2) The reading on a voltage meter connected to a test circuit is uniformly distributed over the interval (0,0+1), where is the true but unknown voltage of the circuit. Suppose that Y₁,..., Yn denotes a random sample of such readings. a) Show that Y is a biased estimator of 0, and compute the bias. b) Find a function of Y that is an unbiased estimator of 0.
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- Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data are at or below for respiratory rates in breath per minute during the first 3 years of infancy are given by y=101.82411-0.0125995x+0.00013401x2 for awake infants and y=101.72858-0.0139928x+0.00017646x2 for sleeping infants, where x is the age in months. Source: Pediatrics. a. What is the domain for each function? b. For each respiratory rate, is the rate decreasing or increasing over the first 3 years of life? Hint: Is the graph of the quadratic in the exponent opening upward or downward? Where is the vertex? c. Verify your answer to part b using a graphing calculator. d. For a 1- year-old infant in the 95 th percentile, how much higher is the walking respiratory rate then the sleeping respiratory rate? e. f.EX7.8) Let Y be a random variable having a uniform normal distribution such that Y U(2,5) 2 Find the variance of random variable Y.Consider a set of data (x;, y;), i = 1,2, .,n and an intercept model given by: 1. Yi = Bo + Ei with E(s;) = 0, Var(ɛ;) = o? and all ɛ,'s are independent and normally distributed. Derive the standard error of the point estimator Bo.
- Suppose that @, and ô, are unbiased estimators of the parameter 0 and that V@,) = 15 and V(@2) = 4. What is the relative efficiency of the two estimators?The value of the correlation r = (x,y) for the information would be: n = 8, Sum(x) = 609, Sum(y) = 504, Sum (x)^2 = 47995, Sum (y)^2 = 32954, and Sum (xy) = 395650 Choose answer from the following: a. 0.855 b.0.174Let X ~ U(a,b). a Derive X mean and variance. b Use the method of moments to construct an estimator for a and for b. c Check whether those estimators are unbiased.
- Let I be an indicator variable with p(I) = {p , I=1 } {1-p , I=0}. Show that the variance of I is p(1-p)obtain the MVB estimator for u in normal population №.(482), where 2 is knowAn environmental chemist is performing a study of iron in atmospheric particulate measured downwind from a steel mill. She is concerned that wind velocity at the time of measurement may affect the readings. So, she decided to obtain observations on 30 randomly chosen days during the period of peak operation of the mill. She then, compares measurements taken on days when the wind is calm (velocity ≤ 5 knots) with measurements taken on windy days (velocity > 5 knots). The data are presented as the following: Table 1: Measurement of iron Calms Days Windy Days 0.68 0.74 0.88 0.25 0.29 0.3 0.43 0.45 0.5 0.89 0.97 1 0.65 0.91 0.92 0.93 0.95 1.01 1.17 1.25 1.27 0.69 0.74 0.8 0.87 0.87 0.89 8.85 1.54 0.6 1.03 1.16 By using SPSS or Minitab only: (1) Generate a descriptive table and summarize the data. (ii) Synthesize and discuss the different on the iron level between two days using boxplots. (iii) Propose the appropriate test to the study.
- A random sample of n, 20 winter days in Denver gave a sample mean pollution index x, = 43. Previous studles show that o,- 10. For Englewood (a suburb of Denver), a random sample of n, - 19 winter days gave a sample mean pollution Index of x, = 37. Previous studies show that g, -13. Assume the pollution Index is normally distributed in both Englewood and Denver. Do these data indicate that the mean population pollution index of Englewood is different (either way) from that of Denver in the winter? Use a 1% level of significance. (a) what is the level of significance? State the null and alternate hypotheses. (b) what sampling distribution will you use? What assumptions are you making? O The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations. O The Student's t. We assume that both population distributions are approximately normal with known standard devtations. O The standard normal. We assume that both population…EXER 6.3 Find the covariance and the correlation coefficient between X and Y, if X and Y are jointly discrete random variables, with joint PMF given by: SHOW SOLUTIONS X\Y 0 1 6 0 28 6 1 28 2 0 333333 28 28 28 2120 28 04. Suppose you are given the following information: = 49.6870 – 2.1586X; r2 = 0.9757 (0.7463) (0.12113) df=8 T= (66.578) (-17.821) p-value = (0.000) (0.000) a) Conduct a two tailed hypothesis test that: i) The intercept is 49 and the slope coefficient is -2, use 1% significance. ii) Confirm your deductions to 4(a) (i) using a confidence interval approach b) What does the r2 tell you? c) Perform a hypothesis using r2, such that p=D0, what do you conclude?