MT 158 MATH

MT221 Final’s study guide. b. There seems to be a contradiction in this information. How is it possible for all three of these statements to be correct? Explain. Name_________________________ • • • • Study this study guide and turn in on Wednesday. Bring this paper to Monday to review. Midterm is not group test. Individual test. When you take midterm, you can not use personal smartphone and computer. Midterm’s study guide 1) International Communications Research (IRC) conducted the 2008 Spring Cleaning Survey for the Soap and Detergent Association. IRC QUESTIONED 777 American adults who springclean about which spring- cleaning chore they would like to hire someone to do for them. The “chore” results were: 47% washing windows, 23% cleaning the bathroom, 12% cleaning the kitchen, 8% dusting, 7% mopping, 3% other. The survey has a margin of error of plus or minus 3.52%. a. What is the population? b. How many people were polled? c. What information was obtained from each person? d. e. Using the information given, estimate the number of surveyed adults who would gladly hire someone to wash the windows, if they could. What do you think the “margin of error of plus or minus 3.52%” means? f. How would you use the “margin of error” in estimating the percentage of all adults who would like to hire out the spring- cleaning chore of “Cleaning the kitchen”? 2) During a radio broadcast a few years ago, David Essel reported the following three statistics: (1) the U.S divorce rate is 55%, and when married adults were asked if they would remarry their spouse, (2) 75% of the women said yes, and (3) 65% of the men said yes. a. What is the “stay married” rate? 3) A drug manufacturer is interested in the proportion of persons with hypertension (elevated blood pressure) whose condition can be controlled by a new drug the company has developed. A study involving 500 individuals with hypertension is conducted, and it is found that 80% of the individuals are able to control their hypertension with the drug. Assuming that the 5000 individuals are representative of the group that has hypertension, answer the following questions: a. What is the population? b. What is the sample? c. Identify the parameter of interest. d. Identity the statistic and give its value. e. Do we know the value of the parameter? 4) In July 8, 2019, USA Today article titled “Couples saying’ I don’t to expensive weddings,” the cutbacks may not be extending to the number of bridesmaids was as follows: 7 6 5 2 3 7 6 13 6 3 2 7 8 9 a. Construct a dotplot of these data. b. What are the most common numbers of bridesmaids? How does the dotplot show this? 4) Some cleaning jobs are disliked more than others. According to the July 17,2009, USA Today Snapshot on a survey of women by Consumer Reports National Research Center, the cleaning tasks women dislike the most are presented in the following Pareto diagram. a. How many total women were surveyed? b. Verify the 15% listed for “Cleaning refrigerator.” c. Explain how the “cu, % for dusting” value of 80% was obtained and what it means. d. What three tasks would make no more than 75% of the women surveyed happy if those tasks were eliminated? 5) Fill in the Blank: (Statistics, Data, Descriptive Statistics, Inferential Statistics, Probability, Population, Samples, Parameter) a) ( )-the science of planning studies and experiments, obtaining data, and then organizing, summarizing, presenting, analyzing, interpreting, and drawing conclusions based on the data. b) ( ) – collections of observations. c) ( ) – organizing and summarizing data; by graphing and by numerical values (such as an average). d) ( )- uses methods that take a result from a sample, extend it to the population, and measure the reliability of the result. e) ( )- the chance of an event occurring. f) ( )- the complete collection of all individuals to be studied. g) ( ) – a subcollection of members selected from a population. Example: h) ( ) – a numerical measurement describing some characteristic of a population. Ex: The average age of students enrolled as an undergraduate at Baylor University is 21.1 years. 6) Identify each of the following as examples of (1) attribute (qualitative) or (2) numerical (quantitative) variables: a) Th breaking strength of a given type of string b) The hair color of children auditioning for the musical Annie c) The number of stop signs in towns of fewer than 500 people d) Whether or not a faucet is defective e) The number of questions answered correctly on a standardized test f) The length of time required to answer a telephone call at a certain real estate office. 7) What is the mean weekly pay if 5 employees earn $425 per week, 3 earn $750 per week, and 1 earns $1340? 8) A city police officer, using radar, checked the speed of cars as they were traveling down the main street in town. Construct a stem-and-leaf plot for this data: 41 31 33 35 36 37 39 49 33 19 26 27 24 32 40 39 16 55 38 36 9) A survey of 100 resort club managers on their annual salaries resulted in the following frequency distribution. Ann. 15Sal.($1000) 25 No. Mgrs 12 2535 37 3545 26 4555 19 5565 6 a. Prepare a cumulative frequency distribution for the annual salaries. b. Prepare a cumulative relative frequency distribution for the annual salaries. c. Construct an ogive for the cumulative relative frequency distribution found above. d. What value bonds the cumulative relative frequency of 0.75? e. 75% of the annual salaries are below what value? Explain the relationship between parts d and e. 10) For those seventh graders with cell phones, the number of programmed numbers in their phones are: 100 37 12 20 53 10 20 50 35 30 a) Find the mean number of programmed numbers on a seventh grader’s cell phone. 25 26 b) Find the median number of programmed numbers on a seventh grader’s cell phone c) Explain the difference in values of the mean and median. d) Remove the most extreme value and answer parts a through c again. e) Did removing the extreme value have more effect on the mean or the median? Explain why. 27 30 33 30 32 30 34 30 27 25 29 31 31 32 34 32 33 30 a) Find the mean. b) Find the range. c) Find the variance. d) Find the standard deviation e) Using the dot-plot from part a, draw a line representing the range. Then draw a line starting at the mean with the length that represents the value of the standard deviation f) Describe how the distribution of data, the range, and the standard deviation are related. 13) Following are the American College Test (ACT) scores attained by the 25 members of a local high school graduating class: 21 24 23 17 31 19 19 20 19 25 17 23 16 21 28 25 25 21 14 19 17 18 28 20 20 a) Using the concept of depth, describe the position of 24 in the set of 25 ACT scores in two different ways. b) Find 𝑃5 ,𝑃10, and 𝑃20 for the ACT scores. 11) Consider the sample 2,4,7,8,9. Find the following c) Find 𝑃99 , 𝑃90 and 𝑃80 for the ACT scores. A research study of manual dexterity involved determining the time requires to complete a task. The time required for each of 49 individuals with disabilities is shown here (data are ranked): A. mean, 𝑥̅ B. median, 𝑥̃ C. mode D. midrange 11) Consider the sample 6,8,7,5,3,7. Find the following A. Range 13) B. Variance𝑠 2 , using formula (2.5) 7.1 7.2 7.2 7.6 7.6 7.9 8.1 8.1 8.1 8.3 8.3 C. Standard deviation, s 8.4 8.4 8.9 9.0 9.0 9.1 9.1 9.1 9.1 9.4 9.6 12) Recruits for a police academy were required to undergo a test that measures their exercise capacity. The exercise capacity (in minutes) was obtained for each 20 recruits: 9.9 10.1 10.1 10.1 10.2 10.3 10.5 10.7 11.0 11.1 11.2 11.2 11.2 12.0 13.6 14.7 14.9 15.5 standard deviation is 6.8 hours. Assume the empirical rule is appropriate. a) What proportion of the time will it take the cleanup crew 97.6 hours or more to clean the plant? b) B. Within what interval will the total cleanup time fall 95% of the time? a) Find 𝑄1. b) Find 𝑄2 c) Find 𝑄3 d) Find 𝑃95 e) Find the 5- number summary. f) Draw the box-and-whisker display. 14) A sample has a mean of 50 and a standard deviation of 4.0. Find the z-score for each value of x: A. x = 54 B. x=50 C. x =59 D. x = 45 15) An exam produced grades with a mean score of 74.2 and a standard deviation of 11.5. Find the z-score for each test score x: A. x=54 B. x=68 C. x=79 D. x=93 16) A sample has a mean of 120 and a standard deviation of 20.0. Find the value of x that corresponds to each of these standard scores: A. Z =0.0 B. Z=1.2 C. Z= -1.4 D. Z=2.05 17) The empirical rule indicates that we can expect to find what proportion of the sample included between the following : A. 𝑥̅ – s and 𝑥̅ + s B. 𝑥̅ -2s and 𝑥̅ + 2s C. 𝑥̅ – 3s and 𝑥̅ + 3s MT221 Test III 1) The empirical rule indicates that we can expect to find what proportion of the sample included between the following : A. 𝑥̅ – s and 𝑥̅ + s ___________ B. 𝑥̅ -2s and 𝑥̅ + 2s __________ C. 𝑥̅ – 3s and 𝑥̅ + 3s _______ 2) The average cleanup time for a crew of a medium-sized firm is 84.0 hours and the 3) A random sample of plum tomatoes was selected from a local grocery store and their weights were recorded. The mean weight was 6.5 ounces with a standard deviation of 0.4 ounces. If the weights are normally distributed: a) What percentage of weights fall between 5.7 and 7.3? b) What percentage of weights fall above 7.7? c) 4) Consider the accompanying contingency table, which presents the results of an advertising survey table, which presents the results of an advertising survey about the use of credit by Martin Oil Company customers. *finish chart Preferred Method of Payment 0-4 Cash 150 100 25 0 0 275 Oil-Company Card 50 35 115 80 70 350 Bank Credit Card 50 60 65 45 5 225 Sum 250 195 205 125 75 850 5-9 10-14 15-19 ≥ 20 Sum a) How many customers were surveyed? b) Why are these bivariate data? What type of variable is each one? c) d) How many customers preferred to use an oil company card? e) How many customers made 20 or more purchases f) How many customers preferred to use an oil-company card and made between 5 and 9 purchases last year? 5) Cell phones and iPods are necessities for the current generation. Does the use of one indicate the use of the other? Seven junior high students who own both a cell phone and an iPod were randomly selected, resulting in the following data: Cell, n(phone #’s ) iPod, n(songs saved) 42 7 75 78 126 22 23 303 212 401 500 536 200 278 f) Find r. 6) The federal Highway Administration annually reports on state motor-fuel taxes. Based on the latest report, the amount of receipts, in thousands of dollars, can be estimated using the equation: Receipts= -5359+0.9956 Collections. a) If a state collected $500,000, what would you estimate the receipts to be? 7) The values of x used to find points for graphing the line y= 14.9 +0.66X in figure are arbitrary. Suppose you choose to use x = 20 and x = 50 A. Find a) SS(x) a) What are the corresponding 𝑦̂ values? b) SS(y) c) SS(xy) d) Find the slope e) Find the y-intercept b) Locate these two points on above Figure. Are these points on the line of best fit? Explain why or why not. 8) A group of files in a medical clinic classifies the patients by gender and by type of diabetes (type 1 or type 2). The grouping may be shown as follows. The table gives the number in each classification. TYPE OF DIABETES Gender 1 2 Male 3 15 FEMALE 35 20 a) “Strongly agrees” that sustainability is important to her. . . If one file is selected at random, find the probability of the following: b) Is a member of Generation X. a) The selected individual is female. b) The selected individual has type 2 diabetes 9) During the Spring 2009 semester at Monroe Community College, a random sample of students was questioned on their knowledge of the meaning of “sustainability”. The primary motivation for the survey was to investigate how interested students might be in a Sustainability CERTIFICATE AND to discover the best means of informing them of this option. The following table lists how much 224 students agreed with the statement “Sustainability is important to me.” d) Is a member of the Baby Boomers, given that she “agrees” with the importance of sustainability Levels of Agreement with Statement “Sustainability is important to me” Generation (ages) c) “disagrees “ with the importance of sustainability to her, given that she’s a member of the Millennium Y generation. Dis Strongly Strongly Agree Agree agree Disagree Total Millennium Y (18-29) 74 109 11 1 193 Generation X (30-44) 14 8 1 0 23 Baby Boomers (45+) 2 3 0 1 6 All Respondents 90 120 12 2 224 Find the probability that a random selected student: 10) A and B are events defined on a sample space, with P (A) =0.7 and P (BlA) = 0.4. Find P (A and B) 11) The odds of event A are 13 : 3 in favor. What is P(A)? 12) Definitions problems: Find the given word and put it in the parentheses [Experiment, Event space, Event, Outcomes: Sample Space, Theoretical Probabilities, the same population, Correlation, no correlation. Positive correlation, Negative correlation, linear correlation,] b) If the P(event) = ( ), then it will happen and is called the certain event c) If the P(event) = ( ), then it cannot happen and is called the impossible event d) ∑P(outcome) = ( ) 14) Which of the following experiments have equally likely outcomes? a) Rolling a fair die. b) Flip a coin that is weighted so one side comes up more often than the other. c) Pull a ball out of a can containing 6 red balls and 8 green balls. All balls are the same size. a) ( ) an activity that has specific results that can occur, but it is unknown, which results will occur. b) ( ) is the results of an experiment c) ( ) is a set of certain outcomes of an experiment that you want to have happen d) ( ) is a collection of all possible outcomes of the experiment. Usually denoted as SS. e) ( ) is the set of outcomes that make up an event. f) It is not always feasible to conduct an experiment over and over again, so it would be better to be able to find the probabilities without conducting the experiment. These probabilities are called ( ) g) Bivariate Data is Consists of the values of two different response variables that are obtained from ( of interest. d) Picking a card from a deck. 15) Why linear regression is important Final study guide(this study guide) 30) If P (A) = 0.4, P (B) =0.5, and P (A and B) = 0.1, find P (A or B) 31) If P (A) =0.5, P (B) =0.3, and P (A and B) =0.2, find P (A or B) 32) IF P (A) =0.4, P (B) = 0.5, and P (A or B) =0.7, find P (A and B) 33) If P (A) = 0.4, P (A or B) = 0.9, and P (A and B) =0.1, FIND P (B) 34) A and B are events defined on a sample space, with P (A) =0.7 and P (B/A) = 0.4. Find P (A and B) h) ( ) measures the strength of a linear relationship between two variables and compares the relationship between only two variables. 35) A and B are events defined on a sample space, with P (A/B) = 0.5 and P (B) =0.8 Find P (A and B) i) ( ): x increases, y increases. j) ( ): x increases, y decreases. k) If the ordered pairs follow a straight-line path: linear correlation. 37) A and B are events defined on a sample space, with P (B)=0.5 and P(A and B) = 0.4 Find(A/B) 13) Probability Properties (Fill it in) a) ( ) ≤ P(event) ≤ ( ) 36) A and B are events defined on a sample space, with P (A) =0.6 AND P (A and B) =0.3 Find P (B/A) 38) Juan lives in a large city and commutes to work daily by subway or by taxi. He takes the subway 80% of the time because it costs less, and he takes a taxi the other 20% of the time. When taking the subway, he arrives at work on time 70% of the time, whereas he makes it on time 90% of the time when traveling by taxi. A. What is the probability that Juan took the subway and is at work on time on any given day? B. What is the probability that Juan took a taxi and is at work on time on any given day? 39) Suppose that A and B are events defined on a common sample space and that the following probabilities are known: P (A) =0.3, P (B) =0.4 and P (A/B) =0.2 Find P (A or B) 40) Suppose that A and B are events defined on a common sample space and that the following probabilities are known: P (A or B) =0.7, P (B) =0.5, and P (A/B) =0.2. Find P (A) 41) Suppose that A and B are events defined on a common sample space and that the following probabilities are known: P (A)=0.4, P(B)=0.3 and P(A or B) =0.66. Find P (A/B) 42) Given P (A or B) =1.0, P ̅̅̅̅̅̅̅̅̅̅̅̅̅ (𝐴 𝑎𝑛𝑑 𝐵)=0.7¸ a) P (B) b). P (A) c). P (A/B) 43) Determine whether each of the following sets of events is mutually exclusive. A. Five coins are tossed: “no more than one head is observed,” “two heads are observed,” and “three or more heads are observed.” B. A salesperson calls on a client and makes a sake: the amount of the sale is “less than $100.”is between $100 and $1,000,” is “more than $500” C. One student is selected at random from the student body” the person selected is “female,” is “male,” is “older than 21,” 44) Do people take indoor swimming lessons in the middle of the hot summer? They sure do at the Webster Aquatic Center. During the month of July 2009 alone, 283 people participated in various forms of lessons. Swim Categories Daytime Evenings Preschool 66 80 Levels 69 56 Adult and diving 10 2 Total 145 138 If one swimmer was selected at random from the July participants: a. Are the events the selected participants is “ daytime” and “evening” mutually exclusive? Explain b. Are the events the selected participant is “preschool” and “levels” mutually exclusive? Explain c. Are the events the selected participant is “daytime” and “preschool” mutually exclusive? Explain d. Find P (preschool). e. Find P (daytime) f. Find P ( not levels) g. Find P ( preschool or evening) h. Find P ( preschool and daytime) i. Find P ( daytime/ levels) j. Find P (adult and diving/ evening) 45) Determine whether each of the following pairs of events is independent: a. Rolling a pair of dice and observing a “1” on the first die and a “1” on the second die b. Drawing a “spade” from a regular deck of playing cards and then drawing another “spade” from the same deck without replacing the first card c. Same as part b except the first card is returned to the deck without replacing the first card d. Owning a red automobile and having blonde hair e. Owning a red automobile and having a flat tire today f. Studying for an exam and passing the exam 46) Suppose that P (A) = 0.3, P (B)=0.4, and P (A and B) =0.20 a. What is P (A I B) ? b. What is P (B I A)? c. Are A and B independent? 47) One student is selected at random from a group of 200 students known to consist of 140 full-time (80 female and 60 male) students and 60 part-time (40 female and 20 male) students. Event A is “the student selected is full time,” and event C is “the student selected is female.” a. Are events A and C independent? Justify your answer. b. Find the probability P( A and C) 48) You have applied for two scholarships: a merit scholarship (M) and an athletic scholarship (A). Assume the probability that you receive the athletic scholarship is 0.25, the probability that you receive both scholarships is 0.15, and the probability that you get at least one of the scholarship is 0.37. Use a Venn diagram to answer these questions: A. What is the probability that you receive the merit scholarship? B. What is the probability that you do not receive either of the two scholarships? C. What is the probability that you receive the merit scholarship given that you have been awarded the merit scholarship? D. What is the probability that you receive the athletic scholarship, given that you have been awarded the merit scholarship? E. Are the events “receiving an athletic scholarship” and “receiving a merit scholarship” independent events? Explain. 49 Consider the set of integers 1,2,3,4, and 5. A. One integer is selected at random. What is the probability that it is odd? B. Two integers are selected at random (one at a time with replacement so that each of the five is available for a second selection). Find the probability that neither is odd; exactly one of them is odd; both are odd. 50) A box contains 25 parts, of which 3 are defective and 22 are non-defective. If 2 parts are selected without replacement, find the following probabilities: A P(both are defective) B. P (exactly one is defective) C. P (neither is defective) 52) A card is randomly selected from a deck of cards. Find the probability that the card is a Jack or the card is a heart. 54) Find Expected Value and Standard Deviation Given) Random Guessing; n = 100 questions. Probability of correct guess; p = ¼, Probability of wrong guess; q = 3/4 55) There are 5 red chip, 4 blue chips, and 6 white chips in a basket. Two chips are randomly selected. Find the probability that the second chip is red given that the first chip is blue. (Assume that the first chip is not replaced.) 56)You roll a die. Find the probability that you roll a number less than 3 or a 4. 57) The College Board website provides much information for students, parents, and professionals with respect to the many aspects involved in Advanced Placement (AP) courses and exams. One particular annual report provides the percent of students who obtain each of the possible AP grades (1 through 5). The 2008 grade distribution for all subjects was as follows: AP Grades Percents 1 20.9 2 21.3 3 24.1 4 19.4 5 14.3 a. Express this distribution as a discrete probability distribution. b. Find the mean and standard deviation of the distribution of the AP exam scores for 2008. 58) The random variable A has the following probability distribution: A 1 2 3 4 5 P(A) 0.6 0.1 0.1 0.1 0.1 a. Find the mean and standard deviation of A. b. How much of the probability distribution is within 2 standard deviations of the mean? c. What’s the probability that A is between -2 and +2 59) The random variable x has the following probability distribution: x 1 2 3 4 5 P(x) 0.6 0.1 0.1 0.1 0.1 a. Find the mean and standard deviation of (x). b. What’s the probability that x is between and + 60) Evaluate each of the following. a) 4! b. 0! d.6!/2! 6! e). 5!/2!3! f) 4!(6−4)! g) (73) h) (41)(0.2)1 (0.8)3 Chocol ate 30% No chocol ate Jellybe ans 25% 13% Crea mfilled Marshmal low 11% 8% Malt ed 7% Don ’t Kno w 6% 65) If boys and girls are equally likely to be born, what is the probability that in a randomly selected family of six children, there will be at least one boy? (Find the answer using a formula.) 66) Consider the binomial distribution where n=11 and p-0.05. A. Find the mean and standard deviation using formulas (5.7) and (5.8). B. Using Table 2 in Appendix B, list the probability distribution and draw a histogram 61) Use the probability function for three coin tosses as demonstrated on page 239 and verify the probabilities doe x =0,2, and 3. C. Locate µ and σ on the histogram. 67) Given the binomial probability function 62) If x is a binomial random variable, calculate the probability of x for each case. A. n= 4, x=1, p=0.3 C. n=2, x=0, p = 1 4 E. n=4, x=2, p=0.5 B. n= 3, x=2, p=0.8 D. n=12, x=12, p=0.99 1 6 F. n=3, x=3, p= 63) if x is a binomial random variable, use Table 2 in Appendix B to determine the probability of x for each of the following: A. n=10, x=8, p=0.3 B. n=8, x=7, p=0.95 C. n=15, x=3, p=0.05 D. n=12, x=12, p=0.99 E. n=9, x=0, p=0.5 F. n=6, x=1, p=0.01 G. Explain the meaning of the symbol 0+ that appears in Table 2. 64) Let’s say that 80% of all business startups in the IT industry report that they generate a profit in their first year. If a sample of 10 new IT business startups is selected, find the probability that exactly seven will generate a profit in their first year. First, do we satisfy the conditions of the binomial distribution model? 64-1) Consider the manager of Steve’s Food Markets illustrated in Example 5.9. What would be the manager’s “risk” if he bought “better” eggs, say with P(BAD) = 0.01 using the “more than one” guarantee? 5 1 1 P (X) = 𝑋 ∙ (2)𝑥 ∙ ( 2)5−𝑥 for x= 0, 1, 2,3,4,5 A. Calculate the mean and standard deviation of the random variable by using formulas (5.1), (5.3a), and (5.4.) B. Calculate the mean and standard deviation using formulas (5.7) and (5.8) 68) If the binomial ( q + p) is squared, the results is (𝑞 + 𝑝)2 = 𝑞 2 + 2qp + 𝑝2 . For the binomial experiment with n=2, the probability of no successes in two trials is 𝑞 2 (the first term in the expansion), the probability of one success in two trails is 2qp (the second term in the expansion). And the probability of two successes in two trials is 𝑝2 (the third term in the expansion). Find (𝑞 + 𝑝 ) 3 and compare its terms to the binomial probabilities for n+3 trails. 69) A binomial random variable has a mean equal to 200 and a standard deviation of 10. Find the values of n and p. 70) The probability of success on a single trial of a 1 4 binomial experiment is known to be . The random variable x, the number of successes, has a mean value of 80. Find the number of trials involved in this experiment and the standard deviation of x.

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