KIN 4310 1st Edition Lecture 13Outline of Last Lecture I. Positive ResultII. Positive vs. NegativeIII. Workshop #1IV. Workshop #2V. Important ConceptsVI. Test StatisticVII. Research HypothesisVIII. Null HypothesisIX. Scientific MethodX. Scientific Method ExampleXI. Scientific Method XII. Test StatisticXIII. Significance LevelXIV. Critical Region, Critical Value, Test Statistic XV. Critical ValueXVI. Conclusions in Hypothesis TestingXVII. Decision CriterionXVIII. p-valueThese notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.XIX. Decision Criterion: p-value methodXX. Decision Criterion: another optionOutline of Current Lecture I. Decision Criterion: another optionII. Non-Directional or Two-tailed TestIII. Directional Right-tailed TestIV. Directional Left-tailed TestV. Type 1 ErrorVI. Type 2 ErrorVII. Type 1 and Type 2 ErrorsVIII. Controlling Type 1 and Type 2 ErrorsIX. Fish Oil StudyX. HypothesesXI. Hypothesis TestsXII. One-sample Z TestXIII. One-sample Z Test ExampleXIV. One-sample Z Test cont.XV. ExampleCurrent LectureI. Decision Criterion: another optiona. Instead of using a significance level such as 0.05, simply identify the p-value and leave the decision to the readerb. This is another option but its more common to just report p-valueII. Non-Directional or Two-tailed Testa.b. Note: for this type of test, you use the non-equal sign and you have 2 rejection zonesIII. Directional Right-tailed Testa.b. Note: Just 1 rejection zoneIV. Directional Left-tailed Testa.V. Type 1 Errora. A Type 1 error is the mistake of rejecting the null hypothesis when it is trueb. The symbol alpha is used to represent the probability of a type 1 errori. Alpha is usually 5%c. AKA False Positivesd. Note: Ex is that you do a clinical trial on a drug and it says that it works but it appeared to work by random chance. Its when the data tells you to reject the nullbut the data occurred randomly.VI. Type 2 Errora. Type 2 Error is the mistake of failing to reject the null hypothesis when it is false.b. The symbol beta is used to represent the probability of a type 2 errorc. AKA: False Negatived. Note: when you have a hypothesis, the null is wrong, but you fail to reject the nullhypothesis. So basically you fail to reject the null hypothesis when you should have.VII. Type 1 and Type 2 Errors (TQ*)a.b. Table 9.1VIII. Controlling Type 1 and Type 2 Errorsa. For any fixed alpha, an increase in the sample size n will cause a decrease in betab. For any fixed sample size n, a decrease in alpha will cause an increase in beta. Conversely, an increase in alpha will cause a decrease in betac. To decrease both alpha and beta, increase the sample sized. Note: use a large n if you do no want errorsIX. Fish Oil Studya.b. Null hypothesis – fish oil diet has no effect on blood pressurec. Are these data likely to result when the null hypothesis is true?d. What is our test statistic?i.e. What level of significance should we use?i. Alpha = 0.05f. Traditional method:i. The test statistic was in the rejection zone. ii. Therefore, we reject the null hypothesisg. p-value methodi. p = 0.0088 < 0.05ii. Therefore, we reject the null hypothesisX. Hypothesesa. Determine a test statisticb. Use an appropriate statistical method to test the hypothesisc. Two possible outcomes:i. Reject null hypothesisii. Fail to reject null hypothesisXI. Hypothesis Testsa. Critical value methodi. Based on a statistical model, find a value that partitions 95% of the “usual” values from 5% of the “unusual” valuesb. p-value methodi. The probability of getting your test statistic or one more extreme if the null hypothesis is trueii. A low p-value means strong datac. Type 1 Errorsi. False positiveii. When you reject a null hypothesis that is trueiii. Level of significance, alphad. Type 2 Errorsi. False negativeii. When you fail to reject a null hypothesis that is falseiii. Betaiv. Note: here, the skeptic is wrong but you failed to prove him wrongXII. One-sample Z Testa. A special hypothesis test for comparing a sample to a populationb.c. Requires a priori knowledge of the population mean (e.g., census data)d. Useful for questions like:i. Do left-handed people have higher IQs than the general population?ii. Do UH students consume more energy drinks than other college students?iii. Do teenage drivers get into more traffic collisions than all drivers?e. Random sample of a populationi.ii. Note: here, the x bar values are a little bit different from mu. The questionthat we are asking is: is that difference significant?XIII. One-sample Z Test Examplea. Assume the null hypothesis. The sample was randomly selected from the population in questionb. Therefore, any difference between x bar and mu is due to random effects (Sampling error)c. What is the probability of getting a difference equal to or more extreme than our data?i. AKA what is the p-value?XIV. One-sample Z Test cont.a. Test Statistic:i.b. wherei.ii. SEM stands for standard error measure and it is the population s.d / sample sizec. If the null hypothesis is true, z has a normal distribution d. So, use Table B1XV. Examplea. In the U.S, the mercury concentration in blood is normally distributed with a mean of 2.55 microg/L and standard deviation of 0.43 microg/Lb. 15 residents of Grand Isle, LA, were tested. Their mean blood mercury concentration was 2.79 microg/L.c. Are the residents of Grand Isle significantly different than the general population?i. “significantly different than” is a non directional phrased.e. Critical valuesi. Use Table B1f. Area under the curve between the mean and z = 1.96 is 47.50%g. Critical values: z = -1.96 and z = 1.96h.i.j. From the datai. SEM = 0.111ii. z = 2.16iii. Since z is greater than both critical values, we reject