Section 11.1: Significance Tests: Basics AP Statistics Section 11.1: Significance Tests: Basics

Objective: To be able to understand the basic concepts of significance tests. Significance Tests (Hypothesis Tests): Second type of formal inference Basic Idea: First we assume that some claim about a population is true. Second we collect a sample and calculate the probability of such an example occurring. If that probability is small, then that is good evidence that the claim is false. Outline for any Significance Test: (5 Steps) State the name of the test and check the conditions. These will vary from test to test. Generally speaking: SRS, Normality, Independence

State the hypotheses. Define the parameter!!! 𝐻 0 : (Null Hypothesis) This is the claim about the population that we assume to be true. 𝐻 𝑎 : (Alternative Hypothesis) This is the claim against the population that we are trying to prove. Always stated in terms of a parameter. Types of alternative hypotheses: (stated in terms of means) One sided alternative. 𝐻 0 :𝜇= 𝜇 0 Where 𝜇 is a parameter and 𝜇 0 is a numeric value. 𝐻 𝑎 :𝜇> 𝜇 0 Upper tailed test OR 𝐻 𝑎 :𝜇< 𝜇 0 Lower tailed test

B. Two-sided alternative. 𝐻 0 :𝜇= 𝜇 0 Where 𝜇 is a parameter and 𝜇 0 is a numeric value. 𝐻 𝑎 :𝜇≠ 𝜇 0 Points: Easier to find 𝐻 𝑎 first. Ask yourself “What am I trying to prove?” 𝐻 0 always has an equal sign in it. Should state prior to collecting.

Ex. State the null and alternative hypotheses Ex. State the null and alternative hypotheses. The national average SAT-Math score for a particular exam was 520. I think that AP Stat students excel at math and would average a score higher than this value. State the rejection region. This is the value by which we make our decision about the null hypothesis. We can state it in terms of A. p-value or B. Critical value

“I will reject 𝐻 0 if my p-value is less than 𝛼.” Significance level: the alpha level at which we are going to make a decision regarding the null hypothesis. Unless otherwise stated, 𝛼=0.05. The more important the decision the lower 𝛼 should be. Should state prior to collecting the data. In practice you choose 𝛼. p-value: the probability, assuming that 𝐻 0 is true, that we observe a statistic at least as extreme from the claim as that actually observed in our sample. The smaller the p-value the stronger the evidence AGAINST the null hypothesis.

Diagrams for p-values:

Calculate the test statistic and p-value. Vary from test to test In general: 𝑇𝑆= 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒 −𝐶𝑙𝑎𝑖𝑚 𝑆𝐸 p-values will vary depending on the test. State the conclusion in the context of the problem. I will reject 𝑯 𝟎 if my p-value < 𝛼. I will fail to reject 𝑯 𝟎 if my p-value ≥𝛼. We don’t say “accept 𝐻 0 ” because we don’t know for sure if 𝐻 0 is true. We just don’t have evidence to say that it is not true.

General Diagram for Significance levels and decisions based on critical values: