The purpose of estimating the true mean is to determine the average value of a population based on a sample. This allows us to make inferences and draw conclusions about the population without having to collect data from every single individual.
The confidence interval for estimating the true mean is calculated using the sample mean, the sample standard deviation, the sample size, and the desired level of confidence. The formula is: sample mean ± (critical value * (sample standard deviation / √sample size)), where the critical value is determined based on the level of confidence and the degrees of freedom.
The confidence level represents the probability that the calculated confidence interval contains the true mean of the population. For example, a confidence level of 95% means that if we were to take multiple samples from the same population and calculate confidence intervals, 95% of those intervals would contain the true population mean.
A larger sample size generally leads to a narrower confidence interval, which means we can be more confident in our estimate of the true mean. This is because a larger sample size reduces the variability of the sample and provides a more accurate representation of the population.
Some potential sources of error in estimating the true mean include biased sampling, measurement error, and outliers in the sample. Biased sampling occurs when the sample is not representative of the population, while measurement error can occur due to human error or faulty equipment. Outliers can also skew the sample mean and lead to an inaccurate estimate of the true mean.