PracticeFinalExam

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Practice Final Exam: STATS 2MB3 Date: April 2023 Instructor: Dr. Katherine Davies Email address . . . . . . . . . . . . . . . . . . . . . . . . . . . . . YOU ARE RESPONSIBLE FOR ENSURING THAT YOUR COPY OF THE EXAM IS COM- PLETE. BRING ANY DISCREPANCY TO THE ATTENTION OF YOUR INVIGILATOR. Examination Duration: 150 minutes Materials Permitted In The Exam Venue: (No electronic aids are permitted e.g. laptops, phones) • McMaster Standard Calculator (Casio FX-991MS/MS+) Materials To Be Supplied To Students: • Scrap Paper (will be recycled immediately) Instructions to Students: • This exam consists of 18 questions in two parts: Part A (15 multiple choice questions) and Part B (3 long answer questions). • There are a total of 50 marks available; marks are as indicated. • Please write neatly and do not write on the QR code in the top right corners. 1

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The next three questions refer to the following. The frequency distribution below summa- rizes the batting averages of a sample of Major League Baseball players: Batting Average Frequency 0.160-0.180 2 0.180-0.200 6 0.200-0.220 7 0.220-0.240 11 0.240-0.260 14 0.260-0.280 32 0.280-0.300 39 0.300-0.320 23 0.320-0.340 9 Question 1. [2 pts.] What is the relative frequency (i.e., proportion) of observations falling in the 0.260-0.280 interval? (A) 72 (B) 0.503 (C) 32 (D) 0.224 (E) 0.273 Question 2. [2 pts.] Which statement is correct? (A) Since the distribution of batting averages is skewed to the left, the median is larger than the mean. (B) Since the distribution of batting averages is skewed to the right, the median is larger than the mean. (C) Since the distribution of batting averages is approximately symmetric, the median is approximately equal to the mean.. (D) Since the distribution of batting averages is skewed to the right, the mean is larger than the median. (E) Since the distribution of batting averages is skewed to the left, the median is less than the mean. Question 3. [2 pts.] Which interval contains the first quartile of the data values? (A) 0.180-0.200 (B) 0.200-0.220 (C) 0.220-0.240 (D) 0.240-0.260 (E) 0.260-0.280

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Question 4. [2 pts.] We take a random sample of 200 Canadians and ask them whether they attend a place of worship (church, temple, mosque, synagogue, etc). We would like to conduct an hypothesis test of H 0 : p = 0 . 25 versus H a : p < 0 . 25 to determine if there is evidence that less than one-quarter of Canadians attend a place of worship. It is decided that H 0 will be rejected if ˆ p < 0 . 20. The level of significance of the test is closest to: (A) 0.044 (B) 0.052 (C) 0.956 (D) 0.945 (E) 0.0001 Question 5. [2 pts.] Consider testing the hypotheses: H 0 : µ = 10 versus H a : µ ̸ = 10, where µ is some population mean. Based on a sample, we find an estimate of µ as ˆ µ = x =10.36. Furthermore, a 90% confidence interval for µ , based on this sample, is found to be (10.45, 11.86). Based on this, we would: (A) fail to reject H 0 at the 5% level of significance since the value 10 is contained in the 90% confidence interval. (B) fail to reject H 0 at the 5% level of significance since the value 10 is not contained in the 90% confidence interval. (C) fail reject H 0 at the 10% level of significance since the value 10.36 is not contained in the 90% confidence interval. (D) reject H 0 at the 10% level of significance since the value 10 is contained in the 90% confidence interval. (E) reject H 0 at the 10% level of significance since the value 10 is not contained in the 90% confidence interval. Question 6. [2 pts.] We take a random sample of 16 adults and ask them how long they sleep on a typical night. The sample mean is calculated to be 7.2 hours. Suppose it is known that the number of hours adults sleep at night follows a normal distribution with standard deviation 1.8 hours. An 88% confidence interval for the true mean time adults sleep at night is: (A) (6.80, 7.60) (B) (6.50, 7.90) (C) (6.60, 7.80) (D) (6.70, 7.70) (E) (6.30, 8.10)

Question 7. [2 pts.] The sizes of farms in a U.S. state follow a normal distribution with standard deviation 30 acres. Suppose we measure the sizes of a random sample of farms and calculate a 98% confidence interval for µ , the true mean size in a U.S. state, to be (295, 305). What is the correct interpretation of this interval? (A) Approximately 98% of farms have a size between 295 and 305 acres. (B) Approximately 98% of samples of 30 farms will have a mean size between 295 and 305 acres. (C) The probability that the population mean µ is between 295 and 305 is 98%. (D) In repeated samples of the same size, 98% of similarly constructed intervals will contain the sample mean. (E) In repeated samples of the same size, 98% of similarly constructed intervals will contain the population mean µ . Question 8. [2 pts.] The Hamilton Street Railway (HSR) claims that the average time taken by Number 51 bus to travel from McMaster University to downtown is 23 minutes. A student who often takes this route believes the true mean is greater than 23 minutes. The student records the times for a sample of five trips. These trips have a mean time of 25 minutes and a standard deviation of 3.5 minutes. Suppose it is known that trip times follow a normal distribution. The p -value for the appropriate test of significance to test the student’s suspicion is: (A) less than 0.0005 (B) between 0.10 and 0.15 (C) between 0.05 and 0.10 (D) between 0.025 and 0.05 (E) between 0.01 and 0.015 Question 9. [2 pts.] Suppose we have two random variables X ∼ N ( µ 1 , σ 2 1 ) and Y ∼ N ( µ 2 , σ 2 2 ). We collect two samples, x 1 , . . . , x 8 and y 1 , . . . , y 16 and based on the first sample we find x =3.5 and s 2 1 =1.8, and based on the second sample, y =3.3 and s 2 2 =2.1. A 98% confidence interval for σ 2 1 /σ 2 2 is: (A) (0.207, 5.409) (B) (0.136, 3.549) (C) (0.218, 4.876) (D) (0.244, 5.248) (E) (0.220, 4.964)

The next two questions refer to the following. Two supermarket managers of two large stores would like to compare the proportion of employees who are considered part-time, in particular, if there is any difference between the two sets of employees. The first manager, of Store A, sampled 30 employees and found 40% were part-time, and the second manager, of Store B, sampled 40 employees and found 30% were part-time. Question 10. [2 pts.] If we were to conduct a test at 5% level of significance, what would be the appropriate test statistic? (A) z = (0 . 4 − 0 . 3) − 0 q (0 . 35)(0 . 65) ( 1 30 + 1 40 ) (B) z = (0 . 4 − 0 . 3) − 0 q (0 . 343)(0 . 657) ( 1 30 + 1 40 ) (C) t = (0 . 4 − 0 . 3) − 0 q (0 . 35)(0 . 65) ( 1 30 + 1 40 ) (D) t = (0 . 4 − 0 . 3) − 0 q (0 . 346)(0 . 654) ( 1 30 + 1 40 ) (E) t = (0 . 4 − 0 . 3) − 0 q (0 . 343)(0 . 657) 30+40 − 2 Question 11. [2 pts.] Suppose we had decided to conduct the test at 5% level of significance and further suppose we find the p -value of the test to be greater than 0.05. We would interpret this information as: (A) If in fact the two stores had the same proportion of part-time employees, the chance of observing a difference of 0.1 between the two sample proportions (defined as Store A-Store B), is less than 0.05. (B) If in fact the two stores had the same proportion of part-time employees, the chance of observing a difference 0.1 or more extreme between the two sample proportions (defined as Store A-Store B), is larger than 0.05. (C) If in fact the Store A had 10% more part time employees, the chance of observing a difference of 0 between the two sample proportions (defined as Store A-Store B), is larger than 0.05. (D) If in fact the Store A had 10% more part time employees, the chance of observing a difference of 0 between the two sample proportions (defined as Store A-Store B), is less than 0.05. (E) If in fact the two stores had the same proportion of part-time employees, the chance of observing a difference of 0.1 or more extreme between the two sample proportions (defined as Store A-Store B), is larger than 0.025.

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Question 12. [2 pts.] Two popular movie streaming services are Netflix and Prime Video. The question is: which service do viewers prefer? Do they prefer one over the other? A matched pairs experiment was carried out as follows: a sample of 30 volunteers was given a free subscription to both services for a six-month period. At the end they had to assign scores to the two streaming services on a scale of 0 to 100. Below is the data that was collected: Scores for Netflix Scores or Prime Video Difference (Netflix-Prime Video) mean=80 mean=72 mean=8 std.dev.=23 st.dev.=21 std.dev=12 Consider the following statements: (I) The scores for Netflix and Prime Video are independent. (II) The scores for Netflix and Prime Video are dependent. (III) We must assume, in order to use matched-pairs t procedures, that scores for Netflix and for Prime Video both follow normal distributions, with possibly different means and variances. (IV) We must assume, in order to use matched-pairs t procedures, that the differences in scores (Netflix Score-Prime Video Score) follows a normal distributions. Which of the above statements is true? (A) I only (B) I and III only (C) I and IV only (D) II and III only (E) II and IV only The next three questions refer to the following. We would like to use the weight of a car to predict its fuel economy. Weights (in 1000s of pounds) and fuel efficiency (in miles per gallon) are shown in the table below for a sample of ten car models: Weight 3.1 3.7 2.2 3.4 2.0 3.9 3.0 2.5 3.6 2.7 Fuel Efficiency 25 21 31 22 35 16 24 27 17 29 It can be shown that the weights have mean 3.01 and standard deviation 0.65. Also, the fuel efficiencies have mean 24.70 and standard deviation 6.02. We also can show that ∑ 10 i =1 ( x i − x ) 2 =3.8025 and ∑ 10 i =1 ( y i − ˆ y i ) 2 =24.92. The least-squares regression line is found to be: ˆ y = 51 . 47 − 8 . 89 x . Question 13. [2 pts.] What is the correct interpretation of the slope of the least-squares regression line? (A) When the weight increases by 8.89lbs, the predicted fuel efficiency increases by 1. (B) When the fuel efficiency increase by 8.89, the predicted weight decreases by 1000lbs. (C) When the weight increases by 1000lbs, the predicted fuel efficiency decreases by 8.89. (D) When the fuel efficiency increase by 1, the predicted weight decreases by 8.89lbs. (E) When the weight increases by 1lb, the predicted fuel efficiency decreases by 8.89.

Question 14. [2 pts.] What is the fitted value corresponding to a weight of 3500lbs? (A) 21.8 (B) 144.52 (C) 20.36 (D) -245.23 (E) 170.26 Question 15. [2 pts.] A 95% confidence interval for β 1 , the true slope of the regression line, is: (A) (-10.91, -6.87) (B) (-53.13, -49.81) (C) (-10.55, -7.23) (D) (-11.03, -6.75) (E) (-53.56, -49.38)

Part B Question 1. [8 pts.] Let X 1 , . . . , X n be a random sample from U (0 , θ ) with probability density function as: f ( x ; θ ) = 1 /θ for 0 < x < θ, θ > 0 , 0 otherwise. (a) Derive the method of moments estimator of θ (b) Derive the maximum likelihood estimator of θ . (c) Without doing any derivation/calculation, determine if the maximum likelihood es- timator of θ is biased or not. Explain. Question 2. [7 pts.] We would like to conduct an hypothesis test at 5% level of significance to determine whether hockey players weigh more, on average, than baseball players. We record the weights of a random sample of 12 professional hockey players and 7 professional baseball players. Weights of such athletes in both sports are known to be normally distributed and past data tell the weights of the two types of athletes have difference variance. Some summary statistics are shown below: Sport n x s Baseball 7 185.571 7.72 Hockey 12 195.583 18.39 (a) If we let µ 1 denote the true mean weight of hockey players and µ 2 denote the true mean weight of baseball players, construct a 95% confidence interval for µ 1 − µ 2 . (b) Conduct an appropriate test of significance at 10% level of significance to determine whether the mean weight of baseball players is less than that of hockey players. Using the p -value method and show all your steps. (c) Provide an interpretation of the p -value found in ( c ).

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Question 3. [5 pts.] Battery life of MP3 players is of great concern to customers. A consumer group tested four brands of MP3 players to determine battery life. Samples of MP3 players of each brand were fully charged and left to run on medium volume until the battery died. The follow table displays the number of hours each of the batteries lasted? Brand A B C D 24 26 18 27 19 28 20 24 22 24 21 22 25 24 mean 21.67 25.75 19.67 24.25 std.dev. 2.52 1.17 1.53 2.06 We would like to conduct an analysis of variance (ANOVA) to compare the performance of the four brands. Suppose all the necessary assumptions have been satisfied. The ANOVA table, with some missing values, is shown below: Source of Variance df Sum of Squares Mean Square F Treatments Error 3.88 Total 113.71 Find and enter all the missing values in the table. Conduct the analysis of variance using the rejection region method at 1% level of significance. Show all your steps, including making a properly worded conclusion. Exam end.

Stats 2MB3 —Formula Sheet • ¯ x = 1 n ∑ n i =1 x i s 2 = 1 n − 1 ∑ n i =1 ( x i − ¯ x ) 2 = 1 n − 1 ∑ n i =1 x 2 i − n ¯ x 2 • E ( ∑ n i =1 a i X i ) = ∑ n i =1 a i E ( X i ) • V ar ( ∑ n i =1 a i X i ) = ∑ n i =1 a 2 i V ar ( X i ) + ∑ ∑ i ̸ = j a i a j Cov ( X i , X j ) • If X i and X j are independent, then Cov ( X i , X j ) = 0. • If X 1 , . . . , X n are normal random variables then ∑ n i =1 a i X i is a normal random variable • If X is the mean of a random sample then E ( X ) = µ and V ar ( X ) = σ 2 /n • The Central Limit Theorem says that for any population distribution with mean µ and variance σ 2 < ∞ , the distribution of the sample mean is approximately normal, with mean and variance as given above. In other words, Z = X − µ σ/ √ n · ∼ N (0 , 1) . One sample inference • Normal( µ, σ 2 ) population with σ known ∗ Confidence interval for the population mean µ is ¯ x ± z α/ 2 σ/ √ n . ∗ Test statistic for testing H 0 : µ = µ 0 is Z = X − µ 0 σ/ √ n which has a N (0 , 1) under H 0 • Large sample size ∗ Confidence interval for the population mean µ is ¯ x ± z α/ 2 s/ √ n ∗ Test statistic for testing H 0 : µ = µ 0 is Z = X − µ 0 S/ √ n which has an approximate N (0 , 1) under H 0 • Normal( µ, σ 2 ) population with σ unknown ∗ Confidence interval for the population mean µ is ¯ x ± t α/ 2 ,n − 1 s/ √ n . ∗ Test statistic for testing H 0 : µ = µ 0 is T = X − µ 0 S/ √ n which has a Student’s t -distribution with n − 1 degrees of freedom under H 0 • Confidence interval for normal variance σ 2 is ( n − 1) s 2 χ 2 α/ 2 ,n − 1 , ( n − 1) s 2 χ 2 1 − α/ 2 ,n − 1 . • Proportion (large sample size) ∗ Confidence interval for p (by hand) is ˆ p ± z α/ 2 ˆ p (1 − ˆ p ) n ∗ Test statistic to test H 0 : p = p 0 is Z = ˆ p − p 0 p 0 (1 − p 0 ) n which has an approximate N (0 , 1) distribution under H 0 1

Two sample inference • Large sample sizes ∗ Confidence interval for µ 1 − µ 2 is ¯ x − ¯ y ± z α/ 2 s 2 1 m + s 2 2 n ∗ Test statistic to test H 0 : µ 1 − µ 2 = ∆ 0 is Z = X − Y − ∆ 0 S 2 1 m + S 2 2 n which has an approximate N (0 , 1) distribution under H 0 • Independent normal populations ∗ Confidence interval for µ 1 − µ 2 is ¯ x − ¯ y ± t α/ 2 ,k s 2 1 m + s 2 2 n where k = s 2 1 m + s 2 2 n 2 1 m − 1 s 2 1 m 2 + 1 n − 1 s 2 2 n 2 ∗ Test statistic to test H 0 : µ 1 − µ 2 = ∆ 0 is T = X − Y − ∆ 0 S 2 1 m + S 2 2 n which has an approximate Student’s t distribution with k degrees of freedom under H 0 , where k is as above ∗ Confidence interval for σ 2 1 /σ 2 2 is 1 F α/ 2 ,ν 2 ,ν 1 s 2 1 s 2 2 , F α/ 2 ,ν 1 ,ν 2 s 2 1 s 2 2 • Independent normal populations with equal variances ∗ s 2 p = ( m − 1) s 2 1 +( n − 1) s 2 2 m + n − 2 ∗ Confidence interval for µ 1 − µ 2 is ¯ x − ¯ y ± t α/ 2 ,m + n − 2 s p 1 m + 1 n ∗ Test statistic to test H 0 : µ 1 − µ 2 = ∆ 0 is T = X − Y − ∆ 0 S p √ 1 m + 1 n which has a Student’s t distribution with m + n − 2 degrees of freedom under H 0 • Paired data ∗ Differences d i = x i − y i ∗ Confidence interval for µ D = µ 1 − µ 2 is ¯ d ± t α/ 2 ,n − 1 s D √ n . ∗ Test statistic to test H 0 : µ d = ∆ 0 is T = ¯ D − ∆ 0 S d / √ n which has a Student’s t distribution with n − 1 degrees of freedom under H 0 . • Two proportions – Confidence interval for p 1 − p 2 is ˆ p 1 − ˆ p 2 ± z α/ 2 ˆ p 1 (1 − ˆ p 1 ) m + ˆ p 2 (1 − ˆ p 2 ) n – Test statistic to test H 0 : p 1 = p 2 is Z = ˆ p 1 − ˆ p 2 ˆ p (1 − ˆ p ) ( 1 m + 1 n ) which follows an approximate N (0 . 1) distribution under H 0 where ˆ p = ( X + Y ) / ( m + n ) • Analysis of Variance • Sums of squares SST = ∑ I i =1 ∑ J i j =1 ( X ij − X ·· ) 2 SSTr = ∑ I i =1 ∑ J i j =1 ( X i · − X ·· ) 2 = ∑ I i =1 J i ( X i · − X ·· ) 2 SSE = ∑ I i =1 ∑ J i j =1 ( X ij − X i · ) 2 = ∑ I i =1 ( J i − 1) S 2 i 2

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• The ANOVA table Source of Sum of Mean Variation df Squares Square F Treatments I − 1 SSTr MSTr = SSTr I − 1 F = MSTr MSE Error I ( J − 1) SSE MSE = SSE I ( J − 1) Total IJ − 1 SST • Simple Linear Regression • ˆ β 1 = ∑ ( x i − ¯ x )( y i − ¯ y ) ∑ ( x i − ¯ x ) 2 = S xy S xx , ˆ β 0 = ¯ y − ˆ β 1 ¯ x • SSE= ∑ ( y i − ˆ y ) 2 = ∑ ( y i − ˆ β 0 − ˆ β 1 x i ) 2 , σ 2 = SSE / ( n − 2) • Confidence interval for ˆ β 1 is ˆ β 1 ± t n − 2 ,α/ 2 se( ˆ β 1 ) • Test statistic to test H 0 : β 1 = β 1 , 0 is T = ˆ β 1 − β 1 , 0 se( ˆ β 1 ) • Correlation – Population Correlation Coefficient: ρ = Cov( X,Y ) √ Var( X )Var( Y ) – Sample Correlation Coefficient: r = S xy / S xx S yy 3

Table A .3 Norma l Probabil ity Table 73 5 Tabl e A.3 Areas under the Normal Cur ve z .oo .01 .02 .03 .04 . 05 .06 .07 .08 .09 - 3 .4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002 - 3 .3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003 - 3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005 - 3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007 - 3.0 0.0013 0.0013 0.0013 0 .0 012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010 - 2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 - 2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019 - 2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026 - 2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036 - 2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048 - 2.4 0.00 82 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064 - 2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084 - 2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 O.Q113 0.0110 - 2.1 O.Ql 79 0.0174 O.Ql 70 0.0166 0.0162 0.0158 0.0154 0. 01 50 0. 01 46 0. 01 43 - 2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0. 01 88 0.0183 - 1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233 - 1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294 - 1.7 0.0446 0.0436 0.0427 0.0418 0.04 09 0.04 01 0.0392 0.0384 0.0375 0.0367 - 1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 - 1.5 0.0668 0.06 55 0.0643 0 .0 630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559 - 1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0. 06 81 - 1.3 0.0968 0. 09 51 0.0934 0.0918 0.0901 0.0885 0.0869 0.08 53 0.0838 0.0823 - 1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.10 56 0.1038 0.1020 0.1003 0.0985 - 1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170 - 1.0 0.1 58 7 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.14 01 0.1379 - 0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611 - 0.8 0.2119 0 .2 090 0 .2 061 0. 2033 0 .2 005 0.1977 0.1949 0.1922 0.1894 0.1 86 7 - 0.7 0.24 20 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148 - 0.6 0.27 43 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 - 0.5 0.30 85 0 .3 050 0.3015 0.2981 0. 29 46 0.2912 0.28 77 0.2843 0.2810 0.2776 - 0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.31 56 0.3121 - 0 .3 0.38 21 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483 - 0.2 0.4207 0 .4 168 0 .4 129 0.4090 0.40 52 0. 4013 0.3974 0.3936 0.3897 0.3859 - 0.1 0.46 02 0.4562 0.4522 0.4483 0.4443 0 .44 04 0 .4 364 0.4325 0.4286 0. 4247 - 0.0 0.5000 0. 4960 0.4920 0.4880 0.4840 0. 48 01 0 .4 761 0.4721 0.4681 0.4641

736 App e ndix A Statistical Tabl es and Proofs Table A.3 (co ntinu ed) Ar eas und er the N orm al C ur ve z .00 .01 .02 . 03 .04 .05 .0 6 .07 .08 .09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.57 93 0.5832 0.5871 0.5910 0.59 48 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0 .4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 07580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0 .8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0. 8106 0. 8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0. 8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.85 77 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0 .8 749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 0.9 45 2 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 3 .4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998

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Table A. 4 Stud e nt t-Dist ri but ion Probabili ty Ta bl e 737 Table A.4 Criti ca l Values of the t-Di st ri bution a v 0 .4 0 0.30 0.20 0. 15 0. 10 0.05 0.0 25 1 0 .3 25 0.727 1.376 1.963 3.078 6.314 12.706 2 0.289 0.617 1.061 1.386 1.8 86 2.920 4.303 3 0. 2 77 0.584 0.978 1.250 1.638 2.353 3.182 4 0. 2 71 0.569 0.941 1.190 1.533 2.132 2.776 5 0.267 0.5 59 0.920 1.1 56 1. 4 76 2.015 2.5 71 6 0.265 0.553 0.906 1.134 1. 440 1.943 2.447 7 0.263 0.549 0.896 1.119 1. 415 1.895 2.365 8 0.262 0. 546 0. 889 1.108 1.397 1. 860 2.306 9 0.261 0.5 43 0. 883 1.100 1.383 1. 833 2.262 10 0.260 0. 542 0 .8 79 1.093 1.372 1. 8 12 2.228 11 0.260 0.540 0. 876 1.088 1.363 1.796 2.201 12 0.259 0.539 0 .8 73 1.083 1.356 1.7 82 2.179 13 0.259 0.538 0 .8 70 1.079 1.350 1.771 2.160 14 0.2 58 0. 537 0.868 1.076 1.3 45 1.761 2.145 15 0.258 0. 53 6 0.866 1.074 1.341 1.7 53 2.131 16 0.258 0.535 0.865 1.071 1.337 1.746 2.120 17 0.257 0.534 0.863 1.069 1.333 1.740 2.110 18 0.257 0. 534 0.862 1.067 1.330 1.734 2. 101 19 0.257 0.533 0.861 1.066 1.328 1.729 2.093 20 0.257 0.533 0.860 1.064 1.325 1.725 2.086 21 0.257 0. 532 0. 859 1.063 1.323 1.721 2.080 22 0.256 0. 532 0.858 1.061 1.321 1.717 2.074 23 0.2 56 0. 532 0.8 58 1.060 1.319 1.714 2.069 24 0.256 0.531 0.857 1.059 1.318 1.711 2.064 25 0.256 0. 53 1 0.856 1.058 1.316 1.708 2.060 26 0.256 0 .5 31 0.856 1.0 58 1.315 1.706 2.056 27 0.256 0.531 0.855 1.057 1.314 1.703 2.052 28 0.256 0. 53 0 0.855 1.0 56 1.313 1.701 2.048 29 0.256 0. 530 0. 854 1.0 55 1.311 1.699 2.045 30 0.256 0. 53 0 0.854 1.0 55 1.310 1. 697 2.042 40 0.2 55 0. 529 0.851 1.050 1.303 1.6 84 2.021 60 0.2 54 0.527 0. 848 1.045 1.296 1.671 2.000 120 0.254 0. 5 26 0.845 1.041 1.289 1.6 58 1.980 00 0.253 0. 524 0. 842 1.036 1.2 82 1.645 1.960

738 A ppendi.'C A Statistical Tables and Proofs Table A.4 (continued) Critical Values of the t-Distribution a v 0.02 0.015 0.01 0.0075 0.005 0.0025 0.0005 1 15.894 21.205 31.821 42.433 63 .656 127.321 636.578 2 4.849 5.643 6.965 8.0 73 9.925 14.089 3 1. 600 3 3.482 3.896 4.541 5.047 5.841 7.453 12.924 4 2.999 3.298 3.747 4.088 4.604 5.598 8.610 5 2.757 3.003 3.365 3.634 4.032 4.773 6.869 6 2.612 2.829 3.143 3.372 3.707 4.317 5.959 7 2.517 2.715 2.998 3.203 3.499 4. 029 5.408 8 2.449 2.634 2.896 3.085 3.355 3.833 5.041 9 2.398 2.574 2.821 2.998 3.250 3.690 4.781 10 2.359 2.527 2.764 2.932 3.169 3.581 4.587 11 2.328 2.491 2.718 2.8 79 3.106 3.497 4.437 12 2.303 2.461 2.681 2.836 3.055 3.428 4.318 13 2.282 2.436 2.650 2.801 3.012 3.372 4.221 14 2.264 2.415 2.624 2.771 2.977 3.326 4.140 15 2.249 2.397 2.602 2.746 2.947 3.286 4.073 16 2.235 2.382 2.583 2.724 2.921 3.252 4. 01 5 17 2.224 2.368 2.567 2.706 2.898 3.222 3.965 18 2.214 2.356 2.552 2.689 2.878 3.197 3.922 19 2.205 2.346 2.539 2.674 2.861 3.174 3.883 20 2.197 2.336 2.528 2.661 2.845 3.153 3.850 21 2.189 2.328 2.518 2.649 2.831 3.135 3.819 22 2 .1 83 2.320 2.508 2.639 2.819 3.119 3.792 23 2.177 2.313 2.500 2.629 2. 807 3.104 3.768 24 2.172 2.307 2.492 2.620 2.797 3.091 3.745 25 2.167 2.301 2.485 2.612 2.787 3.078 3.725 26 2 .1 62 2.296 2.479 2.605 2. 77 9 3.067 3.707 27 2 .1 58 2.291 2.473 2.598 2.771 3.057 3.689 28 2. 154 2.286 2.467 2.592 2.763 3.047 3.674 29 2. 150 2.282 2.462 2.586 2.756 3.038 3.660 30 2.147 2.278 2.457 2.581 2.750 3.030 3.646 40 2.123 2.250 2.423 2.542 2.704 2.971 3.551 60 2.099 2.223 2.390 2.504 2.660 2.915 3.460 120 2.076 2.196 2.358 2.468 2.617 2.860 3.373 00 2.054 2.170 2.326 2.432 2.576 2.807 3.290

Table A. 5 Chi-Sq'Uared Distrib'Ution Probability T ab le 739 Table A.5 Critical Values of th e Chi-Squared Di s tribution 0 x: a v 0.995 0.99 0.98 0.975 0.95 0.90 0.80 0.75 0.70 0.50 1 0.0 4 393 0.0 3 157 0.0 3 628 0.0 3 982 0.00393 0.0158 0.0642 0.102 2 0.0lUO 0.0201 0.0 4 0'1 0.148 0.455 0 .0.501) 0.103 0. 211 3 0. 07 17 0.115 0.185 0.446 0 .5 75 0.713 1.386 0.216 0.352 0. 584 l.UU 5 1.213 l.42-1 2. 36fi 4 0.207 0.297 0.429 0.484 0.711 1.064 1.649 1.923 2.195 3.357 5 0. 412 0.554 0.752 0.831 1.1 45 1.610 2.343 2.675 3.000 4.351 6 0.676 0.872 1.134 1.237 1.635 2 .2 04 3.070 3.455 3.828 5.348 7 0.989 1.239 1. 564 1.690 2.167 2.833 3.822 4.255 4. 671 6.346 8 1.344 1.647 2.032 2.180 2.733 3.490 4.594 5. 071 5.527 7.344 9 1.735 2.088 2.532 2.700 3.325 4. 16 8 5.380 5.899 6.393 8.343 10 2.156 2.558 3.059 3.247 3.940 4.865 6.179 6.737 7.267 9.342 11 2.603 3.053 3.609 3.816 4.575 5.578 6.989 7.584 8.148 10 .341 12 3.074 3.5 71 4.178 4.404 5.226 6.304 7.807 8.438 9.034 11.340 13 3.565 4. 107 4. 76 5 5. 009 5.892 7.041 8.634 9.299 9.926 12 .340 14 4.075 4.660 5.368 5.629 6.5 71 7.790 9.467 10.165 10.8 21 13 .339 15 4.6 01 5.229 5 .9 85 6.262 7.261 8.547 10.307 11.037 11. 721 14.339 16 5. 142 5.8 12 6.614 6.908 7.962 9.312 11.1 52 11.912 12 .624 15.338 17 5.697 6.408 7.255 7. 564 8.672 10.085 12 .002 12 .792 13 .5 31 16 .338 18 6.265 7.015 7.906 8.231 9.390 10 .865 12.857 13.675 14.440 17.338 19 6.844 7.633 8.567 8.907 10.117 11.651 13.716 14.562 15.352 18.338 20 7. 434 8.260 9.237 9.591 10.851 12 .443 14.578 15.452 16.266 19.337 21 8 .0 34 8.897 9.915 10.283 11.591 13.240 15.445 16.344 17.182 20.337 22 8. 643 9.542 10.600 10.982 12 .338 1 4. 041 16 .3 14 17.240 18. 101 21.337 23 9.260 10.196 11.293 11.689 13 .091 14.848 17.187 1 8. 137 19.021 22.337 24 9.886 10.856 11 .992 12 .4 01 13 .848 15.659 18.062 19 .037 19.943 23.337 25 10.520 11.524 12.697 13.120 14.6 11 16.4 73 18.940 19.939 20.867 24.337 26 11 .160 12 .198 13.409 13 .844 15.379 17.292 19.820 20.843 21.792 25.336 27 11.808 12.878 14. 12 5 14.5 73 16.151 1 8. 11 4 20.703 21.749 22 .719 26.336 28 12.461 13.565 14.847 15.308 16.928 18.939 21.588 22.657 23.647 27.336 29 13 . 121 14.256 15.574 16.047 17.708 19. 76 8 22.475 23.567 24 .5 77 28.336 30 13 .787 14.953 16.306 16. 791 18.493 20.599 23.364 24.478 25.508 29.336 40 20.707 22.164 23.838 24.433 26 .509 29.051 32.345 33.66 34.8 72 39.335 50 27.991 29.707 31.664 32.357 34.764 37.689 4 1. 449 42.942 44.313 49 .335 60 35.534 37.485 39.699 40.482 43.188 46.459 50.641 52. 29 4 53 .809 59.335

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740 Appendix A Statistical Tables and Proofs Table A.5 (continued) Criti ca l Values of the Chi-Squared Di st ributi on a v 0.30 0.25 0.20 0.10 0.05 0.025 0.02 0. 01 0. 005 0.001 1 1.074 1.323 1.642 2.706 3.841 5.024 5.412 6.635 7.879 10.827 2 2.408 2.773 3.219 4.605 5.991 7.378 7.824 9.210 10.597 13.815 3 3.665 4.108 4.642 6.251 7.815 9.348 9.837 11.345 12.838 16.266 4 4.878 5.385 5.989 7.779 9.488 11.143 11.668 13.277 14.860 18.466 5 6.064 6.626 7.289 9.236 11.070 12.832 13.388 15.086 16.750 20.515 6 7.231 7.841 8.558 10.645 12.592 14.449 15.033 16.812 18.548 22.457 7 8.383 9.037 9.803 12.017 14.067 16.013 16.622 18.475 20.278 24.321 8 9.524 10.219 11.030 13.362 15 .507 17.535 18.168 20.090 21.955 26.124 9 10.656 11.389 12.242 14.684 16.919 19.023 19.679 21.666 23.589 27.877 10 11.781 12.549 13.442 15.987 18.307 20.483 21.161 23.209 25.188 29.588 11 12.899 13.701 14.631 17.275 19.675 21.920 22.618 24.725 26.757 31.264 12 14.011 14.845 15.812 18.549 21.026 23.337 24.054 26.217 28.300 32.909 13 15.119 15.984 16.985 19.812 22.362 24.736 25 .471 27.688 29.819 34.527 14 16.222 17.117 18.151 21.064 23.685 26.119 26 .873 29.141 31.319 36.124 15 17.322 18.245 19.311 22.307 24.996 27.488 28.259 30.578 32.801 37.698 16 18.418 19.369 20.465 23.542 26.296 28.845 29.633 32.000 34.267 39.252 17 19.511 20.489 21.615 24.769 27.587 30.191 30.995 33.409 35.718 40.791 18 20.601 21.605 22.760 25 .989 28.869 31.526 32.346 34.805 37.156 42.312 19 21.689 22.718 23.900 27.204 30.144 32.852 33.687 36.191 38.582 43.819 20 22.775 23.828 25.038 28.412 31.410 34.170 35.020 37.566 39.997 45.314 21 23.858 24.935 26.171 29.615 32.671 35.479 36.343 38.932 41.401 46.796 22 24.939 26.039 27.301 30.813 33.924 36.781 37.659 40.289 42.796 48.268 23 26.018 27.141 28.429 32.007 35.172 38.076 38.968 41.638 44.181 49.728 24 27.096 28.241 29.553 33.196 36.415 39.364 40.270 42.980 45.558 51.179 25 28.172 29.339 30.675 34.382 37.652 40.646 41.566 44.314 46.928 52.619 26 29.246 30.435 31. 795 35.563 38.885 41.923 42.856 45.642 48.290 54.051 27 30.319 31.528 32.912 36.741 40.113 43.195 44.140 46.963 49.645 55.475 28 31.391 32.620 34.027 37.916 41.337 44.461 45.419 48.278 50.994 56.892 29 32.461 33.711 35.139 39.087 42.557 45.722 46.693 49.588 52.335 58.301 30 33.530 34.800 36.250 40.256 43.773 46.979 47.962 50.892 53.672 59.702 40 44.165 45.616 47.269 51.805 55.758 59.342 60.436 63.691 66.766 73.403 50 54.723 56.334 58.164 63.167 67.505 71.420 72.613 76.154 79.490 86.660 60 65.226 66.981 68.972 74.397 79.082 83.298 84.58 88.379 91.952 99.608

Table A.6 F-Distribution Probability Table 743 Tabl e A .6 (co nt inued) CriLic al Values of the F-DisLribuLion f o.01(v1, v2) V1 V2 1 2 3 4 5 6 7 8 9 1 4052.18 4999.50 5403.35 5624.58 5763.65 5858.99 5928.36 5981.07 6022.47 2 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99. 39 3 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35 4 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66 5 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16 6 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7. 98 7 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 8 11.26 8.65 7.59 7.01 6. 63 6.37 6.18 6.03 5.91 9 10.56 8.02 6.99 6. 42 6. 06 5.80 5.61 5.47 5.35 10 10 .04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94 11 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63 12 9.33 6.93 5.95 5.41 5. 06 4.82 4.64 4.50 4.39 13 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19 14 8.86 6 .5 1 5.56 5.04 4.69 4.46 4.28 4.14 4.03 15 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89 16 8.53 6.23 5.29 4. 77 4.44 4.20 4.03 3.89 3.78 17 8.40 6.11 5.18 4.67 4.34 4.10 3.93 3.79 3.68 18 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60 19 8.18 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52 20 8.10 5.85 4. 94 4.43 4. 10 3.87 3.70 3.56 3.46 21 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.40 22 7.95 5. 72 4.82 4.31 3.99 3. 76 3.59 3.45 3.35 23 7.88 5.66 4. 76 4.26 3.94 3.71 3.54 3.41 3.30 24 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26 25 7.77 5.57 4.68 4. 18 3. 85 3.63 3.46 3.32 3.22 26 7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3. 18 27 7.68 5.49 4.60 4. 11 3.78 3.56 3.39 3. 26 3.15 28 7.64 5.45 4. 57 4. 07 3. 75 3.53 3.36 3.23 3.12 29 7.60 5.42 4.54 4.04 3. 73 3.50 3.33 3.20 3. 09 30 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07 40 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2. 89 60 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72 120 6.85 4.79 3.95 3.48 3.17 2.96 2. 79 2.66 2.56 00 6.63 4. 61 3.78 3.32 3.02 2.80 2.64 2.51 2 .4 1

744 App e ndix A Statistica l Tables and Proofs Table A.6 (conLinued) C riLi ca l Values of the F-Dis tribuli on f o.0 1(v 1,v2 ) V1 V2 10 12 15 20 24 30 40 60 120 00 1 6055.85 6106.32 6157.28 6208.73 6234.63 6260.65 6286.78 6313.03 6339.39 6365 . 86 2 99.40 99.42 99.43 99.45 99.46 99.47 99.47 99.48 99.49 99.50 3 27.23 27.05 26.87 26.69 26.60 26 . 50 26.41 26.32 26.22 26.13 4 14. 55 14.37 14. 20 14.02 13.93 13.84 13.75 13.65 13.56 13.46 5 10.05 9.89 9.72 9.55 9.47 9.38 9.29 9.20 9.11 9.02 6 7.87 7. 72 7.56 7.40 7.31 7.23 7.14 7.06 6.97 6.88 7 6.62 6.47 6.31 6 .1 6 6.07 5.99 5.91 5. 82 5.74 5.65 8 5. 81 5.67 5.52 5.36 5.28 5.20 5.12 5.03 4.95 4.86 9 5.26 5.11 4.96 4.81 4.73 4. 65 4.57 4.48 4.40 4.31 10 4.85 4.71 4.56 4.41 4.33 4. 25 4.17 4.08 4.00 3.91 11 4.54 4.40 4.25 4. 10 4.02 3.94 3.86 3.78 3.69 3.60 12 4.30 4.16 4.01 3.86 3.78 3.70 3.62 3. 54 3.45 3.36 13 4.10 3.96 3.82 3.66 3.59 3.51 3.43 3.34 3. 25 3.17 14 3.94 3.80 3.66 3.51 3.43 3.35 3.27 3.18 3. 09 3.00 15 3.80 3.67 3.52 3.37 3.29 3.21 3. 13 3.05 2.96 2.87 16 3.69 3.55 3.41 3.26 3.18 3.10 3.02 2.93 2.84 2.75 17 3.59 3.46 3.31 3.16 3.08 3.00 2.92 2.83 2.75 2.65 18 3.51 3.37 3.23 3.08 3.00 2.92 2.84 2. 75 2.66 2.57 19 3.43 3.30 3.15 3.00 2.92 2.84 2. 76 2.67 2.58 2.49 20 3.37 3.23 3.09 2.94 2.86 2.78 2.69 2.61 2.52 2.42 21 3.31 3.17 3.03 2.88 2.80 2.72 2. 64 2. 55 2.46 2.36 22 3.26 3.12 2.98 2.83 2.75 2.67 2.58 2.50 2.40 2.31 23 3.21 3.07 2.93 2.78 2.70 2. 62 2.54 2.45 2.35 2. 26 24 3.17 3.03 2.89 2.74 2.66 2.58 2.49 2.40 2.31 2.21 25 3.13 2.99 2.85 2. 70 2.62 2.54 2.45 2.36 2. 27 2.17 26 3.09 2.96 2.81 2.66 2.58 2.50 2.42 2.33 2.23 2.13 27 3.06 2.93 2. 78 2.63 2.55 2.47 2.38 2.29 2.20 2.10 28 3.03 2.90 2.75 2.60 2 .5 2 2.44 2. 35 2. 26 2.17 2.06 29 3.00 2.87 2.73 2.57 2.49 2.41 2.33 2.23 2.14 2.03 30 2.98 2.84 2.70 2.55 2.47 2.39 2.30 2.21 2.11 2. 01 40 2.80 2.66 2.52 2.37 2.29 2.20 2.11 2.02 1.92 1. 80 60 2.63 2.50 2.35 2. 20 2. 12 2. 03 1.94 1.84 1. 73 1.60 120 2.47 2.34 2.19 2.03 1.95 1.86 1.76 1.66 1.53 1. 38 00 2. 32 2.18 2.04 1. 88 1.79 1.70 1.59 1.47 1.32 1.00

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