## Inference: Confidence Intervals ### So based on the Central Limit Theorem and the three sigma rule, we know that approximately 95% of sample means will be within 2 (1.96 to be more exact) standard errors of the true population mean.  So we could write this out as: ### Why is our multiplier (z*) 1.96?  Because the area under the standard normal curve between -1.96 and 1.96 is 0.95 (the probability that z is greater than 1.96 is 0.025, as is the probability that z<-1.96.  Together this probability adds to 0.05, or 1 minus the confidence level).  You can calculate a confidence interval with any level of confidence although the most common are 95% (z*=1.96), 90% (z*=1.65) and 99% (z*=2.58).   The generalized confidence interval form, when we know the population standard deviation ( σ) is: ## Interpretation through a Simulation

### Using the population distribution we used to demonstrate the CLT, we will now sample (size n = 60) 100 times from this distribution and calculate 100 distinct confidence intervals (95%).  How many confidence intervals would be expect to cover the true population (2.25)?  We would expect about 95% of the intervals to cover (include) the population mean.  As a reminder, here is the population distribution (remember, we typically do NOT know this distribution). ### And  here are the confidence intervals of the 100 randomly generated samples (sample size = 60).   Each vertical bar is a confidence interval, centered on a sample mean.  The intervals all have the same length, but are centered on different sample means as a result of random sampling.  The confidence  intervals in red DO NOT cover the true population mean (the horizontal red line μ=2.25).  This is what we would expect using a 95% confidence level. ### Now what would happen if repeat this process, but calculate 68% confidence intervals?  We would expect approximately 68% of the confidence intervals to cover the true population mean.  As you can see the length of each interval has decreased in comparison to the 95% confidence intervals.  Why?  Because we have changed our multiplier (z*) from 1.96 to 1. # Sample Confidence Interval Problems

7. ### True or False:  With all else constant, an increase in population standard deviation will shorten the length of a confidence interval. 