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Ch 11: Testing a Clain 11.1 Significance Tests: The Basics T Copyright © 2008 by W. H. Freeman & Company Activity 11B: I’m a Great FreeThrow Shooter Hoops Testing a Claim: Getting Started • Example 11.2, page 688, Call the paramedics! Stating the Hypothesis Stating the Hypothesis • Example 11.3, page 692 Studying Job Satisfaction Conditions for Significance Test • Example 11.4 Checking Conditions – SRS – Normality – Independence Test Statistic estimate - hypothesized value test statistic = standard deviation of the estimate Calculating The Test Statistic • Example 11.5, page 695 P - Value • The probability , computed assuming that H0 is true, that the observed outcome would take a value as extreme as or more extreme than that actually observed is called the P-Value of the test. The smaller the P-Value is, the stronger the evidence against H0 provided by the data. Computing the P-Value Example 11.6 Page 696 Two-Sided Test Example 11.7, page 697 Determining Statistical Significance Example 11.8, page 700 Deciding to Reject or Fail to Reject H0 • Example 11.9, page 701 Homework • Read Section 11.2 • Complete Homework Exercises 1 – 18 All Ch 11: Testing a Claim 11.2 Carrying Out Significance Test 11.3 Use and Abuse of Tests T Copyright © 2008 by W. H. Freeman & Company Significance Test: Inference Toolbox To test a claim about an unknown population parameter: Step 1: Hypotheses: Identify the population of interest and the parameter you want to draw conclusions about. State the hypothesis Step 2: Conditions: Chose the appropriate inference procedure. Verify the conditions for using it. Step 3: Calculations: If the conditions are met, carry out the inference procedure. •Calculate the test statistic. •Find the P-Value. Step 4: Interpretation: Interpret your results in the context of the problem. •Interpret the P-value or make a decision about H0 using statistical significance. •Don’t forget the 3 C’s conclusion, connection, and context. Z Test for a Population Mean • To test the hypothesis H0: μ = μ0 based on an SRS of size n from a population with unknown mean μ and known standard deviation σ, compute the onesample z statistic. z x 0 n Two-Sided Z Test for a Population Mean Example 11.10, page 706 One Sided, Two Sample Z Test for a Population Mean Example: 11.11, page 707 Duality: Confidence Intervals and Significance Test, Example 11.12, page 711 Using Calculator to Conduct a OneSample z Test • Technology Toolbox Example Page 715 Choosing a Level of Significance • How plausible is H0? If H0 represents an assumption that the people you must convince have believed for years, strong evidence (small Pvalue) will be needed to persuade them. • What are the consequences of rejecting H0? If it means making an expensive change, you need strong evidence. • There is no sharp boarder (ie: α = .05) between statistically significant and statistically insignificant , only increasingly strong evidence as the P-value decreases Statistical Significance is Not the Same Thing as Practical Importance • Example 11.13, Page 717 Additional Heads Up • Don’t Ignore the Lack of Significance, just because it fails. Also consult your confidence interval. Example 11.14, p718 • When planning a study, verify that the test you plan to use has a high probability of detecting an effect of the size you hope to find. (Use a large enough sample size) Additional Heads Up • Inference is not valid on all data sets (Hawthorne effect), Example 11.16, page 719 • Conditions (SRS, Normality, Independence) must be satisfied. The foolish user of statistics who feeds the data to a calculator or computer without exploratory analysis will often be embarrassed. Additional Heads Up • Beware of Multiple Analysis – Remember with α = .05, you will still expect values as extreme 5 out of 100 times in the long run. Homework • Read Section 11.3, 11.4 • Complete Exercises 27-36, 39, 40 • Complete Exercises 43-48 Chapter 11 Testing a Claim 11.4 Using Inference to Make Decisions Definition: Type I and Type II Error The Two Types of Error in Testing Hypothesis Interpreting Type I and Type II Errors • Example 11.19, page 724, Perfect Potatoes: possible Errors Consequences of Type I and II Errors • Example 11.20, page 724, Awful Accidents Type I and II Error Probabilities • Example 11.21 Page 725, Awful Accidents (Continued) Activity 11C Power Applet Four Ways To Increase Power • Increase α. A test at the 5% significance level will have a greater chance of rejecting the alternative than 1% test because the strength of evidence required for rejection is less • Consider a particular alternative that is farther away from μ0. Values of μ that are in Ha but lie close to the hypothesized value μ0 are harder to detect (lower power) than value of μ that are far from μ0. • Increase the sample size. More data will provide more information about x-bar, so we will have a better chance of distinguishing value of μ. • Decrease σ. This has the same effect as increasing the sample size more information about x-bar. Improving the measurement process and restricting attention to a subpopulation are two common ways to decrease σ. Many US Government Agencies Require • 95% Confidence Intervals • 5% Significance Tests • 80% Power Homework • Complete Exercises 49 - 56, 59 – 64 • Complete Take HomeQuiz