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Transcript

• Inference about the mean of a population of measurements (m) 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 m ; and – Standard deviation of Xbar is s/sqrt(n) • Thus we have (Xbar - m )/(s/sqrt(n)) which has a Z distribution if: – Population is normal and s is known ; or if – n is large so CLT takes over… • But what if s 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 s is unknown, the standard deviation of Xbar, s/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 tdistributions 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 m carries through under the assumption of unknown s … • So e.g., a 95% confidence interval for m based on a SRS from a population with unknown s 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 tdistributed standardized Xbar… e.g., to test the H0: m =m0 against Ha: m >m0 we use (Xbar - m0)/(s/sqrt(n)) which has a tdistribution 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