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UA MATH 167 - Lecture Notes

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Review for Test IIKey conceptsExamplesIntroductory Statistics LecturesReview for Test IIRandom variables, probability densities, confidence intervals, hypothesis testingAnthony TanbakuchiDepartment of MathematicsPima Community CollegeRedistribution of this material is prohibitedwithout written permission of the author© 2009(Compile date: Tue May 19 14:51:46 2009)Contents1 Review for Test II 11.1 Key concepts . . . . . . 11.2 Examples . . . . . . . . 21 Review for Test II1.1 Key conceptsThis review is not fully inclusive.Be able to differentiate:• simple random sample, random sample• qualitative variable, quantitative variable• discrete variable, continuous variable• parameter, statistic• biased statistic, unbiased statistic• sampling error, non-sampling error• population distribution, sampling distribution• distribution function, density function, cumulative density function, in-verse cumulative density function• point estimate, confidence intervalBe able to answer questions such as:• What is the easy way to find “the probability of at least one”?• What is a random variable?• What is the binomial distribution used for? What are the requirements?What does it look like? How do you find probability with it? What is itsmean and standard deviation?12 of 8 1.2 Examples• What is the normal distribution used for? What does it look like? Howdo you find probability with it?• What is a z score. What is µzand σzequal to? What is the standardnormal distribution?• What does the Central Limit Theorem state? What are the requirements?Why is it useful?• What does a sampling distribution represent?• If you increase sample size n, would you expect the variance in the sam-pling distribution to increase or decrease?• What do confidence intervals represent? Why are they useful?• What is hypothesis testing?In terms of hypothesis testing:• Know all eight steps.• Know the requirements for the tests.• What is H0and Ha?• What do we assume is true?• Do we use the sampling distribution or population distribution to findthe p-value?• What does the p-value represent?• How do you find the p-value if you have the test statistic?• What are the two types of errors? What do they represent.• What is power? Is it better to have higher or lower power?• If you reject H0, what is the probability you made the wrong decision?• Why do we say a hypothesis test does not prove a hypothesis? How doesproof and statistical evidence differ?1.2 ExamplesGiven the following density function on the left and it’s corresponding CDF forthe χ2distribution, answer the following questions.Anthony Tanbakuchi MAT167Review for Test II 3 of 80 5 10 15 20 250.00 0.05 0.10 0.15Chi−Squared Densityxf(x)0 5 10 15 20 250.0 0.2 0.4 0.6 0.8 1.0Chi−Squared CDFxF(x)Question 1. Find P (x > 5)Question 2. Find P25Question 3. What percent of data lies within ±1.5 standard deviations on anormal distribution? (Check: 0.866)Question 4. What is the probability that a student who randomly guesses ona 10 choice true/false exam will get at least 1 correct? (Check: 0.999)Question 5. What is the probability that a student who randomly guesses on a50 choice true/false exam pass the exam (70% = 35 or more correct)? (Check:0.0036 using normal approx. Using exact: 0.0033 )Anthony Tanbakuchi MAT1674 of 8 1.2 ExamplesIf a researcher is conduction a 1-sample proportion hypothesis test with thehypothesis Ha: p > 0.7. The study finds x = 78 and n = 100.Question 6. What is the test statistic? (Check:1.75)Question 7. What is the p-value? (Check:0.0404)Question 8. What would the p-value have been if Ha: p 6= 0.7A manufacturer of paper used for packaging requires a minimum strengthof 20 lb/in2. A quality control inspector randomly samples 35 pieces of paperfrom the previous hour’s production and tests them in a machine the measuresthe force at which the paper breaks. The standard deviation σ of the strengthmeasurements, computed over many sample, is 2 lb/in2.Question 9. What is the probability distribution of the sample mean strength?Question 10. What is the expected average variation for ¯x? (Check: 0.338lb/in2)Anthony Tanbakuchi MAT167Review for Test II 5 of 8Question 11. If one piece of paper is tested, what is the probability that itsstrength is at least 21 lb/in2? (Assume µ = 20 lb/in2, and the individual valueshave a normal distribution.) (Check: 0.309)Question 12. If 35 pieces of paper are tested, what is the probability that theirmean strength is at least 21 lb/in2? (Assume µ = 20 lb/in2) (Check: 0.00155)Question 13. The mean strength of the paper from the sample is 18.9 lb/in2.Based on the sample data, construct a 98% confidence interval for the truemean strength. (Check: zα/2= 2.33, E = 0.786 lb/in2)Question 14. The mean strength of the paper from the sample is 18.9 lb/in2.Conduct a hypothesis test at the 1% significance level to check the qualityAnthony Tanbakuchi MAT1676 of 8 1.2 Examplescontrol inspector’s concern that the strength is too low. (Check: z = −3.25,p-value= 0.000569)Question 15. The manufacturer changed the process to increase the strength.If the quality control engineer wants to estimate the new strength to within0.25 lb/in2, what sample size should be used? (Check: n = 246)The manufacturer changed the process to increase the strength. However,your boss is a real cheapskate, and he thinks your recommended sample size istoo expensive! A new sample of 5 pieces of paper is measured (in lb/in2):Anthony Tanbakuchi MAT167Review for Test II 7 of 820.4, 22.1, 23.3, 25.6, 23.2Question 16. Since the process is different, assume σ unknown. Test the hy-pothesis Ha: µ > 20 using the sample data. Does the process appear to beok?A researcher who is trying to determine the proportion of people who sup-port increasing the tax on gas guzzlers. Use the output below to answer thefollowing question.R: prop . t e s t ( 87 , 200 , p = 0 . 5 , a l t e r n a t i v e = ” l e s s ”)1− sample p ro p or t io n s t e s t with c o n t i n u i t y c o r r e c t i o ndata : 87 out o f 2 00 , n u l l p r o b a b i l i t y 0 . 5X−s q ua r ed = 3 . 1 2 5 , d f = 1 , p−v a l u e = 0 . 0 3 8 5 5a l t e r n a t i v e hy p o t h e s i s : t r u e p i s l e s s than 0 . 595 p e r c e n t c o n f i d e n c e i n t e r v a l :0. 0 0 0 0 0 0. 4 9 56 5sample e s t i m a t e s :p0 . 4 3 5Question 17. What type of hypothesis test is being conducted?Question 18. What was the study size and number of


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UA MATH 167 - Lecture Notes

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