Chapter 17 Surprised? Testing Hypotheses about Proportions - PowerPoint Presentation

1. Chapter 17Surprised? Testing Hypotheses about Proportions

2. Are the Results Convincing?We keep talking about statistical significance.Now that we know the standard deviation of a proportion, we’re really in business.If a statistic we observe is more than 2 standard deviations from the proportion we expect, we are ready to say we have a statistically significant difference.

3. The Reasoning of Hypothesis TestingThere are four basic parts to a hypothesis test:HypothesesWe begin with an assumption that nothing interesting has happened.This is called the null hypothesis.ModelWe check assumptions and conditions to confirm our choice of model.

4. The Reasoning of Hypothesis Testing (Cont.)There are four basic parts to a hypothesis test:MechanicsWe use our model to ask what should happen if the null hypothesis were true. Then we ask ourselves if our result is surprising.

5. The Reasoning of Hypothesis Testing (Cont.)There are four basic parts to a hypothesis test:ConclusionSome variability is expected, so if our result is not too extreme we can’t claim that anything of interest has occurred.But if our data appears unusual, we reject the null hypothesis and conclude that we have evidence of a change.

6. A Trial as a Hypothesis Test Think about the logic of jury trials: To prove someone is guilty, we start by assuming they are innocent. We retain that hypothesis until the facts make it unlikely beyond a reasonable doubt. Then, and only then, we reject the hypothesis of innocence and declare the person guilty.

7. A Trial as a Hypothesis Test (cont.)The same logic used in jury trials is used in statistical tests of hypotheses: We begin by assuming that a hypothesis is true. Next we consider whether the data are consistent with the hypothesis. If they are, all we can do is retain the hypothesis we started with. If they are not, then like a jury, we ask whether they are unlikely beyond a reasonable doubt.

8. The Process of Hypothesis TestingHypothesesHypotheses are working models that we adopt temporarily.Our starting hypothesis is called the null hypothesis, which we denote by H0.The null hypothesis specifies a particular parameter value to use in our model. We usually write down the null hypothesis in the form H0: parameter = hypothesized value.

9. The Process of Hypothesis Testing (cont.)The alternative hypothesis, which we denote by HA, contains the values of the population parameter that we consider plausible if we reject the null hypothesis.Our question of inquiry determines our HA. For example, if we hope to find evidence that the new medication decreases the percent of infection, our HA will be that p is less than the current percent.

10. The Process of Hypothesis Testing (cont.)ModelTo plan a statistical hypothesis test, specify the model you will use to test the null hypothesis and the parameter of interest.All models require assumptions, so state the assumptions and check any corresponding conditions.Independence AssumptionRandomization Condition10% ConditionSuccess/Failure Condition

11. The Process of Hypothesis Testing (cont.)ModelYour model step should end with a statement suchBecause the conditions are satisfied, I can model the sampling distribution of the proportion with a Normal model.Watch out, though. It might be the case that your model step ends with “Because the conditions are not satisfied, I can’t proceed with the test.” If that’s the case, stop and reconsider.

12. The Process of Hypothesis Testing (cont.)ModelEach test we discuss in the book has a name that you should include in your report.The test about proportions is called a one-proportion z-test.

13. One-Proportion z-TestThe conditions for the one-proportion z-test are the same as for the one proportion z-interval. Wetest the hypothesis H0: p = p0 using the statistic whereWhen the conditions are met and the null hypothesis is true, this statistic follows the standard Normal model, so we can use that model to obtain a P-value.

14. The Process of Hypothesis Testing (cont.)MechanicsUnder “mechanics” we place the actual calculation of our test statistic from the data.We use the sampling model to calculate the z-score for how far our sample proportion lies from the hypothesized value of p.

15. The Process of Hypothesis Testing (cont.)ConclusionOur decision is based on the Rule of 2.A sample result that’s more than 2 standard deviations from what should happen if the null hypothesis were true is strong evidence against the null hypothesis.So if z > 2 (or if z < –2), we reject the null.Notice that we can’t accept the null hypothesis.But we can say that we failed to find evidence to reject the null.

16. Making ErrorsHere’s some shocking news for you: nobody’s perfect. Even with lots of evidence we can still make the wrong decision.When we perform a hypothesis test, we can make mistakes in two ways:The null hypothesis is true, but we mistakenly reject it. (Type I error)The null hypothesis is false, but we fail to reject it. (Type II error)

17. Making Errors (cont.)Which type of error is more serious depends on the situation at hand. In other words, the gravity of the error is context dependent.Here’s an illustration of the four situations in a hypothesis test:

18. Alternative AlternativesThere are three possible alternative hypotheses:HA: parameter < hypothesized valueHA: parameter ≠ hypothesized valueHA: parameter > hypothesized value

19. Alternative Alternatives (cont.)HA: parameter ≠ value is known as a two-tailed alternative because we are equally interested in deviations on either side of the null hypothesis value.When we are looking for changes in either direction, we say we’re doing a two-tailed test.

20. Alternative Alternatives (cont.)HA: parameter < value is known as a one-tailed alternative. A one-tailed alternative focuses on deviations from the null hypothesis value in only one direction.When we are interested in knowing whether the parameter has decreased, we say we’re doing a lower-tail test.

21. Alternative Alternatives (cont.)HA: parameter > value is known as a one-tailed alternative. When we are interested in knowing whether the parameter has increased, we say we’re doing a upper-tail test.

22. Alpha Levels and Critical ValuesOur willingness to risk a Type I error is called the alpha level or significance level of the test.When we use the Rule of 2, the alpha level is about 5%.We write that α = 0.05.The z-score that corresponds to a given alpha level is called the critical value, written z*.

23. Alpha Levels and Critical ValuesSome other commonly used critical values:

24. P-ValuesWe can find the probability of seeing the data we have collected if, in fact, the Null is true.This extra step is the calculation of the P-value.When the P-value is low enough, it tells us that what we observed is very unlikely if the Null is true. This makes our results statistically significant. So we reject the Null.When the P-value is high, we haven’t seen anything surprising. We won’t reject the Null.

25. What Can Go Wrong?Don’t base your null hypothesis on what you see in the data.Think about the situation you are investigating and develop your null hypothesis appropriately.Don’t base your alternative hypothesis on the data, either.Again, you need to Think about the situation.

26. What Can Go Wrong? (cont.)Don’t forget to check the conditions.We need randomization, independence, and a sample that is large enough to justify the use of the Normal model.

27. What Can Go Wrong? (cont.)Don’t accept the null hypothesis. You may not have found enough evidence to reject it, but you surely have not proven it’s true!Don’t forget that in spite of all your care, you might make a wrong decision.We can never eliminate the possibility of a Type I or a Type II error.

28. What have we learned?We can use what we see in a random sample to test a particular hypothesis about the world.Hypothesis testing complements our use of confidence intervals.Testing a hypothesis involves proposing a model, and seeing whether the data we observe are consistent with that model or so unusual that we must reject it.We base our decision on the Rule of 2.

29. What have we learned? (cont.)We’ve learned:Start with a null hypothesis.Alternative hypothesis can be one- or two-sided.Check assumptions and conditions.If the data are far out of line with H0, reject the null hypothesis.If the data are consistent with H0, don’t reject the null hypothesis.State the conclusion in the context of the original question.

30. What have we learned? (cont.)We’ve learned about the two kinds of error we might make.If the Null hypothesis is true and we reject it anyway, we made a Type I error. And alpha is the probability of this event occurring.If the Null hypothesis is false, but we fail to reject it, that’s a Type II error.