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# Statistics for Business and Economics STA 2023

FIU

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This 13 page Class Notes was uploaded by Jonatan Kunde on Monday October 12, 2015. The Class Notes belongs to STA 2023 at Florida International University taught by Dongmei An in Fall. Since its upload, it has received 51 views. For similar materials see /class/221816/sta-2023-florida-international-university in Statistics at Florida International University.

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Date Created: 10/12/15

STA 2023 1 REVIEW 11 not all inclusive True False ilhe standard deviation of the sample mean is always equal to the standard deviation of the population The number of defective vending machines on a college campus is an example of a continuous random variable iEvery normal distribution has a mean 0 and standard deviation l A binomial random variable is a discrete random variable N Which of the following is a valid probability distribution X lpX b X lpX 0 X lpX d X lpX 0 L30 335 0 36 8l24 1 L20 0 L65 3 L52 4l65 2 L40 3 10 6 22 0 11 In a study of the time college freshmen use to study each week it is found that the mean is 68 hours with a standard deviation of 25 hours If 100 freshmen are randomly selected what is the probability that their mean weekly study time exceeds seven hours Suppose in a recent national election 40 of the voters were men If a random sample of 25 voters was selected what is the probability that more than 10 but less than 20 were women A traffic study conducted at one point on an interstate highway shows that vehicle speeds are normally distributed with a mean of 635 miles per hour and a standard deviation of 48 mph a If a vehicle is randomly selected what is the probability that its speed is between 55 and 65 mph b 60 of the speeds in the distribution exceed what value Find the mean and variance of the following probability distribution x 9 1 4 Pxl 3 4 3 Answer Key F F F T d 2119 937 a 5833 b 623 mph a 11 b 2829 9959 STA 2023 Examples on Probability 34 Rules of Probability General Addition Rule For any two events A and B PA U B PA PB PA and B ie PA U B PA PB PA n B For two mutually exclusive events we have the special addition rule PA U B PA PB Hence PA 1 PA WHY Examples 1 Let us start with the tossing the two coins example Recall that S HH HT TH TT In this example Let A getting exactly one head and B getting at least one head and D getting two tails PAPB and PAnB PAUB PAUD 2 In a certain suburb of 500 people 300 are male Also 50 people are colorblind out of which 40 are male If a person is picked at random from the suburb nd the probability of picking either a male or a colorblind person PM or CB PM PCB PM n CB 3150 3 In a certain bank preVious records show that out of all prospective customers walking into the bank 60 open at least a checking account 25 open at least a saVings account and 15 open both accounts Suppose a prospective customer is picked at random Compute the following probabilities a Pcustomer opens at least one of the two accounts 07 b Pcustomer opens neither of the two accounts 03 Conditional Probability In certain situation the sample space S must be changed or altered to take into account new information In such a situation probabilities change and are said to be conditional on the occurrence of the event de ned by this new information In some sense we are basically de ning a new sample space 8 So we de ne the conditional probability of the event A given that the event B has occurred Denoted by PA B it is given by PAnB PB provided PB gt 0 PAIB The formula for computing conditional probabilities leads to a formula for r quot the r 39 39 quotquot ofthe39 quot oftwo events The General Multiplication Rule P A and B PAIBPB PBIAPA Independence of Events Two events A and B are independent if the occurrence of one has no effect on the occurrence of the other In this situation PAB PA and PA and B PAPB An example The following table shows a breakdown of the drivers in a certain city by sex and by eyesight M F Total Colorblind 500 15 515 Not Colorblind 9235 7800 17035 Total 9735 7815 17550 Suppose a driver is chosen at random from the city Compute the probability that a The person is colorblind b The person is female and colorblind c The person is female given that the person is colorblind d A Suppose that we pick a female What is the probability that the person is colorblind e Are the events being a female and being colorblind independent 2 In a certain college 6 of all students are business majors 50 are female and 10 of the females are business majors Suppose a student is selected at random Find the probability that l PBusinesle PBus and F 005 PFlBus 08333 PBus and Male PMl Bus 3 Suppose we select two machines at random from 5 for inspection Suppose also that unknown to the inspector two of the five machines are defective a Find the probability that both machines picked for inspection are defective b Find the probability that at least one of the machines picked for inspection is non defective c Suppose the machines are selected with replacement How do your answers in a and b change 4 A particular airline ies a 10 AM ight to Chicago from Miami and a 10 AM ight to LA from Miami Let C be the event that the Chicago ight is full and let L be the event that the LA ight is full Also PC 06 PL 05 and the two events L and C are independent of each other What is the probability that both ights are full Exercises 323 327 329 330 331 333 335 338 339 343 345 346 350 351 353 355 359 363 364 STA 2023 REVIEW III 1 TrueFalse The width of a con dence interval increases as the sample size decreases everything else held constant It is desirable for the width of a con dence interval to be large A type I error is committed whenever the null hypothesis is incorrectly rejected For an uppertailed hypothesis test if the critical value of z is 233 and the test statistic is 257 the null hypothesis should be accepted 2 Ifa 90 con dence interval for p is 35 45 what is meant by 90 con dence 3 In a random sample of 150 clerical workers 30 exhibit an intense dislike for their jobs Calculate a 94 con dence interval for the proportion of clerical workers who exhibit an intense dislike for their jobs 4 A telephone answering service records the length of each call A random sample of 12 reports yields a mean length of 32 minutes with a standard deviation of 7 minute Calculate a 95 con dence interval for the mean length of phone calls for this answering service 5 A market research rm wishes to determine the mean number of hours families in a particular community watch television per week A random sample of 300 families results in a mean viewing time of 268 hours with a standard deviation of 73 Do you have enough evidence to conclude the mean viewing time in this community exceeds 25 hours Use at 07 6 De ne a Type I and Type II error in terms of problem 5 7 In a random sample of 200 college students 45 had taken Statistics in high school Using 0c 04 do you have enough evidence to conclude the proportion of college students who have not taken Statistics in high school differs from 75 8 Find the pvalue for problem 7 KEY l T F T F 2 In repeated sampling 90 of the intervals constructed will contain p 3 139 261 4 276 364 5 z 427 yes 6 Type I Conclude the mean Viewing time per family in this community is greater than 25 hours when in fact it is less than or equal to 25 hours Type II Conclude the mean Viewing time per family in this community is equal to or less than 25 hours when in fact it is greater than 25 hours 7 z 82 no 4122 STA 2023 SAMPLE REVIEW CHAPTERS 1 3 not allinclusive 1 True False 7 a The Empirical Rule applies to moundshaped data 7 b A sample is the complete set of observations or measurements of interest to an experimenter 7 c The stemandleaf display is a tool used to graph qualitative data 7 d The union of two events A and B is the event containing all the sample points that are either in event A or event B or both events A and B 7 e In a skewed right distribution the mean is greater than the median 2 Use the sample below to nd the following 680436 mean standard deviation median range mode 3 Identify the following variables as Qualitative or Quantitative The declared major of a college student The cost of a fastfood meal The favorite brand of softdrink of a consumer 4 At one retail store the mean time for a person to complete an application for employment is 245 minutes with standard deviation 39 minutes What is the probability a randomly selected applicant takes longer than 37 minutes to fill out the application V39 0 A group of faculty and students were surveyed regarding a campus issue their opinions are recorded below Use the following table to answer the following questions OPINION Faculty Students Opposed No Opinion 34 0 70 23 In Favor 20 93 a What proportion of the responses were in opposition to the issue b What proportion of those opposed were faculty c What proportion of the responses were either from students or in favor of the issue d Are the events SStudent and NNo opinion mutually exclusive events Thoroughly justify your answer e Are the events SStudent and NNo opinion independent events Thoroughly justify your answer As items come to the end of a production line an inspector chooses which items are to go through a complete inspection Five percent of all items produces are defective Seventy percent of all defective items go through a complete inspection and twenty percent of all good items go through a complete inspection If an item goes through a complete inspection what is the probability it is not defective KEY Brief answers on the exam remember to show your work 9959 T F F T T mean 35 median 5 mode 6 st dev 42 range ll Qual Quan Qual at most 11 a 43 b 33 c 86 d No PSnN 10 i 0 e No PS i PSlN 78 i 100 84

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