Constructing Confidence Intervals In Exercises 11 and 12, construct 90% and 95% confidence intervals for the population proportion. Interpret the results and compare the widths of the confidence intervals.
New Year’s Resolutions In a survey of 1790 U.S. adults in a recent year, 816 have a New Year’s resolution related to their health. (Adapted from Finder)

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Step 1: Identify the sample proportion (p̂) and the sample size (n). The sample proportion is calculated as p̂ = x / n, where x is the number of successes (816) and n is the total sample size (1790).

Step 2: Determine the critical z-scores for the desired confidence levels. For a 90% confidence interval, the critical z-score is approximately 1.645. For a 95% confidence interval, the critical z-score is approximately 1.96.

Step 3: Calculate the standard error (SE) of the sample proportion using the formula SE = sqrt((p̂ * (1 - p̂)) / n).

Step 4: Construct the confidence intervals using the formula: CI = p̂ ± (z * SE), where z is the critical z-score for the desired confidence level. Perform this calculation separately for the 90% and 95% confidence levels.

Step 5: Compare the widths of the confidence intervals. The width of a confidence interval is given by 2 * (z * SE). Note that the 95% confidence interval will be wider than the 90% confidence interval because a higher confidence level requires a larger margin of error.

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주요 개념

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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, such as 90% or 95%. It provides an estimate of uncertainty around the sample proportion, indicating how much the sample result might vary if different samples were taken.

Population Proportion

The population proportion is the ratio of members of a population that have a particular characteristic, expressed as a fraction or percentage. In this context, it refers to the proportion of U.S. adults who have a New Year’s resolution related to their health, which is estimated from the sample data.

Width of Confidence Intervals

The width of a confidence interval reflects the precision of the estimate; narrower intervals indicate more precise estimates of the population parameter. The width is influenced by the sample size and the confidence level chosen; higher confidence levels result in wider intervals, while larger sample sizes yield narrower intervals, allowing for more accurate estimates.