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Confidence intervals in statistics are a range of values, derived from the data of a sample, that are believed to contain the value of an unknown population parameter. They provide an estimated range of values which is likely to include the parameter, such as a population mean or proportion, with a certain level of confidence. Key points about confidence intervals: Level of Confidence: The most common confidence level used is 95%, but other levels (like 90% or 99%) can also be used. A 95% confidence interval means that if we were to take 100 different samples and compute a confidence interval for each sample, then approximately 95 of the 100 confidence intervals will contain the population parameter. Calculating Confidence Intervals: The calculation depends on the parameter being estimated and the distribution of the sample data. For a population mean, the confidence interval is typically calculated as: Sample Mean±Margin of Error where the margin of error is determined by the standard error of the mean and the critical value from the distribution (such as the z-score or t-score). Interpretation: A confidence interval provides a range of plausible values for the population parameter. It is not correct to say that the interval has a certain probability of containing the parameter. Instead, the confidence level reflects how often the method used to construct the interval would produce an interval containing the parameter if the study were repeated many times. Factors Affecting Width: The width of a confidence interval is influenced by the sample size, the variability in the data, and the chosen confidence level. Larger sample sizes and lower variability lead to narrower intervals, indicating more precise estimates. Higher confidence levels result in wider intervals, reflecting greater uncertainty but higher confidence that the interval contains the parameter. Applications: Confidence intervals are used in a wide range of applications, from scientific research to business analytics, to indicate the reliability of an estimate. They provide a useful way to convey the uncertainty inherent in sample estimates and are fundamental in hypothesis testing and inferential statistics. In summary, confidence intervals are a key tool in statistics for quantifying the uncertainty of an estimate, allowing us to make inferences about a population parameter based on sample data. Problem: A random sample of 300 diastolic blood pressure measurements are taken. Suppose a 99% confidence interval for the population mean diastolic blood pressure is 68 to 73 mm Hg. If a 95% confidence interval is also calculated, then A) The 95% confidence interval will be wider than the 99%. B) The 95% confidence interval will be narrower than the 99%. * C) 95% and 99% confidence interval will be the same. D) One cannot make a general statement about whether the 95% confidence interval would be narrower, wider or the same as the 99%.
