Fundamentals of Statistics A User’s Guide for Experimental Design and Data Analysis Part II: Inferential Statistics Examples taken from the textbook: Probability and Statistics For Engineering and Sciences by Jay L. Devore Probability and Statistics For Engineering and Sciences by Jay L. Devore

Definitions  Population vs. Sample  Population – complete set of cases  Sample – subset of cases drawn from a population  Parameter vs. Statistic  Parameter – numerical characteristic of population  Statistic – numerical characteristic of a sample

Inferential Statistics  Techniques for analyzing/interpreting data  Hypothesis Tests  Mean  Proportion  Pairwise Comparisons

Hypothesis Testing  Note: Only use if your distribution of data is NORMAL, or your sample size is LARGE!  See me during directed study if you have a hard time determining whether or not you can use these techniques

Hypothesis Testing  Null Hypothesis (H 0 ) – claim initially assumed to be true (“prior belief”)  What is H o in a legal case?  Alternative Hypothesis (H a ) - the complement of the null hypothesis  What is H a in a legal case?

Hypothesis Testing Decisions: Reject the null hypothesis Or Fail to reject the null hypothesis

To Reject or Not to Reject?  Test statistic – function of the sample data on which the decision is based  Probability value (P-value) – smallest value at which we can reject the null hypothesis  Defines the rejection region  Reject the null hypothesis if the test statistic is in this region  Fail to reject the null hypothesis if it does not

Types of Errors  Type I error – null hypothesis is rejected when it is true  Legal system example?  An innocent person is sent to jail  Type II error – null hypothesis is not rejected when actually false  Legal system example?  A guilty person is set free

Examples:  Hypothesis Tests:  Mean  Proportion  Pairwise Mean Comparison  Pairwise Proportion Comparison

Hypothesis Test I: Mean Lightbulbs of a certain type are advertised as having an average lifetime of 750 hours. A random sample of 100 bulbs was selected, the lifetime of each bulb determined. Can we conduct a hypothesis test? Answer: Yes – but why? Relatively large sample How do we set up this hypothesis test? Answer: H 0 : µ ≥ 750H a : µ < 750

Hypothesis Test II: Proportion A manufacturer of nickel-hydrogen batteries randomly selects 100 nickel plates for test cells, cycles them a specified number of times, and determines that 14 of the plates have blistered. Does this provide compelling evidence for concluding that more than 10% of all plates blister? Can we conduct a hypothesis test? Answer: Yes – relatively large sample How do we set up this hypothesis test? Answer: H 0 : p ≤.10 H a : p >.10

Hypothesis Test III: Pairwise Mean An article in Agronomy presented the results of an experiment to compare the yield of Sundance winter wheat and Manitou spring wheat. Data from nine test plots is given in the accompanying table. The claim is that there is no difference, but the researchers believe that the average yield for winter wheat is significantly higher than for spring wheat. How do we set up this hypothesis test? Answer: H 0 : µ 1 - µ 2 = 0 H a : µ 1 - µ 2 > 0

Hypothesis Test IV: Pairwise Proportion Ionizing radiation is being given increasing attention as a method for preserving plants. An article reports that 119 of 180 untreated bulbs were marketable (no external sprouting, rotting, or softening), whereas 153 of 180 irradiated garlic bulbs were marketable. Does this suggest that ionizing radiation is beneficial as far as marketability is concerned? How do we set up this hypothesis test? Answer: H 0 : p 1 – p 2 = 0 H a : p 1 – p 2 < 0

Questions?