Compute (a) ?(? = 1), (b) ?(? ≥ 4), and (c) ?(1 ≤ ? ≤ 3).
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2. Each of 12 refrigerators of a certain type has been returned to a distributor because of the presence of
a high-pitched oscillating noise when the refrigerator is running. Suppose that four of these 12 have
defective compressors and the other eight have less serious problems. If they are examined in random
order, let ? be the number among the first six examined that have a defective compressor. Compute (a)
?(? = 1), (b) ?(? ≥ 4), and (c) ?(1 ≤ ? ≤ 3).
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- *11. Show that for the two-tailed test, the p-value is: Vn(Y – 60) 2 1- ¢If ē , and 6, are two unbiased estimators of 0, w, 0,+ W, 0, is an unbiased 2 estimator of 0 where W, + W, = 1 %3D 1D Question 8 Each of 12 refrigerators of a certain type has been returned to a distributor because of the presence of a high-pitched oscillating noise when the refrigerator is running. Suppose that 5 of these 12 have defective compressors and the other 7 have less serious problems. If they are examined in random order, let X = the number among the first 6 examined that have a defective compressor. Compute P(X= 2). O something else O .4011 O.3788 O.3902 O 4255
- 3. An integer is randomly chosen from the set 1, 2,..., 100. If this integer is divisible by 2 we let Y = 0, and if it is not divisible by 2 but is divisible by 3 then Y = 1. We let Y = 2 in all other cases. Find mean and variance of Y.Suppose Aaron recently purchased an electric car. The person who sold him his new car told him that he could consistently travel 200 mi before having to recharge the car's battery. Aaron began to believe that the car did not travel as far as the company claimed, and he decided to test this hypothesis formally. Aaron drove his car only to work and he recorded the number of miles that his new car traveled before he had to recharge its battery a total of 21 separate times. The table shows the summary of his results. Assume his investigation satisfies all conditions for a one-sample t-test. Mean miles traveled Sample size t-statistic P-value п P 185 21 -0.65 0.262 The results statistically significant at a = 0.05 because P 0.05. are are not >Question 5. A professor decides to run an experiment to measure the effect of time pressure on final exam scores. He gives each of the 400 students in his course the same final exam, but some students have 90 minutes to complete the exam while others have 120 minutes. Each students is randomly assigned one of the examination times based on the flip of a coin. Let Y, denote the number of points scored on the exam by the ith student (0 ≤ Y ≤ 100), let X, denote the amount of time that the student has to complete the exam (X, = 90 or 120), and consider the regression model Yi = Bo + BiXitu. (1) Explain what the term u, represents. Why will different students have different values of ui? (2) Explain why E(u₁|X₁) = 0 for this regression model. (3) The Least Squares Assumptions Yi Bo + B₁Xi + Ui, i = 1,..., n, where I The error term u, has conditional mean zero given X₂: E(ui Xi) = 0; II (X₁, Yi), i 1,, n, are independent and identically distributed (i.i.d.) draws from their joint…
- Consider the following scenario: • Let P(C) = 0.3• Let P(D) = 0.8• Let P(C|D) = 0.3 A. P(C AND D) = [ Select ] ["0.30", "0.24", "0.26", "0.11"] B. Are C and D Mutually Exclusive? [ Select ] ["No, they are not Mutually Exclusive.", "Yes, they are Mutually Exclusive."] C. Are C and D independent events?[ Select ]["No, they are Dependent.", "Yes, they are Independent."] D. P(C OR D) = [ Select ] ["0.92", "0.86", "0.60", "1.1"] E. P(D|C) = [ Select ]["0.30", "0.95", "0.24", "0.80"]On each trial of an experiment, a subject is presented with a constant soft noise, which is interrupted at some unpredictable time by a noticeably louder sound. The time it takes for the subject to react to this louder sound is recorded. The following list contains the reaction times (in milliseconds) for the 19 trials of this experiment: 171, 212, 285, 175, 202, 269, 193, 217, 164, 231, 261, 236, 186, 257, 271, 229, 250, 296, 226 Send data to calculator Find 20th and 75th percentiles for these reaction times. (If necessary, consult a list of formulas.) (a) The 20th percentile: milliseconds (b) The 75th percentile: millisecondsFor Numbers 4 and 5: Dr. Kae Dee, a resident cardiologist who intends to specialize in vascular medicine, wants to verify a claim from a study which states that chronic venous insufficiency affects about 1 in 20 adults. She obtained records from 40 randomly selected hospitals from municipalities with relatively similar population sizes and obtained the number of reported chronic venous insufficiency cases. Let X be the number of reported chronic venous insufficiency cases. 4. Which of the following is(are) TRUE? 1. X is a discrete random variable with possible values x = {0, 1, 2, 3, ..., 40) II. The sum of probabilities of all values of X is 1. A. I only 5. Presented below is the probability distribution table from Dr. Kae Dee's study. X=x P[X=x] A. 1.00 B. II only 19 0.22 20 0.38 B. 4.10 C. Both I and II 21 0.11 The expected number reported chronic venous insufficiency cases is 22 C. 20.57 D. Neither I nor II 23 0.10 on the average. D. 105
- 2. Suppose one has n non-degenerate random variables X1, X2,, X, so that X1+ X2 + · · · + Xn = L for some constant L. (Recall that a random variable is non-degenerate if it is not a constant in disguise.) Show that there must be at least one pair of indices i j so that p(X;, X;) < 0.Q.3. Compute the probability that: (a) their sum is odd; (b) their product is even. Three distinct integers are chosen at random from the first 15 positive integers.