Inference Toolbox! To test a claim about an unknown population parameter: Step 1: State Identify the parameter (in context) and state your hypotheses Step 2: Plan Identify the appropriate inference procedure and verify the conditions for using it (SRS, Normality, Independence) Step 3: Calculations Calculate the test statistic Find the p-value Step 4: Interpretation Interpret your results in CONTEXT Interpret P-value or make a decision about H 0 using statistical significance

Example =) Mel N. Colly is interested in whether or not his new treatment for depressed patients is having any effect on his patients’ rating of depression. Suppose all of his depressed patients have a mean depression score of 8 with a standard deviation of 4. Mel chooses a random sample of 100 depressed patients treated with his innovative approach and determines that the mean depression score for these individuals is 7.5. Does the cream have any effect?

Mel N. Colly is interested in whether or not his new treatment for depressed patients is having any effect on his patients’ rating of depression. Suppose all of his depressed patients have a mean depression score of 8 with a standard deviation of 4. Mel chooses a random sample of 30 depressed patients treated with his innovative approach and determines that the mean depression score for these individuals is 7.5. Does the treatment have any effect?

Step 2: PLAN We will conduct a ______________________________. (1) SRS: The data was collected “at random.” The study does not state that a simple random sample was used, but we will proceed assuming proper sampling methods were used. (2) Normality: We do not know if the population distribution of depression patients’ depression scores is Normal, but the sample size is large enough (n=30) so that the sampling distribution will be approximately normal (by the central limit theorem) (3) Independence: Mel N. Colly selected the patients without replacement, but we will assume that there are more than 30(10) = 300 depressed patients seen in his practice. Also assume that the depression score for the each patient is independent of other patients in the sample.

Mel N. Colly is interested in whether or not his new treatment for depressed patients is having any effect on his patients’ rating of depression. Suppose all of his depressed patients have a mean depression score of 8 with a standard deviation of 4. Mel chooses a random sample of 30 depressed patients treated with his innovative approach and determines that the mean depression score for these individuals is 7.5. Does the treatment have any effect? Step 3: Calculations (1) Test Statistic z = x-bar - μ 0 σ/√n (2) P-value: Draw a picture using the standardized value, then calculate the P-value

Mel N. Colly is interested in whether or not his new treatment for depressed patients is having any effect on his patients’ rating of depression. Suppose all of his depressed patients have a mean depression score of 8 with a standard deviation of 4. Mel chooses a random sample of 30 depressed patients treated with his innovative approach and determines that the mean depression score for these individuals is 7.5. Does the treatment have any effect? Step 4: Interpretation P-value (the problem did not give us an alpha level) A sample mean depression score of 7.5 would happen 49.36% of the time by chance if the true population mean depression score was 8. Because the probability of obtaining these results is so high, we fail to reject our null hypothesis. This is not good evidence that the true mean depression score is not 8.

Di Perrs is the quality control manager for Pampers. A recent ad claimed that the new improved Pampers is more absorbent than the old Pampers. The average absorbency of old pampers was 195 milliliters with a standard deviation of 80 milliliters. A total of 100 new Pampers were selected at random and tested. The average amount of fluid absorbed was x-bar = 210 milliliters. Di Perrs wants to use an α = 0.05 significance level.

Step 2: PLAN We will perform a 1-sample z-test for means (sigma known) (1) SRS: The data was collected “at random.” The study does not state that a simple random sample was used, but we will proceed assuming proper sampling methods were used. (2) Normality: We do not know if the population distribution of Pampers absorbency is Normal, but the sample size is large enough (n=100) so that the sampling distribution will be approximately normal (by the central limit theorem) (3) Independence: Di Perrs selected the diapers without replacement, but we can assume that there are more than 10(100) = 1000 diapers produced at the factory. Also assume that the absorbency of each diaper in the sample is independent of the other daipers.

Di Perrs is the quality control manager for Pampers. A recent ad claimed that the new improved Pampers is more absorbent than the old Pampers. The average absorbency of old pampers was 195 milliliters with a standard deviation of 80 milliliters. A total of 100 new Pampers were selected at random and tested. The average amount of fluid absorbed was x-bar = 210 milliliters. Di Perrs wants to use an α = 0.05 significance level. Step 3: Calculations (1) Test Statistic (2) P-value: Draw a picture using the standardized value, then calculate the P-value

Di Perrs is the quality control manager for Pampers. A recent ad claimed that the new improved Pampers is more absorbent than the old Pampers. The average absorbency of old pampers was 195 milliliters with a standard deviation of 80 milliliters. A total of 100 new Pampers were selected at random and tested. The average amount of fluid absorbed was x-bar = 210 milliliters. Di Perrs wants to use an α = 0.05 significance level. Step 4: Interpretation Using significance Level Since our P-value,

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Choosing a Level of Significance: Things to think about (1) How plausible is H 0 ? A study that finds that smoking increases the risk of Alzheimer's. You read a study that claims to have evidence that smoking is really good for you. (2) What are the consequences of rejecting H 0 ? You find evidence that cats sleep more than dogs. You find evidence that a new drug may have harmful side- effects…but your company has invested millions of dollars in an ad campaign for the drug.

Statistical Significance vs. Practical Importance You decide to run a significance test to see if a particular SAT prep program increases scores on the Math portion. You know from previous research that the average score on the Math section is 510 with a standard deviation of 50. You take a sample of 200 students and find that they have an average score of 515. Use a 5% level of significance. H 0 : μ = 510 H a : μ > 510 P-Value: We can reject the null hypothesis that the prep program does not improve scores…but is a 5 point increase worth anything?

Beware Outliers!!! Pesky little outliers can destroy the significant of otherwise significant data. They can also make data appear significant when it actually is not. Always do a graphical analysis of your data The effect you are searching for should be evidence in your plots Confidence intervals can help you get a better idea

Beware Outliers!!! Be aware of “dropouts” from statistical analysis. Make sure that all the data is represented in the analysis.

Lack of Significance Example In an experiment to compare methods for reducing transmission of HIV, subjects were randomly assigned to a treatment group and a control group. Result: the treatment group and the control group had the same rate of HIV infection. Researchers described this as an “incident rate ratio” of (>1.00 means greater rate of infection among treatment group, <1.00 means greater rate among control). The 95% confidence interval for the incident rate ratio was reported at 0.63 to Can you really say that the treatment has no effect?

Lack of Significance Design a study so that it has a high probability of finding a real effect. What could you do to increase the chances of finding an effect?

Invalid Statistical Inference Hawthorne effect What is the term for a study where neither the subject nor the administrator knows who is getting what treatment?

Invalid Statistical Inference The importance of an SRS from the population of INTEREST.

Multiple Analyses A study using an alpha level of 0.05 is run for 20 different types of soda to see if there is an association between drinking soda and scoring well on a math test. It is found that one soda, Mountain Dew, did increase scores. Why is this not good evidence of an effect?

HOMEWORK!!! 11.43, 11.46, Friday: Chapter 11 TEST

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