Stat 321 – Day 23 Point Estimation (6.1)

Last Time Confidence interval for  vs. prediction interval  One-sample t Confidence interval in Minitab Needs normal population or large n  Prediction interval “by hand” Wider than confidence interval Needs normal population

Last Time Confidence interval for p = population proportion of successes from large population or a process (e.g., coin tossing)  So a binomial random variable underneath  Random sample from population of interest  Large enough sample size:

Example 2 Sample proportion.66; n = 1013 Can assume 95% .66(.34)/ (margin of error 3%) We are 95% confidence that between 63% and 69% of all internet users never read blogs  The interval lies entirely above.5, so we are 95% confident more than half don’t read blogs

Some Precautions Finding voters  Margin-of-error doesn’t measure “non-sampling” errors Alien visits U.S. Senate, wants to estimate proportion of humans who are female  Biased sample  Confidence interval not needed if one’s data is from population, not sample We are 100% confident that p =.16! Claimed to vote…

Choosing a confidence interval formula Were the data collected properly? Is data quantitative (e.g., measurements) or categorical (e.g., blog reader?, binomial)? If categorical: Is n large enough to use the “Wald” procedure?  If not, use “adjusted Wald” If quantitative: Is  known? Hah!  If not, are conditions met for t-interval? Look at behavior of sample, probability plot

Example 5 If given the choice, which would you prefer to hear first: good news or bad?

The Big Picture Want to make a conclusion about unobserved population based on observing a small sample, randomly selected from the population So far have looked at  and p  The sampling distributions of the sample mean and sample proportion behave very predictably and very nicely… What about other parameters? Population Sample Probability Statistics

Example 1: Heroin Addiction Caplehorn and Bell (1991) investigated the times that heroin addicts remained in a clinic Data = time stayed in facility until treatment terminated or study ended What would you like to estimate about the population?

Consideration of other estimators Median  “Typical time”  Less influenced by outliers… But what is the behavior of the median in repeated sampling??  Is the sample median a reasonable predictor of the population median?

Properties of Estimators Unbiased = Expected value of estimator is equal to parameter  Mean of sampling distribution of estimator is at parameter  E( ) =  Variance of estimator  Prefer estimators with smaller variances

Sampling distribution of median Empirically: generate lots of samples from known population, e.g. n=5, calculate median for each sample, look at distribution  Is mean of sampling distribution equal to population median? Mean of medians ≠  ũ

Sampling distribution of median Sample median is not considered an unbiased estimator of the population median With a normal population, Scottish militiamen Mean =  = Median =ũ = sample median is an unbiased estimator of population mean! But with more variability than sample mean

Example 3: Estimating  2 Take population with  2 = Mean = Mean = Unbiased!Biased - too low

Example 3: Estimating  2 S 2 is an unbiased estimator for  2  See proof of E(S 2 )=  2 in text!  Although S is a biased estimator for 

For Thursday Quiz on HW 7 Will collect information for lab 8 in class Hw problem 4, part (d)