ST 305: Exam 3 ( ) = P(A)P(B A) ( ) = P(A) + P(B) ( ) = 1 P( A) ( ) = P(A) P(B) σ X 2 = σ a+bx. σ ˆp. σ X +Y. σ X Y. σ X. σ Y. σ n.

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1 ST 305: Exam 3 By hadig i this completed exam, I state that I have either give or received assistace from aother perso durig the exam period. I have used o resources other tha the exam itself ad the basic mathematical fuctios of a calculator (ie, o otes, electroic commuicatio, otes stored i calculator memory, etc.) Usig your calculator for values from probability distributios like the ormal or t is OK; however, if you are doig a calculatio from a ormal, t, or F distributio show your work all the way to the poit of calculatig the z, t, or F statistic. I have ot copied from aother perso s paper. I uderstad that the pealty if I am foud guilty of ay such cheatig will iclude failure of the course ad a report to the NCSU Office of Studet Coduct. I uderstad that I must show all work/calculatios, eve if they seem trivial, to get credit for my aswers. Name: ID#: x = 1 x i (x s i x) = 1 Z = X µ σ $ x i x ' $ % & s x ( ) & % r = 1 b 1 = r s y s x b 0 = y b 1 x residual = y ŷ P A or B y i y s y ' ) ( ( ) = P(A) + P(B) ( ) = 1 P( A) ( ) = P(A) P(B) P A C P A ad B µ X = x i p i µ a+bx = a + bµ X µ X +Y = µ X + µ Y σ X = σ a+bx σ X +Y σ X Y σ X +Y σ X Y ( x i µ X ) p i = b σ X P A or B = σ X + σ Y = σ X + σ Y = σ X + σ Y + ρσ X σ Y = σ X + σ Y ρσ X σ Y ( ) = P(A) + P(B) P( A ad B) ( ) = P(A)P(B A) ( ) P A ad B P(B A) = P A ad B P(A) µ X = p σ X = ˆp = X / µ ˆp = p p( 1 p) p( 1 p) σ ˆp =! P(X = k) = k!( k)! pk (1 p) k µ X = µ σ X = σ m = z * σ x ± m z = x µ 0 σ # = $ % z* σ m & ' (

2 Simple Liear Regressio b 1 = r s y s x ; b 0 = y b 1 x e i e i = y i ŷ i ; s = b j ± t * SE bj ; t = b j df = SE bj ˆµ y ± t * SE ˆµ ; ŷ ± t * SEŷ SST = SSM = SSE = SE b1 = SE b0 ( y i y ) ( ŷ i y ) ( y i ŷ i ) = s SE ˆµ = s s ( x i x ) 1 + x x i x 1 + ( ) ( x * x ) ( x i x ) SEŷ = s 1+ 1 ( + x * x ) ( x i x ) Multiple regressio chages: Chapter 7 Stuff x ± t * s, df = 1 s ( ) ± t * 1 x 1 x t = x µ 0 s t = x x 1 s 1 + s 1 + s, df = mi( 1, ) 1 1, df = 1, df = mi( 1, ) 1 1 ( x 1 x ) ± t * s p + 1, df = t = x 1 x s p , df = 1 + s p = ( 1 1)s 1 + ( 1)s 1 + s = e i p 1 df = p 1

3 Defiitios. (5 poits each) Clearly defie each of the followig terms. 1. p-value:. Power: 3. Sigificace Level: Multiple Choice. (3 poits each) Select the oe best aswer. 4. A 95% cofidece iterval for the mea umber of available parkig spaces at 10am i the Da Alle parkig deck is (45,60)? Which of the followig is the best iterpretatio of that iterval? a. O 95% of days there are betwee 45 ad 60 available spaces at 10am b. P(45 < mea umber of available spaces at 10am < 60) = c. The mea umber of available spaces at 10am is probably betwee 45 ad Cofidece itervals ted to be arrower whe a. the cofidece level is high b. the sample size is small c. the level of variability i the populatio is low 6. A hypothesis test has a p-value of 0.0 a. the ull hypothesis is rejected if usig a sigificace level of 0.05 b. the ull hypothesis is rejected if usig a sigificace level of 0.01 c. both (a) ad (b) are true d. either (a) or (b) is true 7. Use of t procedures for cofidece itervals for a populatio mea is ever appropriate if a. the populatio distributio is skewed b. the observatios i the data are ot idepedet c. the populatio stadard deviatio is ukow 8. I order to icrease the cofidece level of a hypothesis test oe must a. Icrease the sample size b. Decrease the sample size c. Neither (a) or (b)

4 9. Aswer each of the followig i 15 words or less: (3 poits each) a) Why do we compute a cofidece iterval (CI)? b) Why do we carry out a hypothesis test (HT)? c) What fact about the HT ad CI procedures we leared i Chapter 6 makes them ulikely to be useful i most real-life situatios? 10. Cosider testig the ull hypothesis that a populatio mea is equal to 5 agaist the alterative hypothesis that the mea is greater tha 5. Use a sigificace level of a) If a Type I error occurs, what must the true value of the mea have bee? (3 poits) b) What is the probability of a Type I error for this test? (3 poits) c) Carefully sketch ad label the power curve for this test. (5 poits)

5 For the remaiig questios o the exam: Show all work, eve if the math is trivial Use cofidece level 0.95 or sigificace level 0.05 uless stated otherwise If you coduct a hypothesis test, clearly show your ull ad alterative hypotheses, ad provide the p-value for the test. 11. We coduct a sample of Uber drivers to fid out if the mea umber of hours they work per week is greater tha 0. A sample of 5 drivers has mea 3 hours ad stadard deviatio 6 hours. a) Fid a 95% cofidece iterval for the mea umber of hours drive per week. (5 poits) b) Does this sample provide strog evidece that Uber drivers work more tha 0 hours per week o average? Carry out a appropriate statistical procedure to aswer this questio. (5 poits) 1. Why might you decide to carry out a hypothesis test usig a sigificace level of 0.01 istead of the traditioal value of 0.05? (5 poits)

6 Pick the Procedure. ( poits each) Select the procedure that would be the best approach to aswer each of the followig questios. You oly eed to give the letter of the aswer (to make this easier for me to grade, please write it clearly to the left of each questio or you will lose a poit). Not all aswers must be used, ad some may be used more tha oce. a. CI for a sigle mea b. HT for a sigle mea c. CI for a differece i two meas d. HT for a differece i two meas e. Matched Pairs CI f. Matched Pairs HT g. Simple liear regressio HT h. Simple liear regressio CI 13. Do patiets have higher blood pressure readigs before or after they have a x-ray? 14. O average, how much more do homes cost i Califoria tha i Washigto? 15. As they get older, do U.S. childre sped more time watchig televisio each day? 16. What is the average umber of employees i compaies located i NC? 17. O average, how may hours do NCSU studets sped o homework each week? 18. Is there a correlatio betwee the average speed traveled ad the umber of traffic fatalities o highways? 19. Do Americas ted to sped more o trasportatio tha Europeas? 0. Is the average aual temperature i Greelad showig a upward tred?

7 The followig StatCruch output comes from a study ivestigatig the relatioship betwee populatio desity (populatio per square mile) ad amouts of air pollutio for geographic regios i the US. (Some results have bee removed.) Use it to aswer the remaiig questios. Simple liear regressio results: Depedet Variable: Amt of Air Pollutio (ug/m3) Idepedet Variable: Populatio per Sq mile Amt of Air Pollutio (ug/m3) = XXXXXXX Sample size: 50 R (correlatio coefficiet) = XXXXXXX R-sq = XXXXXX Estimate of error stadard deviatio: XXXXXXX Parameter estimates: Parameter Estimate Std. Err. DF 95% L. Limit 95% U. Limit Itercept Slope Aalysis of variace table for regressio model: Source DF SS MS F-stat P-value Model XXXXXXX Error Total Predicted values: X value Pred. Y s.e.(pred. y) 95% C.I. for mea 95% P.I. for ew ( , ) ( , ) 1. Write a equatio for a lie that ca be used to predict the amout of air pollutio i a regio based o its populatio per square mile. (3 poits). Provide a 95% cofidece iterval for the slope of the lie relatig the amout of air pollutio to the populatio per square mile. (3 poits) 3. Usig a two-sided t-test, test the ull hypothesis that there is o correlatio betwee amout of air pollutio ad populatio per square mile. Show all your work, icludig calculatio of the t statistic! (5 poits)

8 (Cotiue to use the output from the previous page) 4. You are thikig of movig to a specific regio with 70 people per square mile. Give a 95% iterval for the amout of air pollutio i that regio. (3 poits) 5. Compute the correlatio coefficiet betwee amout of air pollutio ad populatio per square mile (3 poits) 6. I the ANOVA table, what do the abbreviatios SS ad MS stad for? ( poits)