Today: Hypothesis testing

Example: Am I Cheating? If each of you pick a card from the four, and I make a guess of the card that you picked. What proportion of guesses should be correct if I am randomly guessing? Should be ¼, because there are only 4 cards.

Example: Am I Cheating?  As we’ve learned, statistics vary from sample to sample  Even if the population proportion is 1/4, not every sample proportion will be exactly 1/4  How do we determine when a sample proportion is far enough above ¼ to provide evidence of cheating?

Let the data speak: Statistical Evidence 3/4 is the highest, so provides the strongest evidence of cheating.

Statistical Hypothesis is a statement about the parameters of one or more populations. i.e. population proportion Statistical tests are framed formally in terms of two competing hypotheses: Statistical Hypotheses

 The alternative hypothesis is established by observing evidence (data) that contradicts the null hypothesis and supports the alternative hypothesis  Hypotheses are always about population paramete rs H o : Null hypothesis H 1 : Alternative hypothesis Competing claims about a population Statistical Hypotheses

 H 0 is usually "Parameter” = a number versus  H 1 is usually "Parameter” ≠, >, or < a number  The inequality in H 1 depends on the question H o : p = 1/4 H 1 : p > 1/4 No Cheating, (No “effect”) “Am I Cheating” Hypotheses Cheating, Claim that need “evidence”

Null and Alternative Hypotheses Construct the null and alternative hypothesis a) b) c) Ans : A)

How Can We Test Hypotheses? Test procedure is a rule, based on sample data, for deciding whether to reject H o Test Statistic : a function of sample data on which the decision (reject H o or do not reject H o ) Rejection (or Critical) region: The region that leads to the rejection of H o Critical points (or values): the boundary points of critical region.

When results as extreme as the observed sample statistic are unlikely to occur by random chance alone (assuming the null hypothesis is true), we say the sample results are statistically significant If our sample is statistically significant, we have convincing evidence against H 0, in favor of H 1 If our sample is not statistically significant, our test is inconclusive Statistical Significance?

Type I Error and Type II Error “Am I Cheating” example: If I am just randomly guessing, but you say that I am cheating What type of error is this? Ans: Type I a) Type I b) Type II H o : p = 1/4 H 1 : p > 1/4 No Cheating, (No “effect”) Cheating, Claim that need “evidence”

Significance Level and Power

See the board

Significance Level and Power

Procedure of Hypothesis testing