Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Inductive probability wikipedia , lookup
Bootstrapping (statistics) wikipedia , lookup
Foundations of statistics wikipedia , lookup
History of statistics wikipedia , lookup
Taylor's law wikipedia , lookup
Resampling (statistics) wikipedia , lookup
Law of large numbers wikipedia , lookup
Practice worksheet 1 In a given year, the average annual salary of a famous football player was $2,000,000 with a standard deviation of $245,000. If a simple random sample of 50 players was taken, what is the probability that the sample mean will exceed $2,100,000? Practice worksheet 2 2. Bill Clinton, the former President of the United States, believes that the proportion of voters who will vote for Hillary Clinton, the former Secretary of State, in the year 2016 presidential elections is 0.51. A sample of 500 voters is selected at random. What is the probability that the number of voters in the sample who will vote for Hillary Clinton in the year 2016 is between 260 and 280? Practice worksheet 3 3. 25% of all children affected by an infectious disease recover from it without any intervention. A big pharmaceutical company develops a new drug for that infectious disease and ten children who have the disease were randomly selected and received the medication. Nine of these children recovered from the disease shortly thereafter. Assume that the medication was ineffective and absolutely worthless. What is the probability that at least nine of ten children receiving the new drug will recover? Based on this probability, explain what your finding would mean. Practice worksheet 4 4. Assume that you just uploaded a video you made on Youtube and further assume that you have observed that the number of hits you get for your video occur at a rate of 2 in one hour. a. What is the probability that your video will get no hits in a one hour period after you uploaded it? b. What is the probability that it will be viewed once in a one hour period after you uploaded your video? c. What is the probability that it will be viewed at least once in a one hour period after you uploaded your video? Practice worksheet 5 5. The length of time patients must wait to see a doctor at an emergency room in a Dubai hospital has a uniform distribution between 30 minutes and 2 hours. a. What is the probability that a patient would have to wait no more than one hour? b. What is the probability that a patient would have to wait exactly one hour? c. What is the probability that a patient would have to wait between one and three hours? Practice worksheet 6 6. Researchers studying the effects of a new diet found that the weight loss over a oneyear period by those on the diet was normally distributed with a mean of 10 kg and a standard deviation of 5 kg. a. What proportion of the dieters lost more than 13 kg? b. What proportion of the dieters gained weight? c. What is the probability that the mean weight loss of a random sample of 36 dieters is greater than 11 kg? d. What is the minimum amount of weight lost by those highest 3% of weight losers on the diet? Practice worksheet 7 7. Times spent studying by students in the week before final exams follow a normal distribution with standard deviation 9 hours. A random sample of 5 students was taken in order to estimate the mean study time for the population of all students. What is the probability that the sample mean exceeds the population mean by more than 2.1 hours? Practice worksheet 8 8. A medical researcher wants to investigate the amount of time it takes for patients' headache pain to be relieved after taking a new prescription painkiller. She plans to use statistical methods to estimate the mean of the population of relief times. She believes that the population is normally distributed with a standard deviation of 25 minutes. How large a sample should she take to estimate the mean time to within 2 minute with 99% confidence? Practice worksheet 9 9. The assembly line that produces an electronic component of a missile system has historically resulted in a 3% defective rate. A random sample of 500 components is drawn. What is the probability that the defective rate is greater than 5%? Suppose that in the random sample the defective rate is 5%. Explain what the result you found suggests about the defective rate on the assembly line? Practice worksheet 10 10. A business student claims that, on average, business students have to take more classes than students in other colleges. She claims that business students take more than 4 classes per semester. To examine this claim, a statistics professor asks a random sample of 10 business students to report the number of classes that they take. The results are exhibited below. Can the professor conclude, at the 5% level of significance, that the claim is true? Assume that the number of classes is normally distributed with a standard deviation of 1.5. 3 6 4 5 5 6 4 3 4 5 a. State the null and alternative hypotheses. b. Test the student’s claim by calculating the p-value. Show all intermediate steps. c. How do you know whether the student’s claim is supported or not? Practice worksheet 11 11. A federal agency responsible for enforcing laws governing weights and measures routinely inspects packages to determine whether the weight of the contents is at least as great as that advertised on the package. A random sample of 18 containers whose packaging states that the contents weigh 8 ounces was drawn. The contents were weighed and the results follow. a. Can we conclude at the 5% significance level that on average the containers are mislabeled? b. How do you know? Explain. 7.8 7.91 7.93 7.99 7.94 7.75 7.97 7.95 7.79 8.06 7.82 7.89 7.92 7.87 7.92 7.98 8.05 7.91 Practice worksheet 12 12. Several students and faculty at AUS make international calls during the academic year. A survey of 12 students and faculty revealed the following information about the number of times an international call is made during the spring semester. Assume that the number of calls made is normally distributed with a standard deviation of 12 calls. 3 41 17 1 33 37 18 15 17 12 29 51 a. Estimate with 90% confidence the average number of calls during the spring semester. b. How large a sample should we take so that the mean number of calls is within 2 calls with 95% confidence? Practice worksheet 13 13. A fast-food company is considering building a restaurant at a certain location. Based on financial analyses, a site is acceptable only if the number of pedestrians passing the location averages more than 100 per hour. The number of pedestrians observed for each of 40 different one-hour periods. The number of pedestrians was then recorded to be 105.7. Assume that the population standard deviation is known to be 16. a. State the null and alternative hypotheses. b. Can we conclude at the 5% significance level that the site is acceptable? How do you know? c. Please state which type of error (Type I or Type II) may be committed here? Consider your conclusion in part (b) of this question and explain your answer. Also, what are the implications of committing such error? Practice worksheet 14 14. A fitness trainer wants to estimate the average weight loss of people who are in his new workout class. In a preliminary study, he guesses that the standard deviation of the population of weight losses is about 10 pounds. a. How large of a sample should he take to estimate the mean weight loss to within two pounds, with 99% confidence? b. How large of a sample should he take to estimate the mean weight loss to within two pounds, with 95% confidence? Practice worksheet 15 15. A statistics practitioner is in the process of testing to determine whether there is enough evidence to infer that the population mean is different from 180. She calculated the mean and standard deviation of a sample of 200 observations as 𝑋̅ = 175 and s = 22. a. Calculate the value of the test statistic (you must decide whether the z or t is appropriate) to determine whether there is enough evidence at the 5% significance level. b. Repeat this with s = 45. c. Discuss what happens to the test statistic when the standard deviation increases. Practice worksheet 16 16. A chain coffee shop owner buys a new espresso machine and wants to determine whether this new machine is generating any profits. To test the profitability of the machine, the owner collects data on the profit made (in thousands of dollars) from 16 randomly selected stores and finds that the sample mean and sample standard deviation are $500 and $200, respectively. If the profit level is normally distributed a. State the null and alternative hypothesis. b. Test the hypotheses using a standardized test statistic at 5% significance level. c. Explain the results you find. Practice worksheet 17 17. A politician claims that the average UAE resident is more than 20 pounds overweight. To test his claim, a random sample of 20 UAE residents was weighed, and the difference between their actual and ideal weights was calculated. The data are listed below. Do these data allow us to infer at the 5% significance level that the politician's claim is true? 16 23 18 41 22 18 23 19 22 15 18 35 16 15 17 19 23 15 16 26 Practice worksheet 18 18. Has the building and allocating Terminal 3 to Emirates Airlines resulted in better on-time performance? Before terminal 3 was built and allocated to Emirates Airlines, Emirates airline claimed that 92% of its flights were on time. A random sample of 165 flights completed this year reveals that 153 were on time. Can we conclude at the 5% significance level that the Emirates airline's on-time performance has improved after allocating terminal 3 only to Emirates airline? Practice worksheet 19 19. A recent survey in Dubai revealed that 60% of the vehicles traveling on highways, where speed limits are posted at 120 km per hour, were exceeding the limit. Suppose you randomly record the speeds of ten vehicles traveling on 311 where the speed limit is 120 km per hour. Let X denote the number of vehicles that were exceeding the limit. a. What is the distribution of X? b. Find P(X = 10). c. Find the probability that the number of vehicles exceeding the speed limit is between 4 and 9(exclusive) d. Find the probability that two of the recorded vehicles were exceeding the speed limit Practice worksheet 20 20. A security department receives an average of 10 telephone calls each afternoon between 2 and 4 P.M. The calls occur randomly and independently of one another. a. Find the probability that the department will receive 13 calls between 2 and 4 P.M. on a particular afternoon. b. Find the probability that the department will receive seven calls between 2 and 3 P.M. on a particular afternoon. c. Find the probability that the department will receive at least five calls between 2 and 4 P.M. on a particular afternoon. Practice worksheet 21 21. A survey among teenagers in the US found that 30% of teenage consumers receive their spending money from part-time jobs. If five teenagers are selected at random, find the probability that at least three of them will have part-time jobs. Practice worksheet 22 22. Statistics show that the survival rate for patients suffering from a certain type of cancer is 25%. What is the probability that of 6 randomly selected patients, 4 will recover? Practice worksheet 23 23. The recent average starting salary for new business college graduates is $47,500. Assume salaries are normally distributed with a standard deviation of $4,500. a. What is the probability of a new graduate receiving a salary between $45,000 and $50,000? b. What is the probability of a new graduate getting a starting salary in excess of $55,000? c. What percent of starting salaries are no more than $42,250? d. What is the cutoff for the bottom 5% of the salaries? e. What is the cutoff for the top 3% of the salaries? Practice worksheet 24 24. The lifetime of an iPhone battery that is advertised to last for 5,000 days is normally distributed with a mean of 5,100 days and a standard deviation of 200 days. a. What is the probability that a battery lasts longer than the advertised figure? b. If we wanted to be sure that 98% of all iPhone batteries last longer than the advertised figure, what figure should be advertised? Practice worksheet 25 25. A new online gaming website has an average random hit rate of 2.9 unique visitors every 4 minutes. What is the probability of getting exactly 50 unique visitors every hour? Practice worksheet 26 26. Several students and faculty at AUS play football. A survey of 12 students revealed the following information. Estimate with 95% confidence the mean number of football matches played. Assume that the number of matches is normally distributed with a standard deviation of 12. 3 41 17 1 33 37 18 15 17 12 29 51 Practice worksheet 27 27. The average person loses an average of 15 days per year to colds and flu. The natural remedy echinacea reputedly boosts the immune system. One manufacturer of echinacea pills claims that consumers of its product will reduce the number of days lost to colds and flu by one-third. To test the claim, a random sample of 50 people was drawn. Half took echinacea, and the other half took placebos. If we assume that the standard deviation of the number of days lost to colds and flu with and without echinacea is 3 days, find the probability that the mean number of days lost for echinacea users is less than that for nonusers. Practice worksheet 28 28. A sample of 50 retirees is drawn at random from a normal population whose mean age and standard deviation are 75 and 6 years, respectively. a. Describe the shape of the sampling distribution of the sample mean in this case. b. Find the mean and standard error of the sampling distribution of the sample mean. c. What is the probability that the mean age exceeds 73 years? d. What is the probability that the mean age is at most 73 years? Practice worksheet 29 29. A survey of 100 businesses revealed that the mean after-tax profit was $80,000. Assume the population standard deviation is $15,000: a. Determine the 95% probability interval estimate of the mean after-tax profit. b. Explain why you can use the probability interval formula here, even though the population is not necessarily normal. Practice worksheet 30 30. A fitness trainer wants to estimate the average weight loss of people who are in his new workout class. In a preliminary study, he guesses that the standard deviation of the population of weight losses is about 10 pounds. a. How large of a sample should he take to estimate the mean weight loss to within two pounds, with 99% confidence? b. How large of a sample should he take to estimate the mean weight loss to within two pounds, with 95% confidence? Practice worksheet 31 31. A fast-food company is considering building a restaurant at a certain location. Based on financial analyses, a site is acceptable only if the number of pedestrians passing the location averages more than 100 per hour. The number of pedestrians observed for each of 40 different one-hour periods. The number of pedestrians was then recorded to be 105.7. Assume that the population standard deviation is known to be 16. a. State the null and alternative hypotheses. b. Can we conclude at the 5% significance level that the site is acceptable? How do you know? c. Define Type I and Type II errors. Then, describe the consequences of Type I and Type II errors. Practice worksheet 32 32. Suppose that 15% of all invoices are for amounts greater than $1,000. A random sample of 60 invoices is taken. What is the mean and standard error of the sample proportion of invoices with amounts in excess of $1,000? What is the probability that the proportion of invoices in the sample is greater than 18%? Practice worksheet 33 33. An advertisement claims that four out of five doctors recommend a particular product. A consumer group wants to test that claim, and takes a random sample of 30 doctors. The consumer group finds that of this group of doctors, only 0.75 would recommend the product. What is the probability that the proportion of doctors in this sample who recommend the product is 0.75 or less? Do you have reason to doubt the manufacturer’s claim? Explain. Practice worksheet 34 34. Examine the data and the graphic below to answer the following questions. a. Is the price of oil interval, ordinal, or nominal data? b. Which variable is independent and which one is dependent? c. What is the slope of the line? d. What is the y-intercept value? e. Which number shown in the graphic below tells you if there is a positive or negative relationship between these two variables? What is that number? Is the relationship positive or negative? f. What is the correlation between the price of oil and the price of gasoline? g. What percentage of variation in the price of gasoline is explained by the price of oil? How do you know? h. If the price of oil increases by $1 per barrel, how much does the price of gasoline increase? i. If the price of oil is $110, what is the corresponding market price of gasoline? j. Is the price of oil a good variable to use to predict the price of gasoline (assuming you are in a country without price controls on gasoline)? Why or why not? Practice worksheet 35 35. Answer the following questions based on the regression output where house size is the dependent variable. Regression Statistics Multiple R 0.865 R Square 0.748 Adjusted R Square 0.726 Standard Error 5.195 Observations 50 a. What percentage of the variability in house size is explained by this model? b. Which of the independent variables in the model are significant at the 1% level? c. What is the predicted house size for an individual earning an annual income of $40,000, having a family size of 4, and having 13 years of education?(Annual income is in thousand $US) d. Suppose the builder wants to test whether the coefficient on income is significantly different from 0. What is the value of the relevant t-statistic? e. At the 0.01 level of significance, what conclusion should the builder draw regarding the inclusion of income in the regression model? Practice worksheet 36 36. A random sample of 15 hourly fees for car washers (including tips) was drawn from a normal population. The sample mean and sample standard deviation were = $14.9 and s = $6.75. Can we infer at the 5% significance level that the mean fee for car washers (including tips) is greater than 12? Practice worksheet 37 37. A company claims that 10% of the users of a certain drug experience depression as a side effect. In clinical studies of this drug, 81 of the 900 subjects experienced depression. We want to test their claim and find out whether the actual percentage is not 10%. a. State the appropriate null and hypotheses. b. Is there enough evidence at the 5% significance level to infer that the this claim is correct? c. Compute the p-value of the test. d. Construct a 95% confidence interval estimate of the population proportion of the users of this drug who experience depression. e. Explain how to use this confidence interval to test the hypotheses Practice worksheet 38 38. The admissions officer of a university is trying to develop a formal system to decide which students to admit to the university. She believes that determinants of success include the standard variables— high school grades and SAT scores. To investigate the issue, she randomly sampled 100 fourth year students and recorded the following variables: GPA for the first 3 years at the university (range: 0 to 12) GPA from high school (range: 0 to 12) SAT score (range: 400 to 1600) The regression output is summarized below. What is the dependent variable? b. What are the independent variables? c. What percentage of variation in the dependent variable is explained by the independent variables? d. What is the coefficient of determination? Interpret its value. e. What is the regression equation? f. Test to determine whether each of the independent variables is linearly related to the dependent variable in this model g. If HS GPA increases by 1 point, how is university GPA affected? h. If a student has a 3.7 has GPA and a 27 SAT, what is the expected university GPA using this model? Practice worksheet 39 39. A researcher believes that energy drink consumption may increase heart rate. Suppose it is known that heart rate (in beats per minute) is normally distributed with an average of 70 bpm for adults. A random sample of 25 adults was selected and it was found that their average heartbeat was 73 bpm after energy drink consumption, with a standard deviation of 7 bpm. a. Formulate the null and alternative hypotheses. b. Test the hypotheses in the previous question at the 10% significance level to determine if we can infer that energy drink consumption increases heart rate. c. If you were to estimate the p-value of the test statistic in part (b), would it be more than or less than 10%? Do not compute the p-value of the test statistic: only state your reason based on your result in part (b). Practice worksheet 40 40. Alaa and Sara are friends. Let X denote the number of cats that Alaa may have in the next two years, and let Y denote the number of cats Sara may have during the same period. The marginal probability distributions of X and Y are shown below. x 0 1 2 y 0 1 2 P(x) 0.5 0.3 0.2 P(y) 0.4 0.5 0.1 a. Compute the mean and variance of X. b. Compute the mean and variance of Y. c. Assume that X and Y are independent and find their bivariate distribution. Practice worksheet 41 41. Alaa and Sara are friends. Let X denote the number of cats that Alaa may have in the next two years, and let Y denote the number of cats Sara may have during the same period. The marginal probability distributions of X and Y are shown below. x 0 1 2 y 0 1 2 P(x) 0.5 0.3 0.2 P(y) 0.4 0.5 0.1 a. Compute the mean and variance of X. b. Compute the mean and variance of Y. c. Assume that X and Y are independent and find their bivariate distribution. Practice worksheet 42 42. The joint probability distribution of variables X and Y is shown in the table below. Ahmed and Hassan are car salespeople. Let X denote the number of cars that Ahmed will sell in a month, and let Y denote the number of cars Hassan will sell in a month. X Y 1 2 3 1 0.3 0.18 0.12 2 0.15 0.09 0.06 3 0.05 0.03 0.02 a. Determine the marginal probability distribution of X. b. Determine the marginal probability distribution of Y. c. Calculate E(X) and E(Y). d. Calculate V(X) and V(Y). e. Develop the probability distribution of X + Y. f. Calculate E(X + Y) directly by using the probability distribution of X + Y. g. Calculate V(X + Y) directly by using the probability distribution of X + Y. h. Verify that E(X + Y) = E(X) + E(Y). i. Verify that V(X + Y) = V(X) + V(Y)? Why did you get this result? Practice worksheet 43 43. A standard certification test was given at three locations. 1,000 candidates took the test at location A, 600 candidates at location B, and 400 candidates at location C. The percentages of candidates from locations A, B, and C who passed the test were 70%, 68%, and 77%, respectively. One candidate is selected at random from among those who took the test. a. What is the probability that the selected candidate passed the test? b. If the selected candidate passed the test, what is the probability that the candidate took the test at location B? c. What is the probability that the selected candidate took the test at location C and failed? Practice worksheet 44 44. The final grades in a statistics course are normally distributed with a mean of 75 and a standard deviation of 10. The professor must convert all marks to letter grades. She decides that she wants 10% A's, 35% B's, 40% C's, 10% D's, and 5% F's. Determine the cutoffs for each letter grade. Practice worksheet 45 45. Assume that a there is a random system of police patrol in Dubai and further assume that a police officer may visit a given location Y=0,1,2,3,… times per half-hour period, with each location being visited an average of once per time period. Also assume that Y is Poisson distributed. a. What is the probability that the police officer will miss a given location during a half hour period? b. What is the probability that it will be visited once? c. Twice? d. At least once? Practice worksheet 46 46. Elizabeth has decided to form a portfolio by putting 30% of her money into stock 1 and 70% into stock 2. She assumes that the expected returns will be 10% and 18%, respectively, and that the standard deviations will be 15% and 24%, respectively. Practice worksheet 47 47. Let X be a normally distributed random variable with a mean of 12 and a standard deviation of 1.5. What proportions of the values of X are: a. less than 14 b. more than 8 c. between 10 and 13 Practice worksheet 48 48. The number of midterm exams taken per semester by AUS students is normally distributed with a mean of 11 and a standard deviation of 3. a. What proportion of students takes more than 13 exams per semester? b. What is the probability that in a random sample of 20 students more than 240 midterms are taken? (Hint: What is the mean number of midterms taken by the sample of 20 students?) Practice worksheet 49 49. The time it takes for a QBA professor to grade his midterm exam is normally distributed with a mean of 26 minutes and a standard deviation of 10 minutes. There are a total of 200 students in the professor's classes. What is the probability that he needs more than 80 hours to mark all the midterm exams? (The 200 midterm exams of the students in this year's classes can be considered a random sample of the many thousands of midterm exams the professor has marked and will mark.) Practice worksheet 50 50. A car dealer in Dubai claims in an advertisement that 3% of all its cars require a service in the first year. An SBM student who takes the QBA class wants to check the claim by surveying 400 people who recently purchased one of the dealer's cars. What is the probability that more than 4.7% require a service within the first year? What would you say about the advertisement's honesty if in a random sample of 400 people 4.7% report at least one service call? Practice worksheet 51 51. Among the most exciting aspects of a student's life are the important calls they make where such critical issues as the make and model of the car he will purchase and whether she will get a new designer handbag are discussed. A sample of 20 students was asked how many hours per month are devoted to these calls. The responses are listed here. Assuming that the variable is normally distributed with a standard deviation of 8 hours, estimate the mean number of hours spent on those calls by all students. Use a confidence level of 90%. 14 17 3 6 17 3 8 4 20 15 7 9 0 5 11 15 18 13 8 4 Practice worksheet 52 52. A statistics professor wants to compare today's students with those 5 years ago. All his current students’ grades are stored on his computer so that he can easily find the population mean. However, the grades 5 years ago are in a random file in his computer. He does not want to retrieve all the grades and will be satisfied with a 95% confidence interval estimate of the mean grade 5 years ago. If he assumes that the population standard deviation is 12, how large a sample should he take to estimate the mean to within 2 grades? Practice worksheet 53 53. A random sample of 12 second-year AUS students enrolled in a business statistics course was drawn. At the course's completion, each student was asked how many hours he or she spent doing the homework in statistics. The data are listed here. It is known that the population standard deviation is σ =16. The instructor has recommended that students devote 6 hours per week for the duration of the 12-week semester, for a total of 72 hours. Test to determine whether there is evidence that the average student spent less than the recommended amount of time. Compute p-value of the test. 62 80 52 60 72 76 58 80 76 60 70 76 Practice worksheet 54 54. A smartphone manufacturer advertises that, on average, its cell phone batteries will last more than 50000 hours. To test the claim, an AUS student took a random sample of 100 smartphones and measured the amount of time until each battery lasted. If we assume that the lifetime of this type of battery has a standard deviation of 4000 hours, can we conclude at the 5% significance level that the claim is true? (Mean for the sample is 50650) Practice worksheet 55 55. A restaurant manager claims that, on average, a chef is required to prepare more than five cakes per week. To examine the claim, the manager asks a random sample of 10 chefs to report the number of cakes they prepare weekly. The results are exhibited here (below). Can the manager conclude at the 5% significance level that the claim is true, assuming that the number of cakes is normally distributed with a standard deviation of 1.5? 2 7 4 8 9 5 11 3 7 4 Practice worksheet 56 56. A travel agent believes that the average number of days traveled per year among international travelers is less than 10 days. From past experience, he knows that the population standard deviation is 3 days. In testing to determine whether his belief is true at α = .01, he uses the sample of 100 travelers. What is the probability of a Type II error, given that the true population average is 9 days? Practice worksheet 57 57. A company that produces Internet services wanted to determine the number of devices connected to the Internet American homes contain. The company surveyed 240 randomly selected homes and determined the number of devices connected to the Internet (the mean for the sample is 4.66 and the standard deviation is 2.37). If there are 100 million households, estimate with 99% confidence the total number of devices connected to the Internet in the United States. Practice worksheet 58 58. Amazon sells ground and whole bean coffee over the Internet. Buyers enter their orders, pay by credit card, and receive delivery by postal services. Amazon analyzed and determined that the average order would have to exceed $85 if the selling coffee on Internet were to be profitable. To determine whether selling coffee on Internet would be profitable in one large city, Amazon offered the service and recorded the size of the order for a random sample of buyers. (Sample mean is 89.27, sample standard deviation is 17.30 and sample size is 85). Can we infer from these data that an e-grocery will be profitable in this city at alpha 5% significance level? Practice worksheet 59 59. Use data file “data”. A Dubai Honda dealer manager has been inspecting weekly advertisement expenditures. They have been giving advertisements to the national newspaper for the past 6 months and the number of ads per week has varied from one to seven. The dealer’s sales staff has been tracking the number of customers who enter the store each week. The number of ads and the number of customers per week for the past 26 weeks are recorded. a. What is the dependent variable? What is the independent variable? b. What is the coefficient of determination? Interpret its value. c. What is the regression equation? Practice worksheet 60 Practice worksheet 61 61. In 2014, the average annual salary of a business school graduate was $50,000 with a standard deviation of $6125. If a simple random sample of 50 business graduates was taken, what is the probability that the sample mean will exceed $52,500? Practice worksheet 62 62. Bill Clinton, the former President of the United States, believes that the proportion of voters who will vote for Hillary Clinton, the former Secretary of State, in the year 2016 presidential elections is 0.52. A sample of 500 voters is selected at random. What is the probability that the number of voters in the sample who will vote for Hillary Clinton in the year 2016 is between 265 and 280? Practice worksheet 63 63. Suppose that the starting salaries of female CFO’s have a positively skewed distribution with mean of $156,000 and a standard deviation of $32,000. The starting salaries of male CFO’s are positively skewed with a mean of $150,000 and a standard deviation of $30,000. A random sample of 50 male CFO’s and a random sample of 40 female CFO’s are selected. a. What is the sampling distribution of the sample mean difference 𝑋̅1−𝑋̅2? Explain. b. What is the probability that the sample mean salary of female CFO’s will not exceed that of the male CFO’s? Practice worksheet 64 64. 30% of all children affected by an infectious disease recover from it without any intervention. A big pharmaceutical company develops a new drug for that infectious disease and ten children who have the disease were randomly selected and received the medication. Eight of these children recovered from the disease shortly thereafter. Assume that the medication was ineffective and absolutely worthless. What is the probability that at least nine of ten children receiving the new drug will recover? Based on this probability, explain what your finding would mean. Practice worksheet 65 65. Assume that you just uploaded a video you made on Youtube and further assume that you have observed that the number of hits you get for your video occur at a rate of 5 in one hour. a. What is the probability that your video will get no hits in a one hour period after you uploaded it? b. What is the probability that it will be viewed once in a one hour period after you uploaded your video? c. What is the probability that it will be viewed at least once in a one hour period after you uploaded your video? Practice worksheet 66 66. The length of time patients must wait to see a doctor at an emergency room in a Dubai hospital has a uniform distribution between 20 minutes and 1.5 hours. If X is the waiting time a. What is the probability that a patient would have to wait no more than 50 minutes? b. What is the probability that a patient would have to wait exactly one hour? c. What is the probability that a patient would have to wait between one and two hours? Practice worksheet 67 67. The final course grades of a mathematics class at AUS are normally distributed with a mean of 82 and a standard deviation of 4. (Total possible points = 100.) a. What is the probability that a randomly selected student will have a score of 86 or higher? b. What is the probability that a randomly selected student will have a score between 80 and 90? c. If four students are selected randomly, what is the probability that they will have an average score of 85 or higher? Practice worksheet 68. Find the following probabilities 68 Practice worksheet 69 69. “Genius” is an organization whose members possess IQs that are in the top 2% of the population. It is known that IQs are normally distributed with a mean of 110 and a standard deviation of 20. Find the minimum IQ needed to be a “Genius” member. Practice worksheet 70 70. The final grades in a statistics course are normally distributed with a mean of 75 and a standard deviation of 10. The professor must convert all marks to letter grades. She decides that she wants 10% A's, 35% B's, 40% C's, 10% D's, and 5% F's. Determine the cutoffs for each letter grade. Practice worksheet 71 71. The assembly line that produces an electronic component of a missile system has historically resulted in a 3% defective rate. A random sample of 500 components is drawn. What is the probability that the defective rate is greater than 5%? Suppose that in the random sample the defective rate is 5%. Explain what the result you found suggests about the defective rate on the assembly line? Practice worksheet 72 72. A business student claims that, on average, business students have to take more classes than students in other colleges. She claims that business students take more than 4 classes per semester. To examine this claim, a statistics professor asks a random sample of 10 business students to report the number of classes that they take. The results are exhibited below. Can the professor conclude, at the 5% level of significance, that the claim is true? Assume that the number of classes is normally distributed with a standard deviation of 1.5. 3 6 4 5 5 6 4 3 4 5 a. State the null and alternative hypotheses. b. Test the student’s claim by calculating the p-value. Show all intermediate steps. c. How do you know whether the student’s claim is supported or not? Practice worksheet 73 73. A federal agency responsible for enforcing laws governing weights and measures routinely inspects packages to determine whether the weight of the contents is at least as great as that advertised on the package. A random sample of 18 containers whose packaging states that the contents weigh 8 ounces was drawn. The contents were weighed and the results follow. 7.8 7.91 7.93 7.99 7.94 7.75 7.97 7.95 7.79 8.06 7.82 7.89 7.92 7.87 7.92 7.98 8.05 7.91 a. Can we conclude at the 5% significance level that on average the containers are mislabeled? b. How do you know? Explain. Practice worksheet 74 74. Several students and faculty at AUS make international calls during the academic year. A survey of 12 students and faculty revealed the following information about the number of times an international call is made during the spring semester. Assume that the number of calls made is normally distributed with a standard deviation of 12 calls. 3 41 17 1 33 37 18 15 17 12 29 51 a. Estimate with 90% confidence the average number of calls during the spring semester. b. How large a sample should we take so that the mean number of calls is within 2 calls with 95% confidence? Practice worksheet 75 75. A fast-food company is considering building a restaurant at a certain location. Based on financial analyses, a site is acceptable only if the number of pedestrians passing the location averages more than 100 per hour. The number of pedestrians observed for each of 40 different one-hour periods. The number of pedestrians was then recorded to be 105.7. Assume that the population standard deviation is known to be 16. a. State the null and alternative hypotheses. b. Can we conclude at the 5% significance level that the site is acceptable? How do you know? c. Please state which type of error (Type I or Type II) may be committed here? Consider your conclusion in part (b) of this question and explain your answer. Also, what are the implications of committing such error? Practice worksheet 76 76. A fitness trainer wants to estimate the average weight loss of people who are in his new workout class. In a preliminary study, he guesses that the standard deviation of the population of weight losses is about 10 pounds. a. How large of a sample should he take to estimate the mean weight loss to within two pounds, with 99% confidence? b. How large of a sample should he take to estimate the mean weight loss to within two pounds, with 95% confidence? Practice worksheet 77 77. A statistics practitioner is in the process of testing to determine whether there is enough evidence to infer that the population mean is different from 180. She calculated the mean and standard deviation of a sample of 200 observations as 𝑋̅ = 175 and s = 22. a. Calculate the value of the test statistic (you must decide whether the z or t is appropriate) to determine whether there is enough evidence at the 5% significance level. b. Repeat this with s = 45. c. Discuss what happens to the test statistic when the standard deviation increases. Practice worksheet 78 78. A chain coffee shop owner buys a new espresso machine and wants to determine whether this new machine is generating any profits. To test the profitability of the machine, the owner collects data on the profit made (in thousands of dollars) from 16 randomly selected stores and finds that the sample mean and sample standard deviation are $500 and $200, respectively. If the profit level is normally distributed a. State the null and alternative hypothesis. b. Test the hypotheses using a standardized test statistic at 5% significance level c. Explain the results you find Practice worksheet 79 79. A politician claims that the average UAE resident is more than 20 pounds overweight. To test his claim, a random sample of 20 UAE residents was weighed, and the difference between their actual and ideal weights was calculated. The data are listed below. Do these data allow us to infer at the 5% significance level that the politician's claim is true? 16 23 18 41 22 18 23 19 22 15 18 35 16 15 17 19 23 15 16 26 a. State the null and alternative hypothesis. b. Test the hypotheses using a standardized test statistic at 5% significance level c. Explain the results you find Practice worksheet 80 80. Has the building and allocating Terminal 3 to Emirates Airlines resulted in better on-time performance? Before terminal 3 was built and allocated to Emirates Airlines, Emirates airline claimed that 92% of its flights were on time. A random sample of 165 flights completed this year reveals that 153 were on time. Can we conclude at the 5% significance level that the Emirates airline's on-time performance has improved after allocating terminal 3 only to Emirates airline? Practice worksheet 81 81. In the United States, voters who are neither Democrat nor Republican are called Independents. It is believed that 10% of all voters are Independents. A survey asked 25 people to identify themselves as Democrat, Republican, or Independent. a. What is the probability that none of the people are Independent? b. What is the probability that fewer than five people are Independent? Practice worksheet 82 82. After analyzing several months of sales data, the owner of an appliance store produced the following joint probability distribution of the number of refrigerators and stoves sold daily. Refrigerators Stoves 0 1 2 0 0.10 0.16 0.14 1 0.07 0.15 0.11 2 0.05 0.14 0.08 a. Find the marginal probability distribution of the number of refrigerators sold daily. b. Find the marginal probability distribution of the number of stoves sold daily. c. Compute the mean and variance of the number of stoves sold daily. Practice worksheet 83 83. Three messenger services deliver to Sharjah. Service A has 50% of all the scheduled deliveries, service B has 30%, and service C has the remaining 20%. Their on-time rates are 65%, 50%, and 45% respectively. Define event O as a service delivers a package on time. a. Calculate P (A and O). b. You are given that P (Ac and O) = 0.24. If a package was delivered on time, what is the probability that it was service A? Practice worksheet 84 84. A math tutor claims that the proportion of students who gets an A after just taking two sessions with him is 55%. What is the probability that in a random sample of 300 students who hires him as a tutor, less than 49% get A? If 49% of the sample actually got an A, what does this suggest about the tutor’s claim? Practice worksheet 85 85. The length of a volleyball game is normally distributed with a mean of 110 minutes and a standard deviation of 12.5 minutes. What is the probability that a randomly selected game is longer than 120 minutes? Practice worksheet 86 86. A uniformly distributed random variable has minimum and maximum values of 25 and 75, respectively. a. Draw the density function. b. Determine P (39 < X < 54) c. Draw the density function including the calculation of the probability in part (b). Practice worksheet 87 87. The following are the ages of a random sample of 8 employees in a bank: 50 66 21 33 31 57 45 47 It is known that the ages are normally distributed with a standard deviation of 10. Determine the 90% confidence interval estimate of the population mean. Interpret the interval estimate. Practice worksheet 88 88. Many hotels that serve the surfers make their forecasts of incomes on the assumption that the average surfer surfs four times per year. To examine the validity of this assumption, a random sample of 96 surfers is drawn and each is asked to report the number of times he or she surfed the previous year (the sample mean is 4.70). If we assume that the standard deviation is 3, can we infer at the 10% significance level that the assumption is wrong? Practice worksheet 89 89. Before the recent changes in the educational system, one private high school claimed that 91% of its graduates were placed in prestigious universities. A random sample of 495 students graduated this year reveals that 459 were placed in prestigious universities. Can we conclude at the 5% significance level that this high school's performance has improved? Practice worksheet 90 90. Use the graph below to answer the following questions. a. When you look at the scatter plot, you can see three numbers. Describe what each number represents (how do you name them) in this graph and how they are interpreted (what do they tell us). b. Write down the deterministic model for the above relationship between SUV sales and price of gas. c. Based on the graph above, if the price is ten, what would be SUV sales? Practice worksheet 91 91. After analyzing several months of sales data, the owner of an appliance store produced the following joint probability distribution of the number of refrigerators and stoves sold daily. Refrigerators Stoves 0 1 2 0 0.06 0.14 0.12 1 0.09 0.17 0.13 2 0.05 0.18 0.04 a. Find the marginal probability distribution of the number of refrigerators sold daily. b. Find the marginal probability distribution of the number of stoves sold daily c. Compute the mean and variance of the number of refrigerators sold daily. d. Compute the mean and variance of the number of stoves sold daily. e. Compute the covariance and the coefficient of correlation. Practice worksheet 92 92. Suppose X is a binomial random variable with n=25 and p=0.7. Use Table 1 to find the following a.P(X=18) b.P(X=15) c.P(X ≤20) d.P(X≥16) Practice worksheet 93 93. A student majoring in accounting is trying to decide on the number of firms to which he should apply. Given his work experience and grades, he can expect to receive a job offer from 70% of the firms to which he applies. The student decides to apply to only four firms. What is the probability that he receives no job offers? Practice worksheet 94 94. In the United States, voters who are neither Democrat nor Republican are called Independents. It is believed 20% of all voters are independents. A survey asked 25 people to identify themselves as Democrat, Republican, or Independent. a. What is the probability that none of the people are Independent? b. What is the probability that fewer than five people are Independent? c. What is the probability that more than two people are Independent? Practice worksheet 95 95. Define/discuss/interpret the following terms: population, sample, parameter, statistic, descriptive statistics(graphical and numeric), inferential statistics, random variable, data, interval data, nominal data, ordinal data, time series data, cross sectional data, variance, standard deviation, covariance, correlation coefficient, coefficient of determination. Practice worksheet 96 96. Create a sample of three numbers whose mean is 10 and standard deviation is 0 Practice worksheet 97 97. Michelin, a tire manufacturer, wants to advertise a mileage interval that excludes no more than 11% of the mileage on tires they sell. Suppose that the average tire mileage is 25000 and the standard deviation is 4000. a. What interval would you suggest for the advertisement? (Use Chebyshev's theorem). b. What would be the advertisement mileage interval that excludes no more than 5% of the mileage on tires they sell if it is known that the mileage follows the normal distribution? Practice worksheet 98 98. The U.S. mint produces dimes with an average diameter of 0.5 and a standard deviation of 0.01. a. Using Chebyshev's theorem, find a lower bound for the number of coins in a lot of 400 coins having diameter between 0.48 and 0.52. b. What would be the number of coins with a diameter between 0.48 and 0.52 if the diameter is normally distributed? Practice worksheet 99 99. The mean grade point average (gpa) for AUS students is 2.5 with a standard deviation of 0.5 a. If the histogram for gpa’s is approximately mounded, what percent of the gpa’s would you expect between 1.5 and 3.5? b. If the histogram for gpa’s is approximately mounded, what percent of the gpa’s would you expect greater than 3.5? c. If the histogram for gpa’s is NOT mounded, what percent of the gpa’s would you expect between 1.5 and 3.5? Practice worksheet 100 100. An investment of $1,000 you made 4 years ago was worth $1,200 after the first year, $1,200 after the second year, $1,500 after the third year, and $2,000 today. a. Compute the annual rates of return. b. Compute the average annual and median of the rates of return. c. Compute the overall (compound) return. d. Discuss which of the above is the best measure of the performance of the investment. Practice worksheet 101 101.In 2007 (the latest year reported) 134,543,000 tax returns were filed in the United States. The Internal Revenue Service (IRS) examined 0.5% of them to determine if they were correctly done. To determine how well the auditors are performing, a random sample of these returns was drawn and the additional tax was reported. The mean return from this sample is $11,000 with a standard deviation of $4,000. a. What is the population of interest? b. What is the sample size? c. What is the main idea to be tested? Practice worksheet 102 102. Are the marks one receives in a course related to the amount of time spent studying the subject? To analyze this mysterious possibility, a student took a random sample of 5 students who had enrolled in an accounting class last semester. He asked each to report his or her mark in the course and the total number of hours spent studying accounting. These data are listed here. a. Calculate the covariance. b. Calculate the coefficient of correlation. c. Calculate the coefficient of determination Practice worksheet 103 103. The number of males and females enrolled at AUS are listed per major in the table below. Use this table to answer questions a. If a student is chosen at random, what is the probability that the student is a female? b. If a student is chosen at random, what is the probability that the student is a male majoring in engineering? c. If one student is chosen to represent the student body, what are the odds in favor of selecting a female? d. If one student is chosen from economics major, which is more likely, selecting a male or selecting a female? Practice worksheet 104 104. Suppose that five good fuses and two defective ones have been mixed up. To find the defective fuses, we test them one-by-one, at random and without replacement. What is the probability that we are lucky and find both of the defective fuses in the first two tests? Practice worksheet 105 105. Let A and B be independent events with P(A) = and P(A ∪ B) = 2P(B) − P(A). Find a) P(B) b) P(A|B) c) P(B’|A) Practice worksheet 106 106. Your favorite football team is in the final playoffs of the UEFA Champions League. You have assigned a probability of 60% that it will win the championship. Past records indicate that when teams win the championship, they win the first game of the series 70% of the time. When they lose the series, they win the first game 25% of the time. The first game is over; your team has lost. What is the probability that it will win the championship? Practice worksheet 107 107. Assume that you have a new gun’s test results. A sample of 4 shots with the following deviations from the target is reported: 25cm, 8cm, 16cm and 19cm. a. Calculate the sample mean, variance, and standard deviation. b. Calculate the coefficient of variation. The old gun’s coefficient of variation was 0.48. Which gun produces more consistent shuts? Practice worksheet 108 108. Consider the following data: X 11 17 18 Y 3 5 11 a. Calculate the covariance for the sample. b. Calculate the coefficient of correlation given the fact that standard deviations for X and Y are 3.78 and 4.16 respectively. c. What does the coefficient of correlation tell you about the relationship between X and Y? Practice worksheet 109 109. A survey of a magazine's subscribers indicates that 40% own a house, 70% own a car, and 65% of the homeowners also own a car. What proportion of subscribers: a. own both a car and a house? b. own a car or a house, or both? c. own neither a car nor a house?