The Null HypothesisThe Alternate HypothesisTesting The Null HypothesisReportingReportingDecidingDecidingDecidingDecidingIs Hanover Water Yummy?INDIRECT, IS ExpOur "Coin" and the Language of Hypothesis TestingMath 5 CrewDepartment of MathematicsDartmouth CollegeOur ”Coin” and the Language of Hypothesis Testing – p.1/11The Null Hypothesis•Question 1. Suppose our "coin" is fair.What do we expect to happen whenwe "flip" it?Our ”Coin” and the Language of Hypothesis Testing – p.2/11The Null Hypothesis•Question 1. Suppose our "coin" is fair.What do we expect to happen whenwe "flip" it?•Null Hypothesis: Is usually somearticulation of "Nothing is going on".Our ”Coin” and the Language of Hypothesis Testing – p.2/11The Null Hypothesis•Question 1. Suppose our "coin" is fair.What do we expect to happen whenwe "flip" it?•Null Hypothesis: Is usually somearticulation of "Nothing is going on".•Our Example: "Heads" and "tails" areboth equally likely.Our ”Coin” and the Language of Hypothesis Testing – p.2/11The Alternate Hypothesis•Question 1. Suppose our "coin" is fair.What do we expect to happen whenwe "flip" it?Our ”Coin” and the Language of Hypothesis Testing – p.3/11The Alternate Hypothesis•Question 1. Suppose our "coin" is fair.What do we expect to happen whenwe "flip" it?•Alternate Hypothesis: Is usually somearticulation of "Something is goingon".Our ”Coin” and the Language of Hypothesis Testing – p.3/11The Alternate Hypothesis•Question 1. Suppose our "coin" is fair.What do we expect to happen whenwe "flip" it?•Alternate Hypothesis: Is usually somearticulation of "Something is goingon".•Our Example: Our "coin" is biasedand either "heads" or "tails" is morelikely.Our ”Coin” and the Language of Hypothesis Testing – p.3/11Testing The Null Hypothesis•Question 2. Suppose we "flip" our "coin" 25times. How might you decide whether or notour "coin" is biased?Our ”Coin” and the Language of Hypothesis Testing – p.4/11Testing The Null Hypothesis•Question 2. Suppose we "flip" our "coin" 25times. How might you decide whether or notour "coin" is biased?•Find a numerical parameter whose behavioryou understand assuming the NullHypothesis.Our ”Coin” and the Language of Hypothesis Testing – p.4/11Testing The Null Hypothesis•Question 2. Suppose we "flip" our "coin" 25times. How might you decide whether or notour "coin" is biased?•Find a numerical parameter whose behavioryou understand assuming the NullHypothesis.•When you perform your test you will observeone of this parameter’s possible values. Callall this value your test statistic.Our ”Coin” and the Language of Hypothesis Testing – p.4/11Testing The Null Hypothesis•Question 2. Suppose we "flip" our "coin" 25times. How might you decide whether or notour "coin" is biased?•Find a numerical parameter whose behavioryou understand assuming the NullHypothesis.•When you perform your test you will observeone of this parameter’s possible values. Callall this value your test statistic.•Our Example: Number of "heads".Our ”Coin” and the Language of Hypothesis Testing – p.4/11Reporting•Question 2. Suppose we "flip" our "coin" 25times. How might you decide whether or notour "coin" is biased?Our ”Coin” and the Language of Hypothesis Testing – p.5/11Reporting•Question 2. Suppose we "flip" our "coin" 25times. How might you decide whether or notour "coin" is biased?•Before performing your test you mustdetermine the values of this test statistic forwhich you will accept the null hypothesis andthe values for which you willreject the NullHypothesisand accept the Alternate Hypothesis.Our ”Coin” and the Language of Hypothesis Testing – p.5/11Reporting•Question 2. Suppose we "flip" our "coin" 25times. How might you decide whether or notour "coin" is biased?Our ”Coin” and the Language of Hypothesis Testing – p.6/11Reporting•Question 2. Suppose we "flip" our "coin" 25times. How might you decide whether or notour "coin" is biased?•We call the values of the test statistic forwhich we will reject the Null Hypothesis theCritical Region.Our ”Coin” and the Language of Hypothesis Testing – p.6/11Reporting•Question 2. Suppose we "flip" our "coin" 25times. How might you decide whether or notour "coin" is biased?•We call the values of the test statistic forwhich we will reject the Null Hypothesis theCritical Region.•Our Example: We will accept if 7 < NumberHeads < 18, and reject otherwise. Hence, thecritical region is the collection of integers Kthat satisfy either 0 ≤ K ≤ 7 or 18 ≤ K ≤ 25.Our ”Coin” and the Language of Hypothesis Testing – p.6/11Deciding•Question 3. Suppose the "coin" is fair. Whatis the chance that you declare the "coin"biased?Our ”Coin” and the Language of Hypothesis Testing – p.7/11Deciding•Question 3. Suppose the "coin" is fair. Whatis the chance that you declare the "coin"biased?•How much risk are you willing to take that theNull Hypothesis is correct but you reject it?.Our ”Coin” and the Language of Hypothesis Testing – p.7/11Deciding•Question 3. Suppose the "coin" is fair. Whatis the chance that you declare the "coin"biased?•How much risk are you willing to take that theNull Hypothesis is correct but you reject it?.•Called the chance of a type 1 error.Our ”Coin” and the Language of Hypothesis Testing – p.7/11Deciding•Question 3. Suppose the "coin" is fair. Whatis the chance that you declare the "coin"biased?•How much risk are you willing to take that theNull Hypothesis is correct but you reject it?.•Called the chance of a type 1 error.•Our Example: Should be near 5 percent.Check it! .Our ”Coin” and the Language of Hypothesis Testing – p.7/11Deciding•Question 3. Suppose the "coin" is fair. Whatis the chance that you declare the "coin"biased?Our ”Coin” and the Language of Hypothesis Testing – p.8/11Deciding•Question 3. Suppose the "coin" is fair. Whatis the chance that you declare the "coin"biased?•The risk of a type 1 error is called theSignificance Level of the experiment.Our ”Coin” and the Language of Hypothesis Testing – p.8/11Deciding•Question 3. Suppose the "coin" is fair. Whatis the chance that you declare the "coin"biased?•The risk of a type 1 error is called theSignificance Level of the experiment.•In order to assure yourself that you can callyour resultsstatistically significant, you must setyou significance level to be less