Statistics Solving - Yosemite National Forest - Math Problem Example

QUESTION 1 A study is under way in Yosemite National Forest to determine the adult height of American pine trees. Specifically, the study is attempting to determine what factors aid a tree in reaching heights greater than 60 feet tall. It is estimated that the forest contains 25,000 adult American pines. The study involves collecting heights from 250 randomly selected adult American pine trees and analyzing the results. Identify the population from which the study was sampled. the 250 randomly selected adult American pine trees V the 25,000 adult American pine trees in the forest all the adult American pine trees taller than 60 feet all American pine trees, of any age, in the forest QUESTION 2 TABLE 2-5  The following are the durations in minutes of a sample of long-distance phone calls made within the continental United States reported by one long-distance carrier. Time (in minutes) Relative Frequency 0 but less than 5 0.37 5 but less than 10 0.22 10 but less than 15 0.15 15 but less than 20 0.10 20 but less than 25 0.07 25 but less than 30 0.07 30 or more 0.02 Referring to Table 2-5, what is the cumulative relative frequency for the percentage of calls that lasted 10 minutes or more? 0.16 0.24 V 0.41 0.90 QUESTION 3 In a perfectly symmetrical distribution the range equals the interquartile range. the interquartile range equals the arithmetic mean. V the median equals the arithmetic mean. the variance equals the standard deviation. QUESTION 4 True or False: In a set of numerical data, the value for Q2 is always halfway between Q1 and Q3. True V False QUESTION 5 If we know that the length of time it takes a college student to find a parking spot in the library parking lot follows a normal distribution with a mean of 3.5 minutes and a standard deviation of 1 minute, find the point in the distribution in which 75.8% of the college students exceed when trying to find a parking spot in the library parking lot. V 2.8 minutes (mean – 0.7 st. dev.) 3.2 minutes 3.4 minutes 4.2 minutes QUESTION 6 True or False: A worker earns $15 per hour at a plant and is told that only 2.5% of all workers make a higher wage. If the wage is assumed to be normally distributed and the standard deviation of wage rates is $5 per hour, the average wage for the plant is $7.50 per hour. True V False – 2 st. dev. below $15 = $5 QUESTION 7 For sample sizes greater than 30, the sampling distribution of the mean will be approximately normally distributed regardless of the shape of the population. only if the shape of the population is symmetrical. only if the standard deviation of the samples are known. V only if the population is normally distributed. QUESTION 8 True or False: Suppose μ = 50 and σ2 = 100 for a population. In a sample where n = 100 is randomly taken, 95% of all possible sample means will fall between 48.04 and 51.96. V True False QUESTION 9 Suppose a 95% confidence interval for μ has been constructed. If it is decided to take a larger sample and to decrease the confidence level of the interval, then the resulting interval width would __________ . (Assume that the sample statistics gathered would not change very much for the new sample.) be larger than the current interval width V be narrower than the current interval width be the same as the current interval width be unknown until actual sample sizes and reliability levels were determined QUESTION 10 True or False: The t distribution approaches the standardized normal distribution when the number of degrees of freedom increases. V True False QUESTION 11 The owner of a local nightclub has recently surveyed a random sample of n = 250 customers of the club. She would now like to determine whether or not the mean age of her customers is over 30. If so, she plans to alter the entertainment to appeal to an older crowd. If not, no entertainment changes will be made. Suppose she found that the sample mean was 30.45 years and the sample standard deviation was 5 years. If she wants to be 99% confident in her decision, what conclusion can she make? V There is not sufficient evidence that the mean age of her customers is over 30. There is sufficient evidence that the mean age of her customers is over 30. There is not sufficient evidence that the mean age of her customers is not over 30. There is sufficient evidence that the mean age of her customers is not over 30. QUESTION 12 True of False: The larger is the p-value, the more likely one is to reject the null hypothesis. True V False QUESTION 13 The residuals represent the difference between the actual Y values and the mean of Y. V the difference between the actual Y values and the predicted Y values. the square root of the slope. the predicted value of Y for the average X value. QUESTION 14 The cyclical component of a time series represents periodic fluctuations which reoccur within 1 year. V represents periodic fluctuations which usually occur in 2 or more years. is obtained by adding up the seasonal indexes. is obtained by adjusting for calendar variation. QUESTION 15 A company that manufactures designer jeans is contemplating whether to increase its advertising budget by $1 million for next year. If the expanded advertising campaign is successful, the company expects sales to increase by $1.6 million next year. If the advertising campaign fails, the company expects sales to increase by only $400,000 next year. If the advertising budget is not increased, the company expects sales to increase by $200,000. Identify the actions in this decision-making problem. V Two choices: (1) increase the budget and (2) do not increase the budget. Two possibilities: (1) campaign is successful and (2) campaign is not successful. Four consequences resulting from the Increase/Do Not Increase and Successful/Not Successful combinations. The increase in sales dollars next year. The following information refers to the next question: The data below represents the amount of grams of carbohydrates in a sample serving of breakfast cereal. 10 18 24 30 19 22 24 20 18 25 20 22 19 QUESTION 16 The coefficient of variation for this data would be ____22.6%. Answer should be consistent with examples in your Levine et al. text and be between one or two decimal places e.g. 12.3 or 12.34 etc. Please do NOT include units (%) in your answer. 0.23 (or 22.6%) The following information refers to the next question: In a local cellular phone area, company A accounts for 70% of the cellular phone market, while company B accounts for the remaining 30% of the market. Of the cellular calls made with company A, 2% of the calls will have some sort of interference, while 3% of the cellular calls with company B will have interference. QUESTION 17 If a cellular call is selected at random, the probability that it will NOT have interference is __________? Answer should be between two and four decimal places e.g. 0.12, 0.123, 0.1234 etc. Please do NOT include any units in your answer. 0.977 (or 97.7%) The following information refers to the next two questions: The manager of a service station is in the process of analyzing the number of times car owners change the oil in their cars. She believes that the average motorist changes his or her car’s oil less frequently than recommended by the owner’s manual (two times per year). In a preliminary survey she asked 15 car owners how many times they changed their car’s oil in the last 12 months. The results are listed below. 0 1 2 0 3 2 0 1 0 1 2 2 1 2 1 QUESTION 18 The value of the test statistic in this problem is approximately equal to __________? Answer should be between two and four decimal places e.g. 1.23, 1.234, 1.2345 etc. Please do NOT include any units in your answer. -3.292 QUESTION 19 What would be your decision if a hypothesis test was conducted on this problem with the null hypothesis given as H0 : µ ≥ 2 and the alternate hypothesis given as H1 < 2? V Reject H0 at the 10%, 5% and 1% level of significance. Reject H0 at the 10% and 5% level of significance but do not reject H0 at the 1% level of significance. Reject H0 at the 10% level of significance but do not reject H0 at the 5% or 1% level of significance. Do not reject H0 at either the 10%, 5% or 1% level of significance. The following information refers to the next two questions: Refer to the following table which contains the sales (in $,000) for a department store for the first ten months of the year. Month Sales January 440 February 480 March 590 April 400 May 500 June 550 July 470 August 500 September 600 October 520 QUESTION 20 Using a three period moving average (i.e. MA(3)) as a forecasting method, what is the MAPE for this forecasting model __________% Answer should be to four decimal places e.g. if your answer was 12.34%, you would enter 0.1234 as your answer. Please do NOT include units (%) in your answer. 0.0827 QUESTION 21 Using simple exponential smoothing (with a smoothing constant of 0.2) as a forecasting method, what is the MAPE for this forecast model __________%? Answer should be to four decimal places e.g. if your answer was 12.34%, you would enter 0.1234 as your answer. Please do NOT include units (%) in your answer. 0.0796 The following information refers to the next question: The ordered array below resulted from taking a sample of 25 batches of 500 computer chips and determining how many in each batch were defective. Defects 1 2 4 4 5 5 6 7 9 10 12 12 14 17 20 21 23 23 24 26 27 27 28 29 29   QUESTION 22 If a frequency distribution for the defects data is constructed, using ‘0 but less than 5’ as the first class, what would be the relative frequency of the ‘10 but less than 15’ class __________%? Answer should be a percentage value to whole numbers e.g. 12, 23, 34 etc. Please do NOT include units (%) in your answer. 0.16 (16%) The following information refers to the next question: A manufacturer of power tools claims that the average amount of time required to assemble their top-of-the-line table saw is fifty (50) minutes with a standard deviation of forty (40) minutes (the very large standard deviation is due to a variety of factors including a large variation in skills amongst the ‘Do it yourself’ home handyman which is traditionally one of the companies customer base). Suppose a random sample of 64 purchasers of this table saw is taken. QUESTION 23 What is the probability that the sample mean will be less than 46 minutes __________? Answer should be to four decimal places which is consistent with the number of decimal places listed in your appendix tables e.g. 0.1234 etc. Please do NOT include any units in your answer. 0.2119 The following information refers to the next question: A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine. The desired length of the insulation is 12 metres. It is known that the standard deviation in the cutting length is 0.15 metres. A sample of 144 cut sheets yield a mean length of 12.14 metres. This sample will be used to obtain a 90% confidence interval for the mean length cut by machine. QUESTION 24 What are the two limits of the confidence interval __________ and __________? Answer should be to four decimal places e.g. 1.2345 and 2.3456. Please do NOT include any units in your answer. min = 12.1194 max = 12.1606 The following information refers to the next four questions: An insurance company wishes to examine the relationship between income (in $,000) and the amount of life insurance (in $,000) held by families. The company drew a simple random sample of families and obtained the following results: Family Income Amount of life insurance A 60 120 B 80 200 C 100 220 D 80 160 E 90 180 F 120 270 G 110 150 H 100 240 I 70 160 J 100 210 QUESTION 25 What is the least squares estimate of the slope? Answer should be to four decimal places e.g. 1.2345. Please do NOT include any units in your answer. 1.7767 QUESTION 26 What is the least squares estimate of the Y intercept? Answer should be to four decimal places e.g. 1.2345. Please do NOT include any units in your answer. 29.3204 QUESTION 27 What is the prediction for the amount of life insurance for a family whose income is $85,000? Answer should be to four decimal places and be consistent with your original data set e.g. if your answer was $75,410.90 you would enter 75.4109 as your answer. Please do NOT include any units in your answer. 180,339.81 QUESTION 28 What would be the residual (error) term for a family income of $90,000? Answer should be to four decimal places and be consistent with your original data set e.g. if your answer was $940.90 you would enter 0.9409, if your answer was $9,400.90 you would enter 9.4009 etc. Please do NOT include any units in your answer. -9.2233 The next six questions refer to the following information: International Pictures is trying to decide how to distribute its new movie ‘Claws’. ‘Claws’ is the story of an animal husbandry experiment at the University of Southern Queensland that goes astray, with tragic results. An effort to breed meatier chickens somehow produces an intelligent, 200 kilogram chicken that escapes from the lab and terrorises the campus. In a surprise ending the chicken is befriended by coach Tim Galvano, who teaches it how to play Rugby and help his team win State, National and World Championships. Because of the movie’s controversial nature, it has the potential to be either a smash hit, a modest success, or a total bomb. International is trying to decide whether to release the picture for general distribution initially or to start out with a ‘limited first-run release’ at a few selected theatres, followed by general distribution after 3 months. The company has estimated the following probabilities and conditional profits for ‘Claws’:     PROFITS (Millions of $) Level of success Probability Limited release General distribution Smash .3 22 12 Modest .4 9 8 Bomb .3 –10 –2 International can run sneak previews of ‘Claws’ to get a better idea of the movies’ ultimate level of success. Preview audiences rate movies as either good or excellent. On the basis of past experiences, it was found that 90% of all smash successes were rated excellent (and 10% rated good), 75% of all modest successes were rated excellent (25% rated good) and 40% of all bombs were rated excellent (60% rated good). The cost of running sneak previews is not cheap. Currently, this stands at $1m. QUESTION 29 What is the opportunity loss for a General Distribution for a Smash level of success? Answer should be to whole numbers only. You do not need to put any units in your answer. 10 QUESTION 30 What would the optimal action be for International before running the sneak preview? V Run a limited release with an expected payoff of $7.20m Run a limited release with an expected payoff of $6.20m Run a general distribution with an expected payoff of $7.20m Run a general distribution with an expected payoff of $6.20m QUESTION 31 What is the maximum amount of money that International would be prepared to pay for an absolutely reliable forecast of the movies’ level of success? $9.6m $7.2m $6.2m V $2.4m QUESTION 32 What would be the joint probability for a ‘smash success’ and excellent preview given that in the past, it was found that 90% of all smash successes were rated excellent? Answer should be to two decimal places e.g. 0.12, 0.23, etc. Please do NOT include any units in your answer. 0.27 QUESTION 33 What is the posterior probability of a bomb given the sneak preview indicates excellent? Answer should be to four decimal places e.g. 0.1234, 0.2345 etc. Please do NOT include any units in your answer. 0.1739 QUESTION 34 What is the maximum amount that should be paid for the sneak preview (i.e. what would be the expected value of sample information (EVSI))? Select the closest correct answer. Use four decimal places in your calculations but you can round off the final answer to 2 decimal places.) V $1.04 million $2.58 million $7.2 million $8.24 million The next six questions refer to the following information: (All calculations should be to at least three decimal places) The tourist industry is subject to enormous seasonal variation. A hotel in North Queensland has recorded its occupancy rate for each quarter during the past 5 years. These data are shown in the accompanying table. Table 1: Occupancy rate   Year   2004 2005 2006 2007 2008 Quarter 1 0.561 0.575 0.594 0.622 0.665 Quarter 2 0.702 0.738 0.738 0.708 0.835 Quarter 3 0.800 0.868 0.729 0.806 0.873 Quarter 4 0.568 0.605 0.600 0.632 0.670 QUESTION 35 What is the centred moving average that would correspond to Quarter 2 in 2006? Answer should be consistent with the data provided and be to three decimal places e.g. 0.123, 0.456 etc. Please do NOT include any units in your answer. 0.653 QUESTION 36 What is the adjusted seasonal index for Quarter 2 __________% Answer should be listed to three decimal places in the form 0.123 i.e. 0.123 represents 12.3%, 1.234 represents 123.4% etc. Please do NOT put units (%) in your answer. 1.077 QUESTION 37 The trend line for this decomposition model has been calculated to be (at 3 decimal places):  Y = 0.650 + 0.004 T where T represents time. What is the coefficient of determination (R2) for this trend line? (Select the closest correct answer.) V 0.0932 (9.32%) 0.3448 (34.48%) 0.4554 (45.54%) 0.7882 (78.82%) QUESTION 38 What would be the forecast in Quarter 2, 2009 using the trend line previously given (i.e. Y = 0.650 + 0.004 T) and the relevant adjusted seasonal index? Answer should be consistent with the data provided and be to three decimal places e.g. 0.123, 0.456 etc. Please do NOT include any units in your answer. 0.791 QUESTION 39 If we exponentially smooth the data in Table 1 with a smoothing constant of 0.1, the smoothed value for Quarter 3 in 2004 would be? Answer should be consistent with the data provided and be to three decimal places e.g. 0.123, 0.456 etc. Please do NOT include any units in your answer. 0.575 QUESTION 40 If we exponentially smooth the data in Table 1 with a smoothing constant of 0.1, the forecast for Quarter 1 2009 would be? Answer should be consistent with the data provided and be to three decimal places e.g. 0.123, 0.456 etc. Please do NOT include any units in your answer. 0.695 Read More