Review for Final ExamCautionsKey concepts to take away from this class.Hypothesis TestsLinear regression and correlationSelecting the statistical methodProblemsIntroductory Statistics LecturesReview for Final ExamAnthony TanbakuchiDepartment of MathematicsPima Community College(Compile date: Fri May 16 06:45:25 2008)Contents1 Review for Final Exam 11.1 Cautions . . . . . . . . . 11.2 Key concepts to takeaway from this class. . . 21.3 Hypothesis Tests . . . . 31.4 Linear regression andcorrelation . . . . . . . . 51.5 Selecting the statisticalmethod . . . . . . . . . 51.6 Problems . . . . . . . . 71 Review for Final Exam1.1 CautionsA researcher developing a new chemotherapy agent conducts an experimentwhere 10 treatment mice receive the new agent during a month period. Thetumor mass in each mouse is measured before and after the treatment. Theresearcher observes that the average tumor mass in the treatment group hasdecreased by 12 grams. The best existing chemotherapy agent in use can onlyreduce tumor mass by 10 grams on average in one month. The researcheris elated and writes a paper submitting it to the New England Journal ofMedicine.Question 1. The paper only gives the above data and states that the newchemotherapy agent is more effective in decreasing tumor mass in comparisonto the standard agent.You are reviewing the paper for the journal. What do you think about theresearcher’s conclusions?ANSWER: This researcher needs to take a statistics class! The differencecould be due to sampling error. We need statistical evidence, they should stateH0and Ha, the hypothesis test used and the p-value!Question 2. What hypothesis test would you suggest the researcher uses?ANSWER: Matched pair hypothesis test. H0: d = 10 tumor mass decreasesby same as standard 10 grams, Ha: d > 10 tumor mass decreases by more than10 grams.12 of 8 1.2 Key concepts to take away from this class.Question 3. You are also unhappy that the researcher simply reported a differ-ence of 12 grams. What should the researcher report to give more informationon their experiment precision?ANSWER: A confidence interval for d. A point estimate is not very mean-ingful without the margin of error.A drug company has developed a new agent that is supposed to reducecholesterol. The drug company conducts a study of 1,000 individuals who havetheir cholesterol measured before and after receiving the drug. The results arenot statistically significant. They cannot support the alternative hypothesisthat the drug reduces cholesterol.Since the survival of the company is dependent on the success of the drug,they get more investor money and repeat the study on another 1,000 patients.Again, they don’t get statistically significant results. After repeating the study10 times, the 10th study has a sufficiently small p-value to support that thedrug reduces cholesterol!The company only reports the good results from the 10th study to the FDA.The drug is approved since they have shown it is effective.1Question 4. What’s wrong with drawing conclusions from only the 10th study?ANSWER: If we reject H0when the p-value ≤ α, then even when H0istrue, we will reject it α proportion of the time. The drug company is simplyforcing a Type I error to occur.1.2 Key concepts to take away from this class.Sampling error• If you don’t measure the whole population, you cannot directly drawconclusions (or prove hypothesis) from study / experiment results.• You must statistically analyze your results to show that they are sig-nificant — indicating that sampling error is not the likely cause of theobserved results.Whether you are designing a study or reading experimental results, all goodstudies and experiments should be composed of the following steps.Design of Experiments and Studies1. Determine your hypothesis and the margin of error you desire. (Better:determine desired power.)2. Determine what to measure. Best to use ratio level of measurement.Worst is categorical.3. Determine the necessary sample size to attain your desired margin oferror.4. Carefully plan your study/experiment and how you will randomize toavoid bias.1This is a fictitious example that exaggerates what has actually happened in recent times.Anthony Tanbakuchi MAT167Review for Final Exam 3 of 85. Collect your data, plot it, look at it, check for outliers, and ensure thetest’s requirements are satisfied.6. Analyze the data, make the formal decision.7. Form the final conclusion and report the results. Make sure to:a) Report the conclusion in clear understandable works.b) Report the p-value.c) Give a confidence interval for the parameter you are estimating.d) Discuss the meaning of results.• Only draw conclusions about populations which your sample represents.• Correlation does not imply causation.We use statistics based on samples to estimate population parameters.Since a statistic is an estimate of a parameter, knowing the distribution of thestatistic — the sampling distribution — tells us how “good” the statistic is.• If the mean of the sampling distribution equals the parameter, the statis-tic is unbiased.• The narrower the sampling distribution (smaller standard deviation of thestatistic), the smaller the confidence interval for the parameter yieldinga more precise estimate.Question 5. What does the CLT state and why is it useful?ANSWER: The CLT states that the sampling distribution of ¯x is normallydistributed with σ¯x= σ/√n.It is valid when the population has a normal distribution or n > 30.Use: We known the sampling distribution of ¯x when it is valid.1.3 Hypothesis TestsKnow the hypothesis testing steps.What you should know for each test• Know the hypothesis H0and Ha.• Know the requirements.Formal decision• Reject H0if p-value ≤ α• Otherwise fail to reject H0Remember,• The p-value is the probability of observing the sample data assuming H0is true. We always assume H0is true unless the sample datasupports rejecting it.• We can’t make statements about supporting H0unless we calculate β.ConclusionReject H0“The sample data supports the claim that [state Hain words].”Fail to reject H0“The sample data does not provide sufficient evidence tosupport the claim that [state Hain words].”Anthony Tanbakuchi MAT1674 of 8 1.3 Hypothesis TestsAlternatively you could say: “The sample data does not contradict thethe claim that [state H0in words].”)Types of errorsType I error :• Occurs when you reject H0but H0is true.• α is the maximum