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Statistical inferences and tests
Inference typeGivenExamplesTypical statistics usedInference about populationPopulation mean and population standard deviationPopulation mean height is 175 cm, with standard deviation of 12 cm. What is the 90th percentile height (i.e., the height at which 10% of people are at or above)?Normal distribution with given mean and standard deviation.
Can use NORMDIST or NORMINV to find probability or value of random variable.Inference about population meanSample mean and population standard deviation Sample mean height is 175 cm, with population standard deviation of 12 cm. The sample size is 50. What is the 95% confidence interval of the population mean?Normal distribution where the mean is the observed sample mean, but the standard deviation of the Mean is the standard error (= EMBED Equation.DSMT4 ). Note that this case almost never happens, because it would be unlikely to know the population standard deviation but no the population mean.Inference about population meanSample mean and population standard deviation Sample mean height is 175 cm, with sample standard deviation of 12 cm. The sample size is 50. What is the 95% confidence interval of the population mean?t distribution where the mean is the observed sample mean, but the standard deviation of the Mean is the standard error (= EMBED Equation.DSMT4 ) , where s is the sample standard deviation.Compare means of two independent samplesActual observations. Or, sample means and sample standard deviations of each sample, and sample sizeCompare the averages heights of females and malesIndependent samples ttestCompare means of paired samplesActual observations.Estimate (or hypothesis testing for) the growth for children between the age of 5 to 8. Two growth measures are obtained for each child.Paired samples ttestCompare means of more than two independent groupsActual observations.Compare the average heights of adults across 5 different regionsANOVA
Also posthoc comparisons between two groups at a time (e.g., Bonferroni)
Note that ANOVA with only two groups gives the same results as two independent samples ttest.Test association between two categorical or ordinal variablesFrequencies of observations in the cells of crosstabulationIs there an association between gender and smoking?Chisquare tests. Typically, the variables (gender and smoking) are categorical variables, or sometimes ordinal variables. E.g., Gender may be coded 1 or 2, but 2 does not mean higher than 1. The variables are not continuous variables. That is, subjects can be classified into cells of a 2 way table. Observations in the cells refer to different subjects. E.g., women who dont smoke, women who smoke, men who dont smoke, men who smoke.Association between two continuous variablesActual observations.Is there an association between Socioeconomic status and Reading test scores?Correlation coefficient. The two variables are continuous, and are two attributes for each subject. Typically the data will have two columns (matched) showing two values for each subject.Explaining values of one variable (dependent variable) using values of a number of other variables (independent variables) Actual observations.How well can height be predicted using weight, age, gender, ethnic group and parental heights?Regression.
Note that a regression with one dependent variable and one independent variable will give the same results as correlation in terms of significance testing.
If the independent variable is categorical with two categories, then the significance test for the variable is the same as carrying out a ttest.
If the independent variable is categorical and has more than two categories, then dummy coding must be carried out before using these as independent variables in a regression.
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