From Theory to Practice: Inference about a Population Mean, Two Sample T Tests, Inference about a Population Proportion Chapters etc

Recently we have Looked at the basic theoretical ideas underpinning inference –Randomness –Samples vs. Populations –The law of large numbers –The central limit theorem –Confidence Intervals –Probability (in regard to an observed statistic or equation and a population parameter or relationship) –Hypothesis testing

Today we are going to move on to making inferences about variables Inference about a population mean with the (t) statistic. Then we are going to move on to two sample problems using a different variation of (t). Then we are going to delve back into theory- land to look at population proportions. (**note for this topic I am going to switch to the publisher’s slides that accompanied the book as they are better than the slides I have produced in the past.)

Inference about a Population Mean Last class we had a bit of fun calculating confidence intervals and probabilities but it was a bit un- realistic as we assumed that even though we did not know the population mean we did know its standard deviation This class we will look at the more realistic situation where both of these parameters of the population (mean and std. deviation) are unknown

We are going to do this by using: –the sample mean as an estimate for the population mean –And confidence levels and P-values derived from the sampling distribution of the mean In order to do this certain conditions need to apply –The data is drawn from a SRS –The variable we are interested in has a distribution in the population that is distributed relatively normally, even symmetrically is probably good enough –The population must be much larger than the sample size (general requirement for almost all work with statistics)

But we don’t know the population standard deviation… Because we don’t know the population standard deviation we estimate it using something called the standard error of the sample mean This is just the sample standard deviation divided by the square root of the sample size

As with many other things we have looked at this term, standard errors for a sample mean of a given size sample, have a known distribution. This is known as the ‘t’ statistic. Sample size n-1 is referred to as the degrees of freedom The distribution of t is almost but not quite normal because the substitution of the sample standard deviation for the population std. d. adds more variability The equation opposite is not something I think you would ever use in the real world but instead explains how the values for table C at the back of the book were calculated

The net result is that we can use the resulting “t” statistic to estimate the confidence interval for our sample mean without knowing our population mean or population standard deviation

The good news is that if you have a computer you don’t need to do this by hand any stats package will calculate this “one sample t procedure”. But since we are practicing the math you can find the table for values of (t) with various degrees of freedom for various confidence intervals at the very back of your book in table “C”

Break

Another slightly different t test is the matched pair test. Here we are trying to figure out if the means are significantly different among the sets of pairs. This is also sometimes used as a before and after test (for example, patients might be asked to rate their pain on a scale of 1 to 10 before and after receiving a drug). Here the question is this: What is the probably that the “t” statistic which was calculated from the scores for matched pairs in our sample would occur if the difference in means among matched pairs in the overall population were actually zero? If it is low, then we can infer the drug had an impact on the pain reported by our sample of patients and we would say our results are statistically significant.

Things to remember about t –As the distribution of the variable you are interested in varies from normal so to will the usefulness of t –In general it is fine as long as the sample is large and the population is even larger (regardless of normality). The book says sample size 40 or greater.

Comparing Means: Two Sample T A third further t test is called the two sample t test Here the assumption is that our two samples are independent of each other and from populations independent from each other. For example, voters in two different provincial elections. The math looks like this

The real trick with t is knowing which of the three procedures to use –1 sample We want to find out how confident we can be that the mean we observe in our sample data will reflect the mean in the population and to what probability the population mean falls within our confidence limits.

–Paired or matched sample We want to find out the significance of the difference in means we observe between pairs. E.g. Do husbands spend less time on average per week cleaning than wives? (or) We want to find out the significance of the difference in means we observe before and after an event or intervention. E.g. did the pain killer produce a meaningful change in the pain experienced by a sample of patients.

–Independent sample t test Similar to the paired sample but our two samples are independent of each other and therefore, we must assume they have different population means and different population standard deviations. Eg. A national public opinion survey asks voters in every province to place themselves on a left-right scale with 0 being extreme left and 100 being extreme right. Do voters in two different provinces have similar mean scores on the left-right continuum?

Break 2

Inference about a Population Proportion Up until now we have been looking at making inferences about population means by looking at sample means Now we are going to shift focus to looking at whether we can predict what portion of a population is going to exhibit some trait or outcome based on the portion of the sample that does so

Please note, for the rest of this lesson I am going to switch to the slides that were provided by the publisher as they do a much better job of explaining this than I do. As they are copyright, I cannot generally distribute them. They are on the secure server for your class.