Many times in statistical analysis, we do not know the TRUE mean of a population of interest. This is why we use sampling to be able to generalize the results of a sample to the entire population. One way that we can estimate the TRUE mean based upon a single sample is through a CONFIDENCE INTERVAL. Criteria for Creating a Confidence Interval 1)Sample must be randomly chosen. 2)CLT (Central Limit Theorem) MUST apply 3)Each subject in the sample must be independent  Most Important

Central Limit Theorem

Sample mean TRUE mean will fall somewhere within this interval The idea of a confidence interval is based upon a statistic (a mean or proportion) of the sample & confidence level (a %). Once you have the statistic and a margin of error based on the confidence level, a range of numbers is established that is believed to enclose the TRUE mean of the population from which the sample originated.

To find this interval, we need to know the z-score (the number of standard deviations) above and below the middle that will enclose the desired confidence level (%). We can get this from the z-score table and our knowledge of the bell curve. The z-score that cuts off the desired confidence level is called a critical value (z*) Look up the lower tail’s % on the table for the z-score This is the critical value for an 80% confidence level

Finding Critical Values for Confidence Levels 1)Using desired confidence level, find % on the TAIL of the normal curve. 2)Look up that % (or as close as you can get) on the INSIDE of the z-score table 3)Follow that value over and out to get the matching critical value (z-score) Tail %

What would be the critical values for the following common confidence levels? Draw a bell curve to help with finding the percentages needed. 90% ………………………………………………………… 95% % …………………………………………………………

Once you have the critical value, you can calculate the interval by using the following formula. Confidence Interval = statistic ± z* ∙ standard error For estimating the population MEAN, the formula would be… This is under the assumption that we know the standard deviation of the population (  )

Example: A simple random sample of 50 Big City University freshmen are collected and given an IQ test. The mean of the sample is 112 and the standard deviation of the population of IQ scores is 15. Create a 95% confidence interval for the TRUE mean IQ score of the population of all Big City freshmen. (107.84, )