bl To claim that a certain die was fair, 1000 rolls of the die were recorded with the following results Outcome #Occurre

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Bl To Claim That A Certain Die Was Fair 1000 Rolls Of The Die Were Recorded With The Following Results Outcome Occurre 1 (69.82 KiB) Viewed 19 times

Bl To Claim That A Certain Die Was Fair 1000 Rolls Of The Die Were Recorded With The Following Results Outcome Occurre 2 (55.9 KiB) Viewed 19 times

Bl To Claim That A Certain Die Was Fair 1000 Rolls Of The Die Were Recorded With The Following Results Outcome Occurre 3 (70.84 KiB) Viewed 19 times

bl To claim that a certain die was fair, 1000 rolls of the die were recorded with the following results Outcome #Occurrences 1 159 2 175 3 160 4 183 5 163 6 160 -n. Test the hypothesis that the die is fair at a = 0.05 significance. Proceed through the following steps. a) State and rigorously formulate the hypotheses H, and H. b) Draw an expanded table. c) Find q and the critical region at a = 0.05. Then make a conclusion. d) Find the P-value and interpret your result. Hint. Use the following formula q = 33 - n. (40 pts) b2 The number of students enrolled in calculus classes at FIT is a Poisson r.v. with parameter X = 144. Use the CLT approximation to find the probability that the new enrollment is going to be 168 or more students. Also show what the exact solution would have been without the CLT approximation. Explain your steps. Hint. Use Table 1 (Gaussian PDF). (35 pts) b3 When a signal having value is transmitted from node A, the value received at node B is normally distributed with unknown mean y and unknown variance o. In other words, when the signal is sent, then its value received is +W. where W represents a Gaussian noise with parameters (0,0%). To reduce the error, 10 signals of the same value je are sent. Upon their receipt at node B, their values were recorded as 7, 12, 13, 15, 7, 14, 6, 9, 16, 11 Construct a 95% confidence interval for . Explain your steps. Hint. Find the sample variance. (40 pts) b4 A coin is tossed until a head occurs and the number X of tosses is recorded. After repeating the experiment 256 times the following results were obtained: Îi 1 2 3 4 5 6 7 8 ni 128 64 32 14 10 3 4 1 where î; is the number of tosses needed to obtain the first head and n, is the number of experiments in which I occurs. Test the hypothesis at the 0.05 significance that the observed distribution of X is geometric with parameter p= . Proceed through the following steps. a) State and rigorously formulate the hypotheses H, and H. b) Form an expanded table. c) Find q and critical value c. d) Interpret your result. e) Find the P-value and interpret your result. Explain your steps. (40 pts)

b5 0 91 It is conjectured that the number of wrong telephone connections is Poisson with parameter X=0.4. A total of 130 days of observations produced the following results: # wrong connections days observed true Poisson distr, with X = 4 0.6703 1 30 0.2681 7 0.0536 2 0.0072 4 0.0007 0.0001 total 130 1.0 2 3 0 0 5 (1) Test the null hypothesis H, that the daily number of wrong telephone connections is indeed Poisson with X = 0.4 at significance level a = 0.05 against the alternative hypothesis H, that it is not. (ii) Find the P-value and make your inference (conclusion). Proceed through the following steps. a) State and rigorously formulate the hypotheses H, and H. b) Draw an expanded table. c) Find q and d) the critical region at a = 0.05. e) Then make a conclusion. 1) Find the P-value. g) Interpret your result. Explain your steps Hint. Use the following formula (40 pts) b6 A sample of 248 people was randomly chosen and were individually identified as to their gender and political affiliation, Democrator Republican. The results were placed in the following table: genderlaffiliation Democrat Republican Women 56 Men 52 72 68 (1) State and rigorously formulate the hypotheses testing. Test the hypothesis that the gender and political affiliation are independent at a = 0.05. (ii) Form an expanded contingency table. () Give the critical region. (i) Calculate and find if falls into the critical region. () Interpret your result. (vi) Find the P-value (usi) Interpret the result. Explain your steps Hint. Use the following formula - ( (40 pts) b7 A random sample of 2100 death certificates of adults were examined in a large metropolitan area and showed the following results: Un 3-1) Cause of Death Heavy Smoker Moderate Smoker Nonsmoker Respiratory Disease 215 72 800 Heart Disease 120 50 843

Test the hypothesis that the cause of death is independent of a person's smoking habit. a) Rigorously formulate hypotheses testing. b) Form an expanded contingency table c) Calculate 9 d) Find the P-value and make your judgment based on the P-value. Explain your steps. Hint. Use the following formula RC 4 = n (Σ Σ - 1). (45 pts) 1) = -(*+*+ ** + + + - n n. n I. b8 Consider a parallel system of two independent unreliable machines. What is the probability that machine 1 fails before machine 2 if the life time X, of machine i is exponential with parameter 5, i = 1, 2? Also draw the integral area. Explain your steps. (30 pts) b9 To test a proportion of defective items one draws a sample X = (X1,...,xn) from a Bemoulli population with an unknown 0 € (0,1). (1) Give the likelihood function of the sample. (ii) Find the m.l.e. and MLE of 0. Explain your steps. (30 pts) b10 Find the mean residual time y(t) = E[X – t|X > t of a serial system of two independent and identically distributed lifetimes of its components, each having the common exponential distribution with parameter 1. Hint. Use formula (+) (t) = Ro SR(u)du. (30 pts) bil Let f(x,y) = { 122*, Osteri elsewhere be the joint pdf of r.v.'s (X,Y). Find the conditional pdf f(x|y). Explain your steps. (25 pts) b12 Using Bayes analysis investigate the proportion of defective items in a large manufactured lot, with its prior density assumed to be uniform in (0,1). Find the posterior pdf E(0|11,..., In). Hint. Use the formula E(3|a, 3) = Flore30-4(1 - 0)3-110,1)(3) for the beta pdf. Explain your Tar (5) steps. (45 pts)