Lane Chapter 104: Why Is A 99% Confidence Interval Wider Tha

Lane Chapter 104 Why Is A 99 Confidence Interval Wider Than A 95 Co

Explain why a 99% confidence interval is wider than a 95% confidence interval. Compute the 95% confidence interval for a person who claims to predict coin flips correctly 16 out of 25 times and interpret the results. Calculate a 99% confidence interval for a population mean based on a sample of 22 with a known population standard deviation, and then recalculate assuming the standard deviation is unknown and estimated from the sample. Determine the 95% and 90% confidence intervals for the average study time of 10 psychology majors with specific data points. Clarify what "90% confident" means in the context of constructing a confidence interval for a mean. Using a sample of 81 jurors with a mean wait time of 8 hours and a standard deviation of 4 hours, construct a 95% confidence interval for the population mean and explain its significance. For a sample of 84 used car sales costs with a mean of $6,425 and standard deviation of $3,156, determine the appropriate distribution, construct a 95% confidence interval, and interpret its meaning. For a sample of 12 soda servings with a mean of 13.30 oz and standard deviation of 1.55, calculate the margin of error assuming normally distributed data. Analyze a survey where 76% of 1,005 U.S. workers expect to work past retirement age, at a 95% confidence level with a ±3% margin of error: find the estimated proportion, sample size, confidence level, error bound, and interpret the confidence interval, including how a news reporter should explain it to an audience.

Paper For Above instruction

Confidence intervals are fundamental tools in statistics used to estimate the range within which a population parameter, such as a mean or proportion, is likely to fall with a certain degree of confidence. The width of a confidence interval is directly influenced by the confidence level chosen; notably, a 99% confidence interval is wider than a 95% interval, primarily due to the increased certainty that the interval contains the true parameter. This difference in width arises because higher confidence levels necessitate capturing more of the distribution's tail, which requires a broader interval.

For instance, considering a person claiming to predict coin flips correctly 16 out of 25 times, the proportion of successes is 0.64. To evaluate this claim, constructing a 95% confidence interval for this proportion involves calculating the sample proportion and applying the formula for a confidence interval for a population proportion, typically using the standard normal distribution. The resulting interval provides a range within which we expect the true success probability to lie with 95% confidence. If the interval contains the hypothesized value (e.g., 0.5 for a fair coin), then there is not enough evidence to conclude the person has predictive ability significantly different from chance.

When estimating a population mean, the choice of standard deviation information is crucial. In a scenario with a known population standard deviation, the Z-distribution is applicable, and the confidence interval is computed using the z-score corresponding to the desired confidence level. For example, with a sample size of 22, a sample mean of 60, and known population standard deviation of 10, the 99% confidence interval can be accurately determined. If the population standard deviation is unknown, the sample standard deviation replaces it, and the t-distribution is used instead. For the same sample size and data, the t-distribution accounts for additional uncertainty, resulting in a slightly wider interval.

In studying the average time psychology majors spend studying, constructing confidence intervals involves calculating the sample mean, standard deviation, and appropriate critical value based on the degrees of freedom and confidence level. The 95% and 90% intervals provide ranges that are likely to contain the true population mean, with the 95% interval being wider than the 90%. These intervals are essential for understanding the precision of the sample estimate and for making informed decisions about the population parameter.

The meaning of "90% confident" relates to the long-term reliability of the interval estimation procedure. Specifically, if numerous samples are taken and intervals calculated similarly, approximately 90% of these intervals will encompass the true population mean. This interpretation aligns with the frequentist perspective, emphasizing the method's reliability rather than the probability that a specific interval contains the parameter.

In a study examining jurors' waiting times, the sample mean and standard deviation inform the construction of a 95% confidence interval. The process involves selecting the correct distribution, often the t-distribution when the population standard deviation is unknown, and computing the margin of error. The resulting interval offers an estimated range within which the actual mean wait time for all jurors resides, providing insight for judicial system efficiency analysis.

Similarly, in the context of used car prices, the assumptions about the distribution (normality for large samples) and the use of the sample mean and standard deviation enable confidence interval construction. This interval quantifies the typical cost and guides consumers and dealers in understanding price variability.

For quality control in a restaurant setting, with a small sample size, calculating the margin of error involves understanding the underlying normality assumption and relevant t-value. The confidence interval then communicates the expected range of soda amounts served, emphasizing consistency and quality assurance.

The survey estimating the proportion of American workers intending to work past retirement illustrates how confidence intervals can be applied to proportions. Calculating the estimated proportion, sample size, and the error bound helps in understanding the population sentiment. Comparing the calculated error bound to the reported margin of error highlights the precision of the estimate, and careful interpretation is critical for accurate communication to the public.

In all these cases, understanding the concepts of confidence level, margin of error, and interval width is essential to correctly interpret and communicate statistical results. Confidence intervals provide a range of plausible values for population parameters, and their widths reflect the degrees of certainty associated with the estimates. Proper application and interpretation of these intervals are fundamental skills in statistical analysis, enabling better decision-making across various fields.

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