Construction Engineering 221 Sampling and Mean Comparison

We have talked about several sampling distributions: –z (normal) distribution used to estimate probability of a measurement occurrence –n (binomial) distribution used to estimate probability of a counting occurrence –r (correlation) distribution used to estimate relatedness between two variables within a population (extension is regression)

Sampling and Mean Comparison Another important distribution is the t-distribution used to estimate population means If you draw a sample from one population (engineers, men, heavy drinkers, truck buyers) and compare them to a different population (accountants, women, non-drinkers, van buyers) on some randomly distributed variable, how will you know if the differences are “real” or merely a fluke of random measurement errors (spurious)

Sampling and Mean Comparison You use the t-statistic for population mean comparisons. Estimating the population mean: –For large samples, the sample mean is an unbiased estimator of population mean –Sample mean is normally distributed –Normally distributed sample means can be used with confidence intervals and margin of error to make judgements about mean comparisons

Sampling and Mean Comparison How big should a sample be: –For 95% confidence interval (2 standard deviations); n= 1/e 2, where e is the margin of error (1%, 2%, etc.) –If you want to be 99% sure that the sample mean will be within 2 standard deviations of the population mean, you must sample 10,000 people. If you can live with being 95% sure, you need only sample 400 people –Usually pick confidence interval and margin of error ahead of time based on criticality and other factors

Sampling and Mean Comparison If you are comparing 2 means, use the t- statistics If you are comparing two percentages, use the z-statistic If you are comparing 3 or more means, use the F-statistic If you are comparing 3 or more percentages, use the Chi Square statistic

Sampling and Mean Comparison The hypothesis that assumes the populations are alike (no differences in the means) is the null hypothesis You test the null hypothesis to determine the likelihood that it is true Mean sample 1 Mean sample 2 Unlikely (95%) that the samples came from the same populations

Sampling and Mean Comparison Assume you are testing an admixture to make concrete more “pumpable”, but don’t want to diminish early strength You test 25 cylinders of regular concrete (control group) and 25 cylinders of concrete with the admixture.

Variable 1Variable 2 Mean Variance Observations25 Pearson Correlation Hypothesized Mean Difference0 df24 t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail