Lane Ch 104: Why Is A 99% Confidence Interval Wider Than A 9

Lane Ch 104 Why Is A 99 Confidence Interval Wider Than A 95 Conf

Why is a 99% confidence interval wider than a 95% confidence interval? A person claims to be able to predict the outcome of flipping a coin, and this person is correct 16/25 times. Compute the 95% confidence interval on the proportion of times this person can predict coin flips correctly. What conclusion can you draw about this test of his ability to predict the future? You take a sample of 22 from a population of test scores, with a sample mean of 60. If the population standard deviation is known, find the confidence interval; if not, use the sample standard deviation of 10 to find the 99% confidence interval. You asked 10 psychology majors about their nightly study times: 2, 1.5, 3, 2, 3.5, 1, 0.5, 3, 2, 4. Find the 95% and 90% confidence intervals for the population mean. Understand what it means to be “90% confident” when constructing a confidence interval for a mean. A committee surveyed 81 jurors, with a mean wait of 8 hours and a standard deviation of 4 hours. Determine the confidence interval for the mean waiting time. A sample of 84 used car prices has a mean of $6,425 and standard deviation of $3,156; construct a 95% confidence interval for the true mean cost. A restaurant chain’s soda serving times from a sample of 12: mean 13.30 oz, SD 1.55 oz. Find the error bound. An article reports survey results about racial groups' openness to interracial marriage. For example, 86% of blacks would welcome a white person into their family. Find the 95% confidence interval for the proportion of all black adults who would welcome a white person. Another survey states that 76% of U.S. workers believe they will work past retirement age; with a confidence level of 95% and a margin of error of ±3%, determine the sample proportion and the actual confidence interval. Explain the significance of the confidence interval to a general audience.

Paper For Above instruction

Confidence intervals are fundamental tools in statistical inference, providing a range within which we expect the true population parameter to lie with a certain level of confidence. Understanding why a higher confidence level, such as 99%, produces a wider interval than a lower level, like 95%, hinges on the principles of statistical variability and the need to account for more uncertainty as confidence increases.

At its core, a confidence interval estimates a population parameter (such as a proportion or mean) based on sample data, incorporating the variability inherent in sampling. Increasing the confidence level from 95% to 99% means we require a broader safety margin to ensure that the true parameter is captured in the interval in 99 out of 100 cases, rather than 95 out of 100. This broader safety margin translates mathematically into a larger critical value or z-score used in the calculation, thereby widening the interval.

Specifically, for proportions, the confidence interval is calculated as:

CI = p̂ ± Z* √(p̂(1 - p̂) / n)

where p̂ is the sample proportion, Z* is the critical z-value corresponding to the confidence level, and n is the sample size.

As the confidence level increases from 95% to 99%, the Z value increases from approximately 1.96 to 2.576. Since the margin of error is proportional to Z, a higher Z* value results in a wider interval, reflecting increased uncertainty that the interval encompasses the true parameter.

This concept applies equally to confidence intervals for means, where the critical value (t-distribution for unknown population standard deviations and small samples, or Z-distribution for known variances) also increases with higher confidence levels, leading to a wider interval.

In practical terms, choosing a higher confidence level indicates a desire for greater assurance in the estimate, at the expense of precision. For example, a public opinion poll might report a 95% confidence interval of 55% to 65%, indicating moderate certainty, while a 99% interval might stretch from 53% to 67%, offering greater confidence but less specificity.

Therefore, the widening of the interval is a natural consequence of the need to control for more uncertainty and achieve the desired confidence level. It's a fundamental trade-off in statistical inference: more confidence entails less precision, but more assurance that the interval contains the true population parameter.

This understanding is crucial when interpreting confidence intervals in research. For instance, in medicine, a wider confidence interval on a drug's effectiveness may reflect uncertainty that warrants further study, whereas in engineering, narrower intervals might be preferred for precise quality control.

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