Textbook Question
True or False? In Exercises 1 and 2, determine whether the statement is true or false. If it is false, rewrite it as a true statement.
The point estimate for the population proportion of failures is 1-p^
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8. Sampling Distributions & Confidence Intervals: Proportion
Confidence Intervals for Population Proportion
1:39 minutesProblem 6.3.10
Textbook Question
In Exercises 7–10, use the confidence interval to find the margin of error and the sample proportion.
(0.087, 0.263)
Verified step by step guidance1
Step 1: Understand the problem. The given confidence interval is (0.087, 0.263). The goal is to find the margin of error (E) and the sample proportion (p̂). The confidence interval is expressed as (p̂ - E, p̂ + E), where p̂ is the sample proportion and E is the margin of error.
Step 2: Calculate the sample proportion (p̂). The sample proportion is the midpoint of the confidence interval. To find it, use the formula: p̂ = (lower bound + upper bound) / 2. Substitute the values: lower bound = 0.087 and upper bound = 0.263.
Step 3: Calculate the margin of error (E). The margin of error is the distance from the sample proportion to either the lower or upper bound of the confidence interval. Use the formula: E = (upper bound - lower bound) / 2. Substitute the values: lower bound = 0.087 and upper bound = 0.263.
Step 4: Verify your calculations. Ensure that the sample proportion (p̂) and margin of error (E) satisfy the confidence interval formula: (p̂ - E, p̂ + E). This step ensures that the calculations are consistent with the given interval.
Step 5: Interpret the results. The sample proportion (p̂) represents the best estimate of the population proportion based on the sample, and the margin of error (E) indicates the range of uncertainty around this estimate within the given confidence level.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Confidence Interval
A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence. For example, a 95% confidence interval suggests that if we were to take many samples and build intervals in this way, approximately 95% of them would contain the true parameter.
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Margin of Error
The margin of error quantifies the uncertainty associated with a sample estimate. It is calculated as half the width of the confidence interval. In the given interval (0.087, 0.263), the margin of error would be (0.263 - 0.087) / 2, indicating how much the sample proportion could vary from the true population proportion.
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Sample Proportion
The sample proportion is the ratio of the number of successes in a sample to the total number of observations in that sample. It is a point estimate of the population proportion. In the context of the confidence interval provided, the sample proportion can be found as the midpoint of the interval, which represents the best estimate of the true proportion based on the sample data.
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