Introduction to Statistical Inference Probability & Statistics April 2014

What is Inference? Suppose we wanted to know the average score of forwards in the NBA? If we had access to all the scores of NBA players and their positions, we could calculate this number. There would be no need for inference!

What is Inference? But suppose we did not have access to all the players’ information, or we didn’t want to spend the time and energy it would take to get it? If we draw a random sample of players, we can measure the sample much more easily. We can then reasonably say the population of all players would likely have the same results.

What is Inference? Statistic inference is the practice of drawing conclusions about a population from sample data. There are two basic types of inference: 1.Estimating a parameter 2.Testing a claim

Estimating a Parameter To estimate a parameter (say, mean or proportion of a characteristic of interest) of a population, we start with a point estimate and then we create an interval of plausible values. This is called a confidence interval. The confidence level is the approximate success rate of the method.

Testing a Claim To test a claim, we form two hypotheses. We assume one is true and look for evidence against it. This is called a test of significance. The significance level of the test is the level of risk we are willing to take for making the mistake of finding evidence against the claim when we shouldn’t.

Procedures for Inference Four Step Procedure: 1.State the population & parameter you are interested in estimating or testing 2.Plan the statistical approach you need & check the conditions for inference 3.Do the procedure: gather sample data, make calculations, analyze the results 4.Conclude by giving an answer in context

Inference about Population Proportion Examples of proportion questions: What proportion of RHS seniors know what college they are going to by the end of January? – This would be a confidence interval Do more than 70% of all RHS seniors plan to be out of school the Monday after prom? – This would be a significance test

Inference about Population Mean Examples of questions regarding mean: What is the average number of colleges that RHS seniors apply to? – This would be a confidence interval Do RHS seniors spend less than $200 on their prom attire, on average? – This would be a significance test

Conditions for Inference General Conditions for Inference: Random: The sample must be a random sample that represents the population OR be from a randomized comparative experiment. 10%: The sample size should not be more than 10% of the population size. Normal: We need to be sure that a Normal distribution is a reasonable assumption for the sample statistic.

Significance Tests There are two hypotheses: – The null hypothesis, denoted H 0, is the assumption we make (it is the equal condition) – The alternative hypothesis, denoted H a, is usually what you are looking for evidence to support (it is the inequality condition) We set the significance level (denoted α, Greek letter alpha) to determine our threshold for statistical significance. We compare the P-Value of the test to alpha. If P-Value < alpha, we reject the null hypothesis. Otherwise, we do not reject the null hypothesis.

Using Minitab For Inference: Quantitative Variables Always do a Graphical Summary of your data before conducting any inference.

Using Minitab For Inference: Quantitative Variables Describe the histogram in terms of shape, center, and spread, and note any possible outliers. Example: Weights are slightly skewed to the right, centered at about 149 pounds and with a spread of 99 to 234 pounds. There are no apparent outliers.

Checking Conditions for Inference Random: This must be designed into your study. 10%: This also must be considered when you design your study, so you do not exceed the 10% limit. (Note: When doing a census, this is not a requirement.) Normal: There are two ways to check this condition…

Checking the Normality Condition 1.If the sample size is very large, say more than 30, we can assume Normal condition is met. 2.If the sample size is less than 30, we look at the Anderson-Darling Normality Test in Minitab. This is in the Graphical Summary report. If the P-Value of that test is larger than 0.05, we can assume the Normality condition is met.  If our conditions are not met, we should not proceed with inference! 

Using Minitab For Inference: Quantitative Variables If conditions are met, proceed to your statistical inference procedure. If you are creating a confidence interval, that is given in the Graphical Summary. Conclude: We are 95% confident that the true mean weight for this population is captured in the interval from pounds to pounds.

Using Minitab For Inference: Quantitative Variables If you are conducting a significance test, state your hypotheses and your alpha level, and then select the appropriate test from the Minitab Stat > Basic Stat menu. Means => t-test Proportions => proportion test Select the “Options” box to select your alternative hypothesis.

Using Minitab For Inference: Quantitative Variables Minitab will display the P-Value of your test in the session panel. If the P-Value is less than your significance level, alpha, then you reject the null hypothesis in favor of your alternative hypothesis. Otherwise, your test is not statistically significant.

Example: