1 Chapter9 Hypothesis Tests Using a Single Sample

2 BASICS In statistics, a hypothesis is a statement about a population characteristic. NEVER about a statistic!!!

3 FORMAL STRUCTURE Hypothesis Tests are based on an reductio ad absurdum form of argument. Specifically, we make an assumption and then attempt to show that assumption leads to an absurdity or contradiction. Hence, the assumption is wrong.

4 FORMAL STRUCTURE The null hypothesis, denoted H 0 is a statement or claim about a population characteristic that is initially assumed to be true. The null hypothesis is so named because it is the “starting point” for the investigation. The phrase “there is no difference” is often used in its interpretation.

5 FORMAL STRUCTURE The alternate hypothesis denoted by H a ) is the competing claim. (I call it the RESEARCH hypothesis.) The alternate hypothesis is a statement about the same population characteristic that is used in the null hypothesis. Generally, the alternate hypothesis is a statement that specifies that the population has a value different, in some way, from the value given in the null hypothesis.

6 FORMAL STRUCTURE Rejection of the null hypothesis will imply the acceptance of this alternative hypothesis. Assume H 0 is true and attempt to show this leads to an absurdity, therefore H 0 is false and H a is true. (Remember proof by contradiction?)

7 FORMAL STRUCTURE Typically one assumes the null hypothesis to be true and then one of the following conclusions are drawn. 1.Reject H 0 Equivalent to saying that H a is correct or true 2.Fail to reject H 0 Equivalent to saying that we have failed to show a statistically significant deviation from the claim of the null hypothesis This is not the same as saying that the null hypothesis is true.

8 AN ANALOGY The Statistical Hypothesis Testing process can be compared very closely with a judicial trial. 1.Assume a defendant is innocent (H 0 ) 2.Present evidence to show guilt 3.Decision: the defendant cannot be innocent given the evidence 4. Reject the assumption of innocence in favor of the alternate—guilty (H a )

9 AN ANALOGY Two Hypotheses are then created. H 0 : Innocent H a : Not Innocent (Guilt)

10 Comments on Hypothesis Form The null hypothesis must contain the equal sign. This is absolutely necessary because the test requires the null hypothesis to be assumed to be true. The numeric value attached to the equal sign is then the value assumed to be true and used in subsequent calculations. The alternate hypothesis should be what you are really attempting to show to be true. This is not always possible.

11 Hypothesis Form The form of the null hypothesis is H 0 : population characteristic = hypothesized value where the hypothesized value is a specific number determined by the problem context. The alternative (or alternate) hypothesis will have one of the following three forms: H a : population characteristic > hypothesized value H a : population characteristic < hypothesized value H a : population characteristic  hypothesized value

12 Examples of Hypotheses You would like to determine if the diameters of the ball bearings you produce have a mean of 6.5 cm. H 0 : µ  =  6.5 H a : µ  ≠  6.5 (Two-sided alternative)

13 The students entering into the math program used to have a mean SAT quantitative score of 525. Are the current students weaker (as measured by the SAT)? H 0 : µ = 525 H a : µ < 525 (One-sided alternative) Examples of Hypotheses

14 Do the “16 ounce” cans of peaches canned and sold by DelMonte meet the claim on the label (on the average)? H 0 : µ = 16 oz H a : µ  < 16 oz Examples of Hypotheses Notice, the real concern would be selling the consumer less than 16 ounces of peaches. We don’t really care if the cans are over 16 ozs!

15 Is the proportion of defective parts produced by a manufacturing process more than 5%? H 0 : p = 0.05 H a : p > 0.05 Examples of Hypotheses

16 Do two brands of light bulb have the same mean lifetime? H 0 : µ Brand A = µ Brand B H a : µ Brand A  µ Brand B Examples of Hypotheses

17 Do parts produced by two different milling machines have the same variability in diameters? or equivalently Examples of Hypotheses

18 Caution When you set up a hypothesis test, the result is either 1.Strong support for the alternate hypothesis (if the null hypothesis is rejected) 2.There is not sufficient evidence to refute the claim of the null hypothesis. (You are stuck with it, because there is a lack of strong evidence against the null hypothesis.)

19 P-value The P-value is the probability of obtaining a test statistic value as extreme (or more so) assuming that H 0 is true!

20 The lower the probability of your sample results happening by chance when H o is true then the less likely H o actually is true!