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A 95% confidence interval is a range of values that you can be 95% confident contains the true population parameter (like the mean or proportion) based on a sample from that population. The concept of a confidence interval is a fundamental aspect of inferential statistics. Here's how to interpret a 95% confidence interval: Confidence Level: The “95%“ in a 95% confidence interval refers to the confidence level. It means that if you were to take many random samples from the population and calculate a confidence interval for each sample, about 95% of these intervals would contain the true population parameter. Range of Values: The interval itself is a range of values. In the context of a population proportion (as in your survey example), the confidence interval provides a range within which the true proportion of the population is expected to lie, based on the sample data. Not a Probability Statement for a Specific Interval: It's important to note that the confidence interval does not say that there is a 95% probability that the true parameter lies within this specific interval. Instead, it means that 95% of intervals constructed in this manner from different samples will contain the true parameter. Margin of Error: The width of the confidence interval is influenced by the sample size and the variability in the data. A larger sample size or less variability will generally result in a narrower confidence interval, implying greater precision in the estimate. In summary, a 95% confidence interval provides a range of values which is likely to contain the population parameter, and this range is derived from the sample data. It is a useful tool for understanding the precision and reliability of an estimate based on sample data. Not a Probability Statement for a Specific Interval: It's important to note that the confidence interval does not say that there is a 95% probability that the true parameter lies within this specific interval. Instead, it means that 95% of intervals constructed in this manner from different samples will contain the true parameter. Margin of Error: The width of the confidence interval is influenced by the sample size and the variability in the data. A larger sample size or less variability will generally result in a narrower confidence interval, implying greater precision in the estimate. n summary, a 95% confidence interval provides a range of values which is likely to contain the population parameter, and this range is derived from the sample data. It is a useful tool for understanding the precision and reliability of an estimate based on sample data. Problem: Answer true or false for the following statements: The 95% confidence interval for the mean: A) Contains the sample mean with 95% certainty. B) Is less likely to contain the population mean than the 99% confidence interval. C) Contains 95% of the observations in the population. D) Is approximately equal to the sample mean plus and minus two standard deviations E) Can be used to give an indication of whether the sample mean is a precise estimate of the population mean.
