a.) If random samples of size n=16 are selected, can we say anything about the x~ distribution of sample means?

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1 Sectio ) Suppose that x has a distributio with u=72 ad r=8. a.) If radom samples of size =16 are selected, ca we say aythig about the x~ distributio of sample meas? 8 X N 72, N 72, 2 16 b.) If the origial x~ distributio is ormal, ca we say aythig about the x~ distributio of radom samples of size 16? Fid P(68=x~=73). Same as part a P68 X 73 P Z P2 Z ) The heights of 18-year-old me are approximately ormally distributed, with mea 68 iches ad stadard deviatio 3 iches. a.) What is the probability that a 18 yr old ma selected at radom is betwee 67 ad 69 iches tall P67 X 69 P Z P Z b.) If a radom sample of ie 18 yr old me is selected, what is the probability that the mea height x~ is betwee 67 ad 69 iches? P67 X 69 P Z P1 Z c.) Compare your aswers for parts (a) ad (b). Is the probability i part (b) much higher? Why would you expect this? Yes, the probability i part b is much higher sice the stadard deviatio of the sample mea is smaller.

2 10.) Let x be a radom variable that represets the icubatio time for Alle hummigbird eggs. The x distributio has a mea of u=16 days. Let us assume that the stadard deviatio is approximately r=2days. The distributio of x values is more or less moud-shaped ad symmetrical but ot ecessarily ormal. Suppose that we have =30 eggs i a icubator. Let x~ be the average icubatio time for these eggs. a.) What ca we say about the probability distributio of x~? Is it approximately ormal? What are the mea ad stadard deviatio? Yes, accordig to the C.L.T, we have 2 X N 16, 30 b.) What is the probability that x~ is betwee 16 ad 17 days? P16 X 17 P Z P0 Z c.) What is the probability that x~ is less tha 15 days? 1516 PX 15 P Z PZ

3 Sectio ) USA today reported that 11% of all books sold are of the romace gere. If the local bookstore sells 316 books o a give day, what is the probability that a.) fewer tha 40 are romaces? PX 40 P Z PZ b.) at least 25 are romaces? PX 25 Z PZ c.) betwee 25 ad 40 are romaces? P25 X 40 Z P1.845 Z d.) I the solutio to this problem, what is? p? q? Does it appear that both p ad q are larger tha 5? Why is this a importat cosideratio? p = 316(0.11) = q = 316(0.89) = both are larger tha 5, this is importat for assurig a good ormal approximatio.

4 6.) It was stated that i the Cozumel regio about 44% of strikes resulted i a catch. Suppose that o a give day a fleet of fishig boats got a total of 24 strikes. What is the probability that the umber of fish caught was a.) 12 or fewer? PX 12 P Z PZ b.) 5 or more? PX 5 P Z PZ c.) betwee 5 ad 12? P5 X 12 P Z P2.492 Z d.) I the solutio to this problem, what is? p? q? Does it appear that both p ad q are larger tha 5? Why is this a importat cosideratio? p = > 5 q = > 5 both are larger tha 5, this is importat for assurig a good ormal approximatio.

5 Sectio ) Over a period of moths, a adult male patiet has take eight blood tests for uric acid. The mea cocetratio was x~= 5.35 mg/dl. The distributio of uric acid i healthy adult males ca be assumed to be ormal with r= 1.85 mg/dl. a.) Fid a 95% cofidece iterval for the populatio mea cocetratio of uric acid i this patiets blood. What is the margi of error? 95% CI: 1.85 x z , Marigi of Error = b.) What coditios are ecessary for your calculatios? Normal distributio, kowig the stadard deviatio ad the observatios eed to be idepedet. c.) Give a brief iterpretatio of your results i the cotext of this problem. We ca be 95% certai that the iterval (4.068, 6.632) cotais the true mea cocetratio of uric acid i his blood stream. d.) Fid the sample size ecessary for a 95% cofidece level with maximal error of estimate E=1.10 for the mea cocetratio of uric acid i this patiets blood. 2 2 z roudedupwards E 1.1

6 6.) Suppose you read i your local ewspaper that 45 officials i studet services eared a average of x~= $50,340 each year. a.) Assume that r=$16,920 for salaries of college officials i studet services. Fid a 90% cofidece iterval for the populatio mea salary of such persoel. What is the margi of error? 90% CI: 16,920 x z 50, , Marigi of Error = z = b.) Assume that r=$10,780 for salaries of college officials i studet services. Fid a 90% cofidece iterval for the populatio mea salary of such persoel. What is the margi of error? 90% CI: 10,780 x z 50, , Marigi of Error = z = c.) Assume that r=$4,830 for salaries of college officials i studet services. Fid a 90% cofidece iterval for the populatio mea salary of such persoel. What is the margi of error? 90% CI: 4,830 x z 50, , Marigi of Error = z = d.) Compare the margi of error for parts (a) through (c). As the stadard deviatio decreases, does the margi of error decrease? The margi of error decreases as the stadard deviatio decreases

7 e.) Compare the legths of cofidece itervals for parts (a) through (c). As the stadard deviatio decreases, does the legth of a 90% cofidece iterval decrease? The legth of the 90% CI decreases as the stadard deviatio decreases

8 10.) The followig data give aual profit per employee (i uits of oe thousad dollars per employee). Assume r=3.8 thousad dollars 4.4, 6.5, 4.2, 8.9, 8.7, 8.1, 6.1, 6.0, 2.6, 2.9, 8.1, , 8.2, 6.4, 4.7, 5.5, 4.8, 3.0, 4.3, -6.0, 1.5, 2.9, , 9.4, 5.5, 5.8, 4.7, 6.2, 15.0, 4.1, 3.7, 5.1, 4.2 a.) Use a calculator or appropriate computer software to verify that, for the precedig data x~=5.1. x = OK! b.) Let us say that the precedig data are represetative of the etire sector of retail sales compaies. Fid a 80% cofidece iterval for u, the average aual profit per employee for retail sales. Sice > 30 we use ormal approximatio: 80% CI: s 3.8 x z , c.) Let us say that you are maager of a retail store with a large umber of employees. Suppose the aual profits per employee are less tha 3 thousad dollars per employee. Do you thik that this might be low compared with other retail stores? Explai by referrig to the cofidece iterval you computed i part (b). Yes, because the 80% C.I does ot cotai 3. So we ca say that the mea aual profit per employee is sigificatly higher tha 3. d.) Suppose the aual profits are more tha 6.5 thousad dollars per employee. As store maager, would you feel somewhat better? Explai by referrig to the cofidece iterval you computed i part (b). Agai, sice the value 6.5 is ot i the 80% C.I, we ca coclude that our store has a higher mea aual profit per employee. e.) Repeat parts (b), (c), ad (d) for a 95% cofidece iterval. 95% C.I: 3.8 x z , The 95% C.I does ot cotai the value 3, so we still coclude that the store mea aual profit per employee is lower tha average. The 95% C.I does ot cotai the value 6.5, so we still coclude that our store has a higher mea aual profit per employee.

9 Sectio ) Adult wild moutai lios (18 moths or older) captured ad released for the first time i the Sa Adres Moutai gave the followig weigh (lb): 68, 104, 128, 122, 60, 64 a.) Use a calculator with mea ad sample stadard deviatio keys to verify that x~=91.0 lb ad s=30.7 lb. OK! b.) Fid a 75% cofidece iterval for the populatio average weight u of all adult moutai lios i the specified regio. Small sample size, we use the t-distributio. df = -1 = 6 1 = 5 75% CI: s 30.7 x t , ) Based o a radom sample of hospital reports from easter states, the followig iformatio was obtaied (uits i percetage of hospitals providig at least some charity care): 57.1, 56.2, 53.0, 66.1, 59.0, 64.7, 70.1, 64.7, 53.5, 78.2 a.) Use a calculator with mea ad sample stadard deviatio keys to verify that x~=62.3% ad s=8.0%. OK! b.) Fid a 90% cofidece iterval for the populatio average u of the percetage of hospitals providig at least some charity care. Small sample size, we use the t-distributio. df = -1 = 10-1 = 9 90% CI: s 8 x t ,