BASIC STATISTICS ECOE 1323

Size: px

Start display at page:

Download "BASIC STATISTICS ECOE 1323"

Constance McDaniel
5 years ago
Views:

1 BASIC STATISTICS ECOE 33 SPRING 007 FINAL EXAM NAME: ID NUMBER: INSTRUCTIONS:. Write your ame ad studet ID.. You have hours 3. This eam must be your ow work etirely. You caot talk to or share iformatio with ayoe. 4. Show all your work. Partial credit will oly be give where sufficiet uderstadig of the problem has bee demostrated ad work is show. DON'T WRITE ON THIS TABLE QUESTION # # #3 #4 BONUS TOTAL POINTS

2 SECTION : MULTIPLE-CHOICE Questio # For each questio i this sectio, circle the correct aswer. (Problem is worth pts.). The distictio betwee descriptive ad iferetial statistics is that a) descriptive statistics are umeric, iferetial statistics are graphic. b) descriptive statistics are mea-based, iferetial statistics are media-based. c) descriptive statistics describe data sets, iferetial statistics ivolve geeralizig to populatios. d) descriptive statistics are used i social sciece, iferetial statistics are used i physical scieces. e) Noe of these.. For which of the followig statistics would oe ot eed to put the data i order from smallest to largest? a) the iterquartile rage b) the trimmed mea c) the media d) the rage e) the variace 3. Suppose that for a set of umeric data, where the umbers are ot all differet, the stadard deviatio is less tha.0. The it must be true that a) the variace < the stadard deviatio. b) the variace the stadard deviatio. c) the variace = the stadard deviatio. d) the stadard deviatio e) the stadard deviatio the variace. < the variace. 4. Which of the followig idicates that a associatio betwee ad y is positive? a) A positive coefficiet of determiatio b) A positive stadard deviatio about the least squares lie c) A positive itercept of the least squares lie d) A positive Pearso s correlatio coefficiet e) A positive residual sum of squares 5. The slope of the regressio lie ad the correlatio betwee two variables is related i the followig way: a) The slope is always greater i absolute value tha the correlatio. b) The slope ad correlatio must be of the same sig. c) The slope ad correlatio must be of differet sig. d) The slope is always less i absolute value tha the correlatio. e) Noe of (a) - (d) is ecessarily true.

3 6. Of the properties below, which is NOT a basic property of probability? ( ) a) For ay evet E, 0 P E. ( ) b) If S is the sample space for a eperimet, P S =. c) If two evets E ad F are disjoit, the ( ) = ( ) + ( ) P E or F P E P F. d) For ay evet E, P ( E ) P ( ot E ) + = e) For ay two DISJOINT evets, E ad F, ( ) P ( E or F ) P E ad F 7. The evet, ot A is called the of evet A. a) egatio b) complemet c) uio d) itersectio e) cojuctio For questios 8-9, let deote the umber of accidets i a give moth at a certai high school parkig lot. Suppose that the probability distributio of is: P() What is the probability that there are fewer tha 3 accidets i a give moth? a).0 b).4 c).585 d).799 e) Noe of these 9. What is the probability that to 4 (iclusive) accidets occur i a give moth? a).6 b).34 c).394 d).608 e) Noe of these 0. Which of the followig statemets about ormal curves is false? a) Every ormal curve is symmetric. b) Every ormal curve is symmetric about 0. c) Every ormal curve is bell-shaped. d) Every ormal curve is cetered at its mea. e) About 0.68 of the area uder a ormal curve is withi stadard deviatio of its mea.

4 . The proportio of values i a ormal populatio distributio that fall withi stadard deviatios of the mea is: a) 0.08 b) c) d) e) Which of the followig is ot a property of a biomial eperimet? a) It cosists of a fied umber of trials,. b) Outcomes of differet trials are idepedet. c) Each trial ca result i oe of several differet outcomes. d) Observatios cosist of the umber of successes for each trial of the eperimet. e) The probability of success is costat for each trial. 3. If is a biomial radom variable with = 0 ad p = 0.5, the a) σ =.875 b) σ =.5 c) σ =.875 d) σ =.5 e) Noe of (a) - (d) 4. Suppose that has a probability distributio with desity fuctio c, if 6 < < 8 f ( ) = 0 otherwise. The the value of c is: a) 0.5 b) 0.6 c) 0.7 d) 0.8 e) Whe costructig a 95% cofidece iterval, the cofidece level is: a) 0.95 b) c) d) e) Caot be determied 6. Which of the followig is ot a statistical hypothesis? a) > 00 b) µ = 00 c) µ > 00 σ = d) 5 e) P.5 3

5 7. A type I error is made by a) rejectig H 0 whe it is true. b) rejectig H 0 whe it is false. c) failig to reject H 0 whe it is true. d) failig to reject H 0 whe it is false. 8. The P-value for a z test of H : P=.5 vs. H : P<.5, where z=.36 is: 0 0 ( > ) b) P( z<.36) P( z> ) ( z> 6 ) e) P(.36 < z ad z>.36) a) P z.36 c).36 d) P.3 9. Suppose you take a simple radom sample from a populatio kow to be ormally distributed, but the value of σ is ukow. Your sample size is = 0. Which formula below should be used to fid the 90% cofidece iterval for the mea? a) ±.645 s σ b) ± d) ±.8 σ σ e) ± c) ±.833 s 0 0. The degrees of freedom of a paired t test based o = 0 pairs is a) 9 b) 0 c) 9 d) 0 e) Noe of these 4

6 Questio () (I) A reporter for a studet ewspaper is writig a article o the cost of attedig college. A portio of the article deals with the cost of off-campus housig. A sample of 0 oe-bedroom uits withi oe-half mile of campus resulted i a sample mea of $350 per moth ad a sample stadard deviatio of $30, assumig that the populatio is ormally distributed. (a) [5 Poits] Provide a 95% cofidece iterval estimate of the populatio mea. (b) [8 Poits] The college ewspaper claims that mea cost per moth for oe-bedroom uits withi oe-half mile of campus is less tha $370. Test this claim at 0.05 level of sigificace. 5

7 (II) [7 Poits] To estimate the proportio of traffic deaths i Florida last year that were alcohol related, determie the ecessary sample size for the estimate to be accurate to withi.05 with probability.99. Based o results of a previous study, we epect the proportio to be about.35. 6

8 Questio (3) (I) [0 Poits] I 990, 5.8% of job applicats who were tested for drugs failed the test. At the 0.0 level, test the claim that the failure rate is ow lower if a radom sample of 50 curret job applicats results i 58 failures. H o : H a : Test Statistic: P-value: Coclusio: (a) State H 0 ad H a. H 0: p=0.058 H a: p<0.058 (b) Calculate the test statistic. z = 50, = ˆp p SE ˆp 0 = = 58 58, ˆp = = , = 3. 3 SE ˆp = p0( p) 0 = ( )( 0. 94) 50 = (c) Fid the P-value or give the rejectio regio. P-value=P(Z<-3.3)=ormalcdf(-E99,-3.3)= (d) State your coclusio. Coclusio: We reject H 0 ad coclude that the failure rate is ow lower tha 5.8%. 7

9 (II) [0 Poits] How large a sample size is eeded to estimate the mea aual icome of Native Americas correct to withi $000 with probability.99? Suppose there is o prior iformatio about the stadard deviatio of aual icome of Native Americas, but we guess that about 68% of their icomes are betwee $0000 ad $40,000 ad that this distributio of icomes is approimately moud shaped. 8

10 Questio (4) (I) [0 Poits] The Motaa Highway Patrol is iterested i determiig whether Motaa residets or oresidets drive faster o a particular stretch of Iterstate 90. Idepedet radom samples of the speeds of cars havig Motaa licese plates ad cars licesed i other states results i the summary data listed below. Group Sample size Sample Mea Sample stadard deviatio Motaa Others Assume the populatio variaces are the same. At 0.05 level of sigificace, is there sufficiet evidece to coclude that oresidets drive faster o this stretch of Iterstate 90 tha residets of Motaa? H o : H a : Test Statistic: P-value Coclusio: 9

11 (II) [0 Poits] A article reports that (4.0, 5.6) is a 95% cofidece iterval for the mea legth of stay, i days, of patiets i hospital for a particular operatio. The article reports the sample size of 50, but ot the sample mea or stadard deviatio. Fid them. 0

12 إضافي Bous: (I) [3 Poits] For a ormally distributed variable, verify that the probability betwee µ +. 67σ equals.50 µ 67σ. ad (II) [3 Poits] Fid the b-value such that the iterval probability for a ormal distributio. µ bσ ad µ + bσ cotais 98% of the (III) A fast food chai sells hamburger that they claim has sodium cotet of 650 milligrams. A simple radom sample of 35 hamburgers was aalyzed for sodium cotet. A 99% cofidece iterval for the populatio mea sodium cotet, µ, of such hamburgers is (65, 67). Aswer the followig questios with yes, o or ca't tell. Give a eplaatio for your aswer. (a) [ Poit] Does the populatio mea lie i the iterval (65, 67)? (b) [ Poit] If we were to use the precedig data to test the hypotheses H o : µ =650 versus H a : µ 650. At a % sigificace level, would we reject the ull hypothesis? Eplai.

13 Formulas: i i = IQR = Q3 Q, =, S = i i = ( ) i y i y S y i = yˆ = a + b, b = r, a = y b r = S i y i y i = i = k k! P ( X = k ) = p ( p ), k = 0,,,, =,! = ( ) 3.., µ = p, σ = p p k k k!( k )! Populatio mea (s) Level C cofidece iterval Hypothesis test Large sample Oe-sample z test Use s ifσ is ukow Small sample ad σ ukow Oe-sample t test Large samples Two-sample z test Use s ad s if σ, σ are ukow Pooled two-sample t test ( ) s + ( ) s s p = ( + ) σ, ukow & equal σ Two-sample t test σ ± z σ s ± t df = ( ) σ, ukow & uequal ( ) Matched pairs t-test Oe-Proportio z test Computig P-values Sample size for desired margi of error m Oe-sample z iterval: z σ = m ± z σ + σ µ z = σ t ( ) µ =, df = s z = σ σ + ( ) ± t sp + t = sp + df = + df = + s s ± t + t = s + s ( ) df = mi, df = mi, t =, =, df = s p~( p~) p p p~ ± z z = + 4 p ( p ) where Wilso Estimate where the Sample proportio + p~ = p ˆ = + 4 Use z -table for z tests ad t-table for t tests Reject H if P-value < α Oe-proportio z iterval: z + 4 = p p m ( ) ( ) ( ), 4 z + = m ( )

14 3

15 4

16 5

Inferential Statistics and Probability a Holistic Approach. Inference Process. Inference Process. Chapter 8 Slides. Maurice Geraghty,

Inferential Statistics and Probability a Holistic Approach. Inference Process. Inference Process. Chapter 8 Slides. Maurice Geraghty, Iferetial Statistics ad Probability a Holistic Approach Chapter 8 Poit Estimatio ad Cofidece Itervals This Course Material by Maurice Geraghty is licesed uder a Creative Commos Attributio-ShareAlike 4.0