• Inference about the mean of a population of measurements () is based on the standardized value of the sample mean (Xbar).

• The standardization involves subtracting the mean of Xbar and dividing by the standard deviation of Xbar – recall that– Mean of Xbar is ; and – Standard deviation of Xbar is /sqrt(n)

• Thus we have (Xbar - )/(/sqrt(n)) which has a Z distribution if:– Population is normal and is known ; or if– n is large so CLT takes over…

• But what if is unknown?? Then this standardized Xbar doesn’t have a Z distribution anymore, but a so-called t-distribution with n-1 degrees of freedom…

• Since is unknown, the standard deviation of Xbar, /sqrt(n), is unknown. We estimate it by the so-called standard error of Xbar, s/sqrt(n), where s=the sample standard deviation.

• There is a t-distribution for every value of the sample size; we’ll use t(k) to stand for the particular t-distribution with k degrees of freedom. There are some properties of these t-distributions that we should note…

• Every t-distribution looks like a N(0,1) distribution; i.e., it is centered and symmetric around 0 and has the same characteristic “bell” shape… however, the standard deviation of t(k) {sqrt(k/(k-2))} is greater than 1, the s.d. of Z so the t-distribution density curve is more spread out than Z. Probabilities involving r.v.s that have the t(k) distributions are given by areas under the t(k) density curve … the pt function in R gives us the probabilities we need…

pt(q, k) = Prob(t(k)<= q)

• The good news is that everything we’ve already learned about constructing confidence intervals and testing hypotheses about carries through under the assumption of unknown …

• So e.g., a 95% confidence interval for based on a SRS from a population with unknown is

Xbar +/- t*(s.e.(Xbar)) Recall that s.e.(Xbar) = s/sqrt(n). Here t* is the

appropriate quantile from the t-distribution so that the area between –t* and +t* is .95

• As we did before, if we change the level of confidence then the value of t* must change appropriately…

• e.g., for 95% confidence with df=12, qt(.975,12) gives the correct t* ….

• Similarly, we may test hypotheses using this t-distributed standardized Xbar… e.g., to test the H0: against Ha: we use

(Xbar - /(s/sqrt(n)) which has a t-distribution with n-1 df, assuming the null hypothesis is true. See the last page of these notes for a summary of hypothesis testing in the case of “the one-sample t-test” …

• HW: Read the online Chapter 10 on Hypothesis Testing with Standard Errors (start with the first 3 sections… the third deals with the t-distribution). Work on the second problem set handout…

• Note: a statistic is robust if it is insensitive to violations of the assumptions made when the statistic is used. For example, the t-statistic requires normality of the population… how sensitive is the t-statistic to violations of normality?? Consider these practical guidelines for inference on a single mean:– If the sample size is < 15, use the t procedures if the

data are close to normal.– If the sample size is >= 15 then unless there is strong

non-normality or outliers, t procedures are OK– If the sample size is large (say n >= 40) then even if

the distribution is skewed, t procedures are OK