = p-hat - ME, Upper limit = p-hat + ME.
After calculations:
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- The sample proportion (p-hat) is 0.739.
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- The standard error (SE) calculated to three decimal places is approximately 0.024.
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- The margin of error (ME) is 1.645 * 0.024 ≈ 0.040.
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- The lower limit of the 90% confidence interval is 0.739 - 0.040 = 0.699.
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- The upper limit is 0.739 + 0.040 = 0.779.
Therefore, we are 90% confident that the true proportion of letters delivered the next day in the United States falls between 0.699 and 0.779 when rounded to two decimal places.