MTH3003 TUTORIAL QUESTIONS FOR UPM STUDENTS JULY 2010/2011(PLEASE TRY ALL YOUR BEST TO ANSWER ALL QUESTIONS)

Preparation tutorial notes for Statistics of Applied Sciences by Dr. Mohd Bakri Adam for MTH3003 students at Mathematics Department, UPM, Semester July 2010/2011. Copyright c 2010 by Mohd Bakri Adam

Contents0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.10 0.11 Tutorial 1 . . . . . Tutorial 2 . . . . . Tutorial 3 . . . . . Tutorial 4 . . . . . Tutorial 5 . . . . . Tutorial 6 . . . . . Tutorial 7 . . . . . Tutorial 8 . . . . . Tutorial 9 . . . . . Tutorial 10 . . . . . Binomial Table and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard Normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 4 7 9 12 14 16 19 20 24

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0.1

Tutorial 1

1. In a recent survey, 1000 adults in Malaysia were asked if they read news on the internet at least once a week. Six hundred of the adults said yes. Identify the population and the sample. Describe the data set. 2. Decide wether the numerical value describes a population parameter or a sample statistic. Explain your reasoning. (a) A survey of a sample of Master of Applied Statistics reported that the average starting salary for Master of Applied Statistics is less than RM3000.00. (b) Starting salaries for the 667 Master of Applied Statistics graduates from UPM increased 8.5% from the previous year. (c) In a random check of a sample of retail stores, the health and consumer ministery found that 34% of the stores were not storing sh a the proper temperature. 3. What is a random variable? Give an example of discrete random variable and a continuous random variable. Justify your answer. 4. Decide which part of the study represent the descriptive branch of statistics. What conclusions might be drawn from the study using inferential statistics? (a) A large sample of women, aged 50, was studied for 18 years. For unmarried women, approximately 70% were alive at the age 65. For married women, 90% were alive at the age 65. (b) In a sample of BSKL analysts, the percentage who incorrectly forecasted hightech earnings in a recent year was 45%.

0.2

Tutorial 2

Table 1: The number of minutes 50 students spent on reading during their leasure time

50 19 72 46 36

40 23 56 31 39

41 37 17 39 30

17 11 7 22 44 28 21 51 54 42 88 41 78 56 7 69 30 80 56 29 33 20 18 29 34 59 73 77 62 54 67 39 31 53 44

1. From Table 1 construct a frequency distribution using Sturges method. Where it used the following formula to estimate the number of the classes, Class Numbers, k = 1 + 3.322 log10 (n) 1

where n is the sample size and k is the round up value from the formula. Use the smallest value in the data as a lower class limit. 2. Construct a frequency distribution that has ve classes from Table 1, include additional features such as midpoint, relative frequency, cumulative frequency, cumulative relative frequency. 3. From question 1 construct a histogram, polygon and ogive. 4. Calculate mean, mode and median of the data from Table 1. 5. Calculate mean, mode and median from frequency table in question 2. Hint: Use formula in Question 7. It will be tested in the examination. 6. Calculate P30 , P45 , P80 for the data in Table 1. Class 7-18 19-30 31-42 43-54 55-66 67-78 79-90 Frequency, Class f Boundaries 6 6.5-18.5 10 18.5-30.5 13 30.5-42.5 8 42.5-54.5 5 54.5-66.5 6 66.5-78.5 2 78.5-90.5

Table 2: Frequency distribution for reading times (in minutes) with boundaries. 7. Calculate P30 , P45 , P80 for the data in frequency table in Table 2. Used the following formula [n(0.p) CF BP C]w Pp = LCBp + fp where LCBp is the lower class boundary which equavalance to the cumulative frequency class equal to 0.p or the nearest value greater than 0.p. CF BP C is the cumulative frequency before percentile class. w is width of the percentile class. fp is a frequency in the percentile class. As a Pp for a grouped calculation is not given in the notes, you need to go to the library of UPM to look for the references. It will be tested in the examination. 8. Calculate a variance for Table 3. As a variance for a grouped calculation is not given in the notes, you need to go to the library of UPM to look for the references. It will be tested in the examination. 2

Table 3: Frequency distribution for reading times (in minutes) with midpoint, rf and cf . Class 7-18 19-30 31-42 43-54 55-66 67-78 79-90 Frequency, f 6 10 13 8 5 6 2 f = 50 Midpoint 12.5 24.5 36.5 48.5 60.5 72.5 84.5 Relative Cumulative frequency frequency 0.12 6 0.20 16 0.26 29 0.16 37 0.10 42 0.12 48 0.04 50 f /n = 1

9. Construct stem-and-leaf for the data from Table 1. 10. Construct box-and-whiskers plot for the data from Table 1 11. Suppose the age of a sample of 10 students are: 20.9, 18.1, 18.5, 21.3, 19.4, 25.3, 22.0, 23.1, 23.9, and 22.5 (a) Calculate the mean, mode and median. (b) Calculate the range, variance, coecient of variation and standard deviation. (c) Find x1 , x3 and x(3) values. (d) Construct a stem-and-leaf. (e) Calculate P40 , D1 and D7 . (f) Construct the box-and-whiskers plot. What is you comment on the plot. 12. What is the statistic(s) that you use in the following scenario (a) You are working in the supermarket and you want to advise you manager in purchasing a new stocks of shampoo. (b) You want to conclude about a performance of MTH3003 in your class. (c) You were asked by your neigbour about planting manggo tree in your backyard. (d) You were asked to choose one bag from 4 dierent bags. (e) Youre now a lecturer for MTH3003, the top management wants to imposed new system of grading i.e. Gred A is given only to ten percents of all students from the highest mark and gred F is given only to the last ve percent of students from the lowest marks. (f) You want to by a new bed and a new furniture for your new rental room in Sri Serdang. 3

(g) You have asked your senior regarding his study times at University. (h) You want to report the everage times of students using the library facilities in a year. 13. The following unordered stem-and-leaf display represents the number of hours spent by 30 MTH3003 students working on assignments during the past month.(1|0 represents 1 units) 1 2 3 4 5 6 7 | | | | | | | 0 6 2 5 9 5 8 7 2 7 1 9 4 5

0 4 0 1 4

8 2 2 2

9 3

6 6

6

4

8

(a) Construct a frequency table and histogram for the about table with a number of classes is 7. (b) Construct an Ogive. (c) Calculate the mean, mode, variance and coecient of variation for the grouped data in (a). How is the dispersion of the data? (d) Calculate the median for the grouped data. 14. For a given data as follows 0 4 3 2 8 9 7 6 6 0

(a) Calculate the sample mean, mode and median (b) Calculate the sample variance, standard deviation and the coecient of variation. (c) Construct a box-and-whiskers plot. What you comment on the skewedness of the data.

0.3

Tutorial 3

1. How many ways to arrange 6 person in a round table. 2. How many ways to arrange 2 pairs of twin in a round table. 3. Identify the sample space of the probability experiment 4

(a) Tossing a coin. (b) Answering a true and false question. (c) Tossing four coins and recording the number of heads. (d) Answering a multiple choice question with A, B, C, and D as the possible answers. (e) Determining the childrens gender for a family of three children. (f) Rolling a single 12-sided die with sides numbered 1-12. (g) A calculator has a function button to generate a random integer from -5 to 5. 4. A coin is tossed. Find the probability that the result is heads. Answer:0.5 5. A single six-sided die is rolled. Find the probability of rolling an even number. Answer:0.5 6. If an individual is selected at random, what is the probability that he or she has a 31 birthday in July? Ignore leap years. Answer: 365 7. A question has ve multiple-choice questions. Find the probability of guessing the correct answer. Answer:0.2 8. Find the probability of selecting two consecutive threes when two cards are drawn without replacement from a standard deck of 52 playing cards. Round your answer to four decimal places. Answer: 0.0045 9. A multiple-choice test has ve questions, each with ve choices for the answer. Only one of the choices is correct. You randomly guess the answer to each question. (a) What is the probability that you answer the rst two questions correctly? Answer:0.04 (b) What is the probability that you answer all ve questions correctly? Answer:0.00032 (c) What is the probability that you do not answer any of the questions correctly? Answer:0.32768 (d) What is the probability that you answer at least one of the questions correctly? Answer:0.67232 10. The events A and B are mutually exclusive. If Pr(A) = 0.3 and Pr(B) = 0.2, what is Pr(A B). Answer:0.51 11. Given that Pr(A B) = 1 , Pr(A) = 1 , and Pr(A B) = 5 , nd Pr(B). Answer: 17 3 4 60

5

12. A tourist in Malaysia wants to visit six dierent cities i.e. Kuala Lumpur, Penang, Shah Alam, Johor Bharu, Seremban and Kota Bharu. (a) How many dierent routes are possible? Answer:720 (b) If the route is randomly selected, what is the probability that the tourist will visit the cities in alphabetical order? Answer: 0.001 13. Four best friends of MTH3003 are invited for a dinner by their lecturer. How many ways can they be seated at a dinner table if the table is straight with seats only on one side. Answer:24 14. Yesterday ve best friends of MTH3003 have been invited for a dinner by their lecturer. How many ways can they be seated at a dinner table if the table is straight with seats on only one side. Answer:120 15. How many ways can jury of six men and six women be selected from twelve men and ten women? Answer: 194,040 16. How many dierent permutations of the letters in the word PROBABILITY are there? Answer: 9,979,200 17. The acces code to a houses security system consist of ve digits. (a) How many dierent codes are available if each digit can be repeated? Answer:100,000 (b) How many dierent codes are availbe if the rst digit cannot be zero and the arrangement of ve ves is excluded? Answer:89,999 18. In Malaysia, each automobile license plate consists of a single digit followed by three letters, followed by three digits. (a) How many distinct license plates can be formed if there are no restrictions on the digits or letters? Answer:175,760,000 (b) How many distinct license plates can be formed if the rst number cannot be zero and the three letters cannot form "GOD"? Answer: 158,175,000 19. An experiment involves tossing two dice. Find the following events (a) List the sample space. (b) The probability that sum is at least 3. (c) The probability that both dice show odd numbers. 6

20. Urn A contains three red balls and seven blue balls. Urn B contains eight red balls and four blue balls. Urn C contains ve red balls and eleven blue balls. An urn is chosen, with each urn equally likely to be chosen, and then a ball is chosen randomly from that selected urn. Calculate the probabilities (a) A red ball is chosen. (b) A blue ball is chosen. (c) A red ball from urn B is chosen.

0.4

Tutorial 4

1. Three objects were selected by random from a container. Each object taken was checked and categorised either good or bad. If X represents the good one, what is the probable values of X? 2. State whether the variable is discrete of continuous (a) The number of cups of coee sold in a cafeteria during lunch. (b) The height of a player on a basketball team. (c) The cost of a statistics book. (d) The blood pressures of a group of students the day before their nal exam. (e) The temperature in degrees Fahrenheit on July 4th in Kuala Lumpur. (f) The number of goal scored in a soccer game. (g) The speed of a car on a Seremban-KL highway during rush hour trac. (h) The number of phone calls to the attendance oce of a high school on any given school day. (i) The age of the oldest student in MTH3003 class. 3. Determine whether the distribution represents a probability distribution. If not, identify the requirement that is not satised. (a) x Pr(x) x Pr(x) x Pr(x) 1 0.2 3 -0.3 1 1.2 2 0.2 6 0.5 2 1.2 3 0.2 9 0.1 3 1.4 4 0.2 12 0.3 4 1.1 5 0.2 15 0.4 5 1.1 7

(b)

(c)

4. Construct a distribution for the formula and determine whether it is a probability distribution. (a) Pr(x) = x , x = 1, 2, 3. 6 (b) Pr(x) = x , x = 3, 4, 7. 6 (c) Pr(x) =x ,x x+2

= 0, 1, 2.

5. The random variable X represents the number of boys in a family of three children. Assuming that boys and girls are equally likely. (a) Construct a probability distribution. (b) Graph the distribution. (c) Find the mean and standard deviation for the random variable X. Answers: 1.50,0.87 6. A twenty-ve-years-old man decides to pay RM325 for a one-year insurance policy with coverage for RM1,000,000. The probability of him living through the year is 0.9995. What is his expected value for the insurance policy? Answer:RM175.16 7. One thousand tickets are sold at RM1 each. One ticket will be randomly selected and the winner will receive a color television valued at RM350. What is the expected value if a person buys one tickets? Answer:-RM0.65 8. If a person rolls doubles when tossing two dice, the roller prots RM5. If the game is fair, how much should the person pay to play the game? Answer:RM1 9. Show that f (x) is pdf. x , for 0 < x < 2 f (x) = 2 0, otherwise . 10. Find the k value if k , for 0 < x < 4 2 f (x) = x 0, otherwise .

is a density probability function. 11. As f (x) is pdf as follow 3 x, for 0 < x < 1 f (x) = 2 0, otherwise .

(a) Find F (x) function 8

(b) Find Pr(0.2 < x < 0.3) (c) Find F (0.3) (d) Find Pr(x > 0.3) (e) Find the expected value. (f) Find the variance, V ar(x). 12. Find the k value if (x2 + 3)k, for 2 < x < 3 f (x) = 0, otherwise . is a density probability function. Then answers the following questions (a) Find Pr(1 < x 2) (b) Find Pr(x > 2) (c) Find the expected value. (d) Find the variance.

0.5

Tutorial 5

1. Decide whether the experiment is a binomial experiment. If it is not, explain why. (a) You observe the gender of the next 100 babies born at a local hospital. The random variable represents the number of girls. (b) You roll a die 100 times. The random variable represents the number that appears on each roll of the die. (c) You spin a number wheel that has 10 numbers 100 times. The random variable represents the winning numbers on each spin of the wheel. (d) Surveying 100 prisoner to see whether they are serving time for their rst oense. The random variable represents the number of prisoners serving time for their rst oense. (e) Selecting ve cards, one at a time without replacement from a standard deck of cards. The random variable is the number of red cards. 2. Assume that male and female births are equally likely and that the birth of any child does not aect the probability of the gender of any other children. Find the probability of at most three boys in ten births. Answer:0.172 9

3. A test consist of 10 true or false questions. To pass the test a student must answer at least eight questions correctly. If the student guesses on each question, what is the probability that the student will pass the test? Answer:0.055 4. A test of consist of 10 true of false questions. If the student guesses on each question, what is the mean and standard deviation of the number of correct answers?5;1.58 5. A company ships computer components in boxes that contain 20 items. Assume that the probability of a defective computer component is 0.2. Find the probability that the rst defect is found in the seventh component tested. Round your answer to four decimal places. Answer:0.0524 6. A sales rm recieves an average of three calls per hour on its toll-free number. For any given hour, nd the probability that it will receive exactly three calls and nd the probability that it will receive at least three calls. Answers:0.2240;0.5768 7. A local re station recieves an average of 0.55 rescue calls per day. Find the probability that on a randomly selected day, the re station will receive fewer than two calls. Answer:0.894 8. Given that a marksman can hit a target on a single trial with probability equal to 0.8. Suppose he res 4 shots at the target. What is the probability that he will hit the target at least once. Binomial,0.1536 9. The mean number of bacteria per ml of liquid is 2. By assuming that the number of bacteria follows a Poisson distribution, nd the probability that in 1 ml of liquid there are 3 bacteria. 0.1804 10. A factory manufactures computer and 100 computers are packed in a box. The probability that a computer is defective is 0.005. Find the probability that there are 2 defective computers in a box. Approximation Poisson, 0.0758 11. The time taken by Aminah to distribute Nasi Lemak everyday to residents staying at Fifth College follows a normal distribution with mean 32 minutes and variance 6.25 minutes2 . Find the probability that on Wednesday, Aminah takes more than 40 minutes.Normal,0.000687 12. A regular tetrahedral shaped die with its fares labelled 1,2,3 and 4 is tossed 200 times. Find the probability of obtaining more than 60 times the digit 4. Approximation Normal,0.0432 13. The waiting time at a clinic MTH3003 has an exponential distribution with an average waiting time of 5 minutes. What is the probability that you have to wait 10 minutes or more at the counter. 0.135 10

14. When a blue bulb is selected at random from a box that contains a 40 watt bulb, a 60 watt bulb, a 75 watt bulb, and a 100 watt bulb. Each elements of the sample space S = {40, 60, 75, 100} occurs ith the probability 1/4. Derived the distribution involved. 15. The complexity of arrivals and departures into an airport are such that computer simulation is often used to model the ideal condition. For a certain airport containing three runways it is known that in the ideal setting the following are the probabilities that the individual runaways are access by a randomly arriving commercial aircraft. Runaway 1: P1 = 2/9 Runaway 2: P2 = 1/6 Runaway 3: P3 = 11/18 What is the probability that 6 randomly arriving aircrafts are distributed in the following pattern. Runaway 1: 2 aircarfts Runaway 2: 1 aircraft Runaway 3: 3 aircraft Multinomial,0.1127 16. We have 100 items of which 12 are defectives. What is the probability that in a sample of 10, 3 are defectives? Hypergeometry,0.08 17. In MTH3003 soccer league series, the team who wins 4 games out of 7 will be the winner. Suppose the team A has probability 0.55 of winning over team B and both teams A and B faced each other in the league series. What is the probability that team A will win the league series in six games? Negative binomial,0.1853 18. In a certain manufacturing process it is know that, on the average, one in every 100 items is defective. What is the probability that the fth item inspected is the rst defective item. Geometric,0.0096 19. During the laboratory experiment the average number of radioactive particle passing through a counter in one millisecond is 4. What is the probability that 6 particles enter the counter in a given millisecond. Poisson,0.1042 20. Suppose we select 5 cards from an ordinary deck of playing cards. What is the probability of obtaining 2 or fewer hearts? Hypergeometry, 0.9072 11

21. (a) A test bank consists of 1000 objective questions. Any student may randomly select any 10 questions to work on. Each question has 5 choices (A, B, C, D, E) and there is exactly one correct answer to a question. Let X be the number of correctly answered questions for a particular student. Calculate Pr(X > 4). (b) A bowl contains 5 blues balls and 6 red balls of the same size. Four balls are selected at random from the bowl. Let Y be the number of blue balls in the sample selected. Find Pr(Y 3) and calculate E(Y ) and V ar(Y ). (c) An external examiner realized that it is usual to observe, on the average, ve typing errors on every page of a thesis submitted for the rst time. if X represents the variable mentioned, calculate Pr(3 < X 6). 22. (a) The weights of newly born baby girls in Hospital Serdang may be assumed to be normally distributed with mean 3.5 kg and standard deviation of 1.2 kg. If a baby girl born at that hospital is randomly selected, what is the probability that its weight is within the range from 2.0 to 4.0 kg? 23. Suppose that a baby girl born at the hospital stated in the previous question. (a) will be incubated if its weight does not exceed 2.5 kg. Previous records indicated that 3 out of 10 baby girls born at that hospital needed incubation. If the records of 100 baby girls born at that hospital are selected at random, approximate the probability that the number of baby girls who had undergone incubation is between 24 and 32. (b) may have congenitials defects. Previous records indicated that only 1% of all baby girls born at that hospital had congenital defects. One hundred records of babies born at that hospital are selected at random, approximate the probability that more than 2 had congenital defects.

0.6

Tutorial 6

1. The weight of the packages acceptance by one big hypermarket is normally distributed with mean 30 kg and the standard deviation is 5 kg. What is the probability that 25 packages which accepted randomly will exceed the maximum limit of 820 kg, for the lift carrier in the building of the hypermarket? Answer:0.0026 2. In one of random sample of size 25 from normally distributed with mean 98.6 and standard devation 17.2. Answers the following questions. (a) Pr(92 < X < 102) (b) Find the probability of the previous question if n now is 36. 12

3. Given X N(5, 36) (a) Find E(X) and V (X) if n = 20. (b) Find Pr(X > 7) if n = 6. 4. The mean for an executive paying for his lunch is RM 6.50 with standard deviation is RM 6.00. If 36 executives are selected at random from the rm (i.e. where the executive working), nd the probability that the mean money spent are between RM 5.00 and RM 10.00. 5. Mr Ali, is an auditor in a credit card company, knows that in average a client have the balance in the account is RM 112 and standard deviation is RM 56. If he audited 50 accounts which he selected by random, what is the probability that the average balance in a month is below RM 100. Answer:0.0643 6. The height for 1000 trees is approximating normal distribution with mean 174.5 cm and standard deviation is 6.9 cm. If 200 sample which each is n = 25 where taken from the population and the mean where recorded (a) Find the mean and standard deviation for X. (b) The sample size, when mean is between 172.5 and 175.5 cm. Answer:151 trees (c) The sample size if th emean is below 172.0. Answer:8 trees 7. The balances of all current accounts at a local bank belong to one of former MTH3003 have a distribution that is skewed to the right with its mean equal to RM12,450 and standard deviation equal to RM4,300. Find the probability that the mean balance of a sample of 50 current accounts selected from this bank will be (a) more than RM11,500. (b) between RM12,000 and RM13,800. (c) within RM1,500 of the population mean. (d) more than the population mean by at least RM1,000. 8. Kuala Lumput city council is planning to build a gigantic solar-power plant. A local newspaper found out that 53% of the voters in the city favour the construction of this plant. Assume that this result holds true for the population of all voters in this city. (a) What is the probability that more than 50% of the voters in a random sample of 200 voters selected from this city will favour the construction of this plant? 13

(b) A politician would like to take a random sample of voters in which more than 50% would favour the plant construction. How large a sample would be selected so that the politician is 95% sure of this outcome. Notes: Tutor(s) should shows carefully to students how to read all the table given in the lectures-room.

0.7

Tutorial 7

1. Given that the margin error is 15 seconds from true mean with 95% condent. Calculate the real expected time if it is normally distributed with standard deviation is 40 seconds. 2. From 100 laundry shops, we observed that 64 from the shops using Brand As detergent. (a) Find the 90% condence interval for proportion of the shop using Brand As detergent. (b) Find the 95% condence interval for proportion of the shop using Brand As detergent. Find the 99% condence interval for proportion of the shop using Brand As detergent. 3. The mean and standard deviation for the achievement of randomly selected 25 graduates is 15 and 0.3 respectively. Find the 95% condence interval for the true mean achievement. 4. An automatic machine that can produce juice products with its volume followed a normal distribution with the standard deviation 15 ml. What is the 95% condence interval if n = 25 and x = 225. 5. A botanist wants to estimate the minima number of the trees in a jungle near to Malaysia National Park. How many of the trees per acre should be checked if he wants 95% condent that the dierences between sample and population means is within 3 trees per acre. From previous experience 2 = 12 trees per acre. 6. In a group of 500 male adults who drink milk, 86 of them like Brand As milk. Find the 90% condence interval for the proportion of adults like to drink Brand As milk. 7. The following data represent the time of previewing the movies by 2 good manner lm companies. 14

Company A Company B

102 86 98 109 92 81 165 97 134 92

Assumed that both population are normally distributed. Get the 95% condence interval for dierence of the mean of the time of casting the movies by the 2 lm companies. We found earlier that the variance for both preview times are similar. 8. The data showed the wear resistance measurement for two type of tyres which put on each side of fronts wheels. Tyre on the right wheel, A Tyre on the left wheel, B 8.8 8.3 9.7 9.1 9.8 9.4 10.6 10.2 12.3 11.8

Find the 95% condence interval for the mean dierence of wear between tyre A and Tyre B? 9. According to the following information. Type A 34.8 34.5 35.0 34.6 34.2 34.6 34.9 Type B 33.8 35.0 33.5 33.3 34.5 33.1 35.4 33.9 Find the 95% condence interval for dierence of the mean if (a) The population variances are similar. (b) The population variances are assumed to be dierent.2 2 (c) We know that A = B = 0.36. 2 2 (d) We know that A = 0.49 and B = 0.36.

33.9

(a) A religous scholar from one good university conducted a survey of 400 of its students and found that the average amount of time spent prayer to GOD by their student was 12.5 days per month with a standard deviation of 5.4 days. i. Estimate the true mean amount of time spent prayer to GOD by their student in the university and nd the margin of error of your estimate. ii. Construct 99% condence interval for . (b) It is interest to know if the average time it takes Ambulance A to reach the scene of an accident diers from that of an Ambulance B to reach the same accident. The summary data is listed below Ambulance A nA = 60 xA = 4.2 s2 = 0.08 A 15 Ambulance B nB = 55 xB = 4.5 s2 = 0.10 B

Estimate the dierence in times (in minutes) between Ambulance A and Ambulance B using 95% condence interval. Do the times dier? Why? (c) In UPM, 500 open-minded and deligent students where asked if they would use public transportation if a new system was implemented. From 500 students, 420 of them say "yes" A proposal for the new public transportation system will be submitted to the authority if it can attract at least 80% of the retired people to use public transport. i. Estimate p the true proportion of retired people that will use public transport and nd the margin of error of your estimate. ii. Construct the 90% condence interval for p. Should the proposal be submitted or not? Why? (d) A good and generous of an aircraft CEO company wants to estimate the late arrival rate for ights of his company. How many ights must he included in a simple random sample if he wants to be 95% condent that the true population proportion of ights that arrive late lies within 0.1 of his sample proportion?

0.8

Tutorial 8

1. The manufacturer of the X-15 steel belted radial truck tyre claims that the mean mileage the tyre can be driven before the tread wears out is 60,000 miles. The standard deviation of the mileages is known to be 5,000 miles. The SimeX Truck Company bought 48 tyres and found that the mean mileage for their trucks is 59,500 miles. (a) Is SimeXs experience dierent from that claimed by the manufacturer at the 0.05 signicance level? (b) What is the p value for the test in (a)? (Modied from Mason, Lind and Marchal, Statistical Techniques in Business and Economics) 2. A new halal food outlet claims that the average time between a customer entering the outlet and the customer being served is no more than 5 minutes. The standard deviation of times to service is known to be 1 minute. A simple random sample of 25 customers had a mean service time of 5 minutes and 30 seconds. (a) Does this sample reject the claim made by the outlet? Use a 5% test. (b) What is the p value for the test in (a)? (c) What assumptions must be made for this test to be valid? Do you think these assumptions will be satised here? 16

3. In a test of the hypotheses: H0 : 20 the reported p value was 0.04. (a) Is H0 rejected at the 5% level? (b) Is H0 rejected at the 1% level? (c) A researcher wants to test the hypotheses H0 : = 20 at the 5% level. Is H0 rejected? 4. A greedy union ocial has claimed that the weekly wages paid to domestic workers in Serdang are normally distributed with a mean of RM40 and a standard deviation of RM8. A good economist accepts that weekly wages are normally distributed and that the standard deviation is RM8 but believes that the quoted gure for the mean is too low. To test the gure for the mean, the good economist plans to take a random sample of 16 domestic workers and to calculate their mean wage. The good economist then proposes to reject the claim made by the greedy union ocial if this calculated sample mean wage is RM42 or more. If the sample mean wage is less than RM42, the good economist will keep quiet and not reject the greedy union ocials claim. (a) What are the null and alternative hypotheses to be tested here? (b) If the union ocials claim is in fact true, what is the probability that the economist will falsely reject the claim. What type of error is this? (c) If in fact the population mean wage is RM45 (and so the quoted union gure for the mean wage is indeed too low), what is the probability that the good economist will not reject the claim? What type of error is this? (d) What decision rule should the good economist adopt if he wants to have only a 5% chance of rejecting the union ocials claim when it is true? Specify this decision rule in the same way as above, i.e. as a range of sample means for which the claim will be rejected. For this decision rule, what is the probability that the economist will not reject the union ocials claim when in fact the population mean wage is RM45? (e) The good economist would prefer to have a decision rule such that the probability of not rejecting the greedy union ocials claim when it is true will be 5% and the probability of not rejecting the claim when the population mean 17 versus HA : = 20 versus HA : < 20

is RM45 will be 10%. How large a sample would this require and what would the decision rule be? 5. If 10% of MTH3003 students are vegetarians, test the hypothesis that students who gamble are less likely to be vegetarians. If the 120 students polled, 10 claimed to be a vegetarian. 6. For randomly selected adults IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. A sample of 24 randomly selected college professors resulted in IQ scores having a standard deviation of 10. Test the claim that the IQ scores for college professors is the same as the general population, that is 15. Use a 0.05 level of signicance. Answer: Reject the null hypothesis 7. A medical researcher wishes to see whether the variances of the heart rates (in beats per minutes) of smokers are dierent from the variances of heart rates of people who do not smoke. Two samples are selected, and the data are shown below. Using = 0.05, is there enough evidence to support the claim? Smokers n1 = 26 s1 = 6 Non-smoker n2 = 18 s2 = 3.16

8. A manufacturer wishes to determine whether there is less variability in the silver plating done by Company 1 than that done by Company 2. Independent random samples yield the following results. Do the populations have dierent variances? solution: reject H0 since 3.14 > 2.82 Sample 1 n s2 12 0.035 mil Sample 2 12 0.062 mil

9. Each respondent in the Current Population Survey of May 2008 was classied as employed, unemployed, or outside the labor force. The results for men in Serdang age 35-44 can be cross-tabulated by marital status, as follows: Widowed, Never Married divorced, Married separated Employed Unemployed Not in labour force 679 63 42 18 103 10 18 114 20 25

Men of dierent marital status seem to have dierent distributions of labor force status. Or is this just chance variation? (you may assume the table results from a simple random sample.)

0.9

Tutorial 9

1. What its mean by negative linear relationship between x and y? Can you guess about the slope and r values? 2. You get y = 96.14, what this result means for you? 3. You get y = 2.75 + 96.14x, what this result means for you? 4. You get y = 2.75 96.14x, what this result means for you? 5. In area of Pasir Mas, Kelantan, records were kept on the relationship between the rainfall (in inches) and the yield of paddy (bushel per acre) as follow

Rain fall (in inches), x Yield(bushel per acre), y

10.5 50.5

8.8 46.2

13.4 58.8

12.5 59.0

18.8 82.4

10.3 49.2

7.0 31.9

15.6 76.0

16.0 78.8

(a) Find the least-squares prediction line. (b) Determine whether there is a signicant linear relationship between x and y. Test at the 5% level of signicance. (c) Find a 95% condence interval estimate of the slope . (d) Calculate r and r2 6. A private agency conducted a survey in 9 regions of the country to determine the average weekly spending in RM per person on "Nasi lemak", x, and "Teh Tarik", y. The data are listed below. Region x y 1 12.8 8.50 2 13.20 7.60 3 9.50 6.90 4 10.30 6.80 5 6.80 6 5.70 7 10.00 6.50 8 8.90 4.90 9 11.60 7.00

9.80 11.70

(a) Construct a scatter plot. What do you see? (b) Find the regression line. Answer: y = 0.449x + 1.87 (c) Find the coecient of determination. What can you conclude? Answer: 0.437 (d) Find the standard error of estimate. Answer: 0.8247 19

(e) Construct a 95% prediction interval for the weekly spending on Nasi Lemak when the amount spent on Teh Tarik is RM9.50. Answer: (3.983,8.279) 7. The data are given below. Student 1 2 3 4 5 6 (a) Dene x and y. (b) Find the least-squares prediction line. (c) Determine whether there is a signicant linear relationship between x and y. Test at the 5% level of signicance. (d) Find a 95% condence interval estimate of the slope . (e) Calculate r and r2 (f) Predict the CGPA for 20-year-old student. CGPA 3.5 2.8 3.9 3.4 2.3 3.3 Age 23 28 22 27 21 26

0.10

Tutorial 10

1. Identify the factor, its levels, the treatments, the response vari- able, the experimental unit, and the observational unit in the following situa- tions: (a) An agricultural experimental station is going to test two varieties of wheat. Each variety will be planted on 3 elds, and the yield from the eld will be measured. (b) An agricultural experimental station is going to test two varieties of wheat. Each variety will be tested with two types of fertilizers. Each combination will be applied to two plots of land. The yield will be measured for each plot. (c) Fish farmers want to study the eect of an anti-bacterial drug on the amount of bacteria in sh gills. The drug is administered at three dose- levels (none, 20, and 40 mg/100L). Each dose is administered to a large controlled tank through the ltration system. Each tank has 100 sh. At the end of the experiment, the sh are killed, and the amount of bacteria in the gills of each sh is measured. 20

2. Four treatments for fever blisters, including a placebo (A), were randomly assigned to 20 patients. The data below show, for each treatment, the numbers of days from initial appearance of the blisters until healing is complete Treatment A B C D Number of days 5 8 7 7 8 4 6 6 3 5 6 4 4 5 4 7 4 6 6 5

Test the hypothesis, at the 5% signicance level, that there is no dierence between the four treatments with respect to mean time of healing. 3. Three special micro-oven in a metal working shop are used to heat steel specimens. All the ovens are supposed to operate at the same temperature. It is known that the temperature of an oven varies, and its is suspected that there are signicance mean temperature dierences between ovens. The table below shows the temperatures, in degrees centigrade, of reach the three ovens on a random sample of heatings. Oven 494 1 2 489 489 3 Temperature (o C) 497 481 496 487 494 479 478 483 487 472 472

477

Stating any necessary assumptions, test for a dierence between mean oven temperatures. 4. Serdang Health Authority has a policy whereby any patient admitted to a hospital with a suspected coronary heart attack is automatically placed in the intensive care unit. The table below gives the number of hours spent in intensive care by such patients at ve hospitals in Serdang area. Hospitals B C D 42 57 47 30 65 46 55 27 67 58 81

A 30 25 12 23 16

E 70 63 80

Use a one factor analysis of variance to test, at the 1% level of signicance, for dierences between hospitals. 21

5. Prior to submitting a quotation for a construction project, companies prepare a detailed analysis of the estimated labour and materials costs required to complete the project. A company which employs three project cost assessors, wished to compare the mean values of these assessors cost estimates. This was done by requiring each assessor to estimate independently the costs of the same four construction projects. These costs, in RM0000s, are shown in the next column. Assessor A B C Project Project Project Project 1 2 3 4 46 62 50 66 49 63 54 68 44 59 54 63

Perform a two factor analysis of variance on these data to test, at the 5% signicance level, that there is no dierence between the assessors mean cost estimates. 6. In an experiment to investigate the warping of copper plates, the two factors studied were the temperature and the copper content of the plates. The response variable was a measure of the amount of warping. The resultant data are as follows. Copper content (%) 100 Temp (o C) 40 60 80 50 75 100 125 17 12 14 17 19 15 19 20 23 18 22 22 29 27 30 30

Stating all necessary assumptions, analyse for signicant eects. 7. A drug is produced by a fermentation process. An experiment was run to compare three similar chemical salts, X, Y and Z, in the production of the drug. Since there were only three of each of four types of fermenter A, B, C and D available for use in the production, three fermentations were started in each type of fermenter, one containing salt X, another salt Y and the third salt Z. After several days, samples were taken from each fermenter and analysed. The results, in coded form, were as follows. Fermenter type A B C D X 67 Z 68 Y 78 y 73 X 72 Z 70 Z 65 Y 80 X 68 X 69 Z 73 Y 69 22

State the type of experimental design used. Test, at the 5% level of signicance, the hypothesis that the type of salt does not aect the fermentation. Comment on what assumption you have made about the interaction between type of fermenter and type of salt.

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0.11

Binomial Table and Standard Normal Tablex 0 1 0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 9 10 0.01 .9900 .0100 .9510 .0480 .0010 0.05 .9500 .0500 .7738 .2036 .0214 .0011 0.10 .9000 .1000 .5905 .3281 .0729 .0081 .0005 .4305 .3826 .1488 .0331 .0046 .0004 0.15 .8500 .1500 .4437 .3915 .1382 .0244 .0022 .0001 .2725 .3847 .2376 .0839 .0185 .0026 .0002 0.20 .8000 .2000 .3277 .4096 .2048 .0512 .0064 .0003 .1678 .3355 .2936 .1468 .0459 .0092 .0011 .0001 .1074 .2684 .3020 .2013 .0881 .0264 .0055 .0008 .0001 p 0.25 .7500 .2500 .2373 .3955 .2637 .0879 .0146 .0010 .1001 .2670 .3115 .2076 .0865 .0231 .0038 .0004 .0563 .1877 .2816 .2503 .1460 .0584 .0162 .0031 .0004 0.30 .7000 .3000 .1681 .3602 .3087 .1323 .0284 .0024 .0576 .1977 .2965 .2541 .1361 .0467 .0100 .0012 .0001 .0282 .1211 .2335 .2668 .2001 .1029 .0368 .0090 .0014 .0001 0.35 .6500 .3500 .1160 .3124 .3364 .1811 .0488 .0053 .0319 .1373 .2587 .2786 .1875 .0808 .0217 .0033 .0002 .0135 .0725 .1757 .2522 .2377 .1536 .0689 .0212 .0043 .0005 0.40 .6000 .4000 .0778 .2592 .3456 .2304 .0768 .0102 .0168 .0896 .2090 .2787 .2322 .1239 .0413 .0079 .0007 .0060 .0403 .1209 .2150 .2508 .2007 .1115 .0425 .0106 .0016 .0001 0.45 .5500 .4500 .0503 .2059 .3369 .2757 .1128 .0185 .0084 .0548 .1569 .2568 .2627 .1719 .0703 .0164 .0017 .0025 .0207 .0763 .1665 .2384 .2340 .1596 .0746 .0229 .0042 .0003 0.50 .5000 .5000 .0313 .1563 .3125 .3125 .1563 .0313 .0039 .0313 .1094 .2188 .2734 .2188 .1094 .0313 .0039 .0010 .0098 .0439 .1172 .2051 .2461 .2051 .1172 .0439 .0098 .0010

n=1 n=5

n=8

.9227 .0746 .0026 .0001

.6634 .2793 .0515 .0054 .0004

n = 10

.9044 .0914 .0042 .0001

.5987 .3151 .0746 .0105 .0010 .0001

.3487 .3874 .1937 .0574 .0112 .0015 .0001

.1969 .3474 .2759 .1298 .0401 .0085 .0012 .0001

Table 4: The table gives the values for Pr(X = x) = b(x; n, p)

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z 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

0.00 0.5000 0.5398 0.5793 0.6179 0.6554 0.6915 0.7257 0.7580 0.7881 0.8159 0.8413 0.8643 0.8849 0.9032 0.9192 0.9332 0.9452 0.9554 0.9641 0.9713 0.9772 0.9821 0.9861 0.9893 0.9918 0.9938 0.9953 0.9965 0.9974 0.9981 0.9987

0.01 0.5040 0.5438 0.5832 0.6217 0.6591 0.6950 0.7291 0.7611 0.7910 0.8186 0.8438 0.8665 0.8869 0.9049 0.9207 0.9345 0.9463 0.9564 0.9649 0.9719 0.9778 0.9826 0.9864 0.9896 0.9920 0.9940 0.9955 0.9966 0.9975 0.9982 0.9987

0.02 0.5080 0.5478 0.5871 0.6255 0.6628 0.6985 0.7324 0.7642 0.7939 0.8212 0.8461 0.8686 0.8888 0.9066 0.9222 0.9357 0.9474 0.9573 0.9656 0.9726 0.9783 0.9830 0.9868 0.9898 0.9922 0.9941 0.9956 0.9967 0.9976 0.9982 0.9987

0.03 0.5120 0.5517 0.5910 0.6293 0.6664 0.7019 0.7357 0.7673 0.7967 0.8238 0.8485 0.8708 0.8907 0.9082 0.9236 0.9370 0.9484 0.9582 0.9664 0.9732 0.9788 0.9834 0.9871 0.9901 0.9925 0.9943 0.9957 0.9968 0.9977 0.9983 0.9988

0.04 0.5160 0.5557 0.5948 0.6331 0.6700 0.7054 0.7389 0.7704 0.7995 0.8264 0.8508 0.8729 0.8925 0.9099 0.9251 0.9382 0.9495 0.9591 0.9671 0.9738 0.9793 0.9838 0.9875 0.9904 0.9927 0.9945 0.9959 0.9969 0.9977 0.9984 0.9988

0.05 0.5199 0.5596 0.5987 0.6368 0.6736 0.7088 0.7422 0.7734 0.8023 0.8289 0.8531 0.8749 0.8944 0.9115 0.9265 0.9394 0.9505 0.9599 0.9678 0.9744 0.9798 0.9842 0.9878 0.9906 0.9929 0.9946 0.9960 0.9970 0.9978 0.9984 0.9989

0.06 0.5239 0.5636 0.6026 0.6406 0.6772 0.7123 0.7454 0.7764 0.8051 0.8315 0.8554 0.8770 0.8962 0.9131 0.9279 0.9406 0.9515 0.9608 0.9686 0.9750 0.9803 0.9846 0.9881 0.9909 0.9931 0.9948 0.9961 0.9971 0.9979 0.9985 0.9989

0.07 0.5279 0.5675 0.6064 0.6443 0.6808 0.7157 0.7486 0.7794 0.8078 0.8340 0.8577 0.8790 0.8980 0.9147 0.9292 0.9418 0.9525 0.9616 0.9693 0.9756 0.9808 0.9850 0.9884 0.9911 0.9932 0.9949 0.9962 0.9972 0.9979 0.9985 0.9989

0.08 0.5319 0.5714 0.6103 0.6480 0.6844 0.7190 0.7517 0.7823 0.8106 0.8365 0.8599 0.8810 0.8997 0.9162 0.9306 0.9429 0.9535 0.9625 0.9699 0.9761 0.9812 0.9854 0.9887 0.9913 0.9934 0.9951 0.9963 0.9973 0.9980 0.9986 0.9990

0.09 0.5359 0.5753 0.6141 0.6517 0.6879 0.7224 0.7549 0.7852 0.8133 0.8389 0.8621 0.8830 0.9015 0.9177 0.9319 0.9441 0.9545 0.9633 0.9706 0.9767 0.9817 0.9857 0.9890 0.9916 0.9936 0.9952 0.9964 0.9974 0.9981 0.9986 0.9990

Table 5: Standard Normal: The values in the table are the areas between zero and the z-score. That is, Pr(Z < z score)

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Conf. Level One Tail Two Tail df 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 50 60 70 80 90 100 z

50% 0.250 0.500 . 1.000 0.816 0.765 0.741 0.727 0.718 0.711 0.706 0.703 0.700 0.697 0.695 0.694 0.692 0.691 0.690 0.689 0.688 0.688 0.687 0.686 0.686 0.685 0.685 0.684 0.684 0.684 0.683 0.683 0.683 0.681 0.679 0.679 0.678 0.678 0.677 0.677 0.674

80% 0.100 0.200 . 3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.303 1.299 1.296 1.294 1.292 1.291 1.290 1.282

90% 95% 98% 0.050 0.025 0.010 0.100 0.050 0.020 . . . . 6.314 12.706 31.821 2.920 4.303 6.965 2.353 3.182 4.541 2.132 2.776 3.747 2.015 2.571 3.365 1.943 2.447 3.143 1.895 2.365 2.998 1.860 2.306 2.896 1.833 2.262 2.821 1.812 2.228 2.764 1.796 2.201 2.718 1.782 2.179 2.681 1.771 2.160 2.650 1.761 2.145 2.624 1.753 2.131 2.602 1.746 2.120 2.583 1.740 2.110 2.567 1.734 2.101 2.552 1.729 2.093 2.539 1.725 2.086 2.528 1.721 2.080 2.518 1.717 2.074 2.508 1.714 2.069 2.500 1.711 2.064 2.492 1.708 2.060 2.485 1.706 2.056 2.479 1.703 2.052 2.473 1.701 2.048 2.467 1.699 2.045 2.462 1.697 2.042 2.457 1.684 2.021 2.423 1.676 2.009 2.403 1.671 2.000 2.390 1.667 1.994 2.381 1.664 1.990 2.374 1.662 1.987 2.368 1.660 1.984 2.364 1.645 1.960 2.326

99% 0.005 0.010 63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.704 2.678 2.660 2.648 2.639 2.632 2.626 2.576

Table 6: Students t Probabilities: The values in the table are the areas critical values for the given areas in the right tail or in both tails.

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df

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 50 60 70 80 90 100

0.995 0.010 0.072 0.207 0.412 0.676 0.989 1.344 1.735 2.156 2.603 3.074 3.565 4.075 4.601 5.142 5.697 6.265 6.844 7.434 8.034 8.643 9.260 9.886 10.520 11.160 11.808 12.461 13.121 13.787 20.707 27.991 35.534 43.275 51.172 59.196 67.328

0.99 0.020 0.115 0.297 0.554 0.872 1.239 1.646 2.088 2.558 3.053 3.571 4.107 4.660 5.229 5.812 6.408 7.015 7.633 8.260 8.897 9.542 10.196 10.856 11.524 12.198 12.879 13.565 14.256 14.953 22.164 29.707 37.485 45.442 53.540 61.754 70.065

0.975 0.001 0.051 0.216 0.484 0.831 1.237 1.690 2.180 2.700 3.247 3.816 4.404 5.009 5.629 6.262 6.908 7.564 8.231 8.907 9.591 10.283 10.982 11.689 12.401 13.120 13.844 14.573 15.308 16.047 16.791 24.433 32.357 40.482 48.758 57.153 65.647 74.222

0.95 0.004 0.103 0.352 0.711 1.145 1.635 2.167 2.733 3.325 3.940 4.575 5.226 5.892 6.571 7.261 7.962 8.672 9.390 10.117 10.851 11.591 12.338 13.091 13.848 14.611 15.379 16.151 16.928 17.708 18.493 26.509 34.764 43.188 51.739 60.391 69.126 77.929

0.90 0.10 0.05 0.025 0.01 0.005 0.016 2.706 3.841 5.024 6.635 7.879 0.211 4.605 5.991 7.378 9.210 10.597 0.584 6.251 7.815 9.348 11.345 12.838 1.064 7.779 9.488 11.143 13.277 14.860 1.610 9.236 11.070 12.833 15.086 16.750 2.204 10.645 12.592 14.449 16.812 18.548 2.833 12.017 14.067 16.013 18.475 20.278 3.490 13.362 15.507 17.535 20.090 21.955 4.168 14.684 16.919 19.023 21.666 23.589 4.865 15.987 18.307 20.483 23.209 25.188 5.578 17.275 19.675 21.920 24.725 26.757 6.304 18.549 21.026 23.337 26.217 28.300 7.042 19.812 22.362 24.736 27.688 29.819 7.790 21.064 23.685 26.119 29.141 31.319 8.547 22.307 24.996 27.488 30.578 32.801 9.312 23.542 26.296 28.845 32.000 34.267 10.085 24.769 27.587 30.191 33.409 35.718 10.865 25.989 28.869 31.526 34.805 37.156 11.651 27.204 30.144 32.852 36.191 38.582 12.443 28.412 31.410 34.170 37.566 39.997 13.240 29.615 32.671 35.479 38.932 41.401 14.041 30.813 33.924 36.781 40.289 42.796 14.848 32.007 35.172 38.076 41.638 44.181 15.659 33.196 36.415 39.364 42.980 45.559 16.473 34.382 37.652 40.646 44.314 46.928 17.292 35.563 38.885 41.923 45.642 48.290 18.114 36.741 40.113 43.195 46.963 49.645 18.939 37.916 41.337 44.461 48.278 50.993 19.768 39.087 42.557 45.722 49.588 52.336 20.599 40.256 43.773 46.979 50.892 53.672 29.051 51.805 55.758 59.342 63.691 66.766 37.689 63.167 67.505 71.420 76.154 79.490 46.459 74.397 79.082 83.298 88.379 91.952 55.329 85.527 90.531 95.023 100.425 104.215 64.278 96.578 101.879 106.629 112.329 116.321 73.291 107.565 113.145 118.136 124.116 128.299 82.358 118.498 124.342 129.561 135.807 140.169

Table 7: Chi-Square Probabilities: The areas given across the top are the areas to the right of the critical value. To look up an area on the left, subtract it from one, and then look it up (ie: 0.05 on the left is 0.95 on the right)

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