We concentrate on the reasoning of confidence intervals, estimating the standard deviation from the data where necessary and without apology. We then consider what to do when the sample size is small and the standard deviation is estimated from the data, introducing Student's t. From this point on, students can and should see the full output of a real statistics program. Estimating with Confidence We renew our attempt to estimate the mean of a population density. Now we recognize the population mean as a parameter, and we know something of the distribution and variability of the sample mean.
Standard inference acts as if the sample comes from a randomized data production design.
We note that it is the deliberately introduced randomness of random sampling or randomized assignment of subjects to treatments that makes inference possible. It is only because we randomize that repeating the sample or experiment yields a sampling distribution. A video introduces the problem of estimating dissolved oxygen levels in Chesapeake Bay, and a case study follows up on the Bay pollution data. We simulate many confidence intervals for the same population in Data Desk, noting that they have the same width, but that because their centers are random, some may not cover the true mean. We note the value of increasing the sample size to simultaneously narrow the confidence interval and improve the confidence level.