8.12 The reading on a voltage meter connected to a test circuit is uniformly distributed over the interval (0, +1), where 0 is the true but unknown voltage of the circuit. Suppose that Y₁, Y2,..., Y, denote a random sample of such readings. a Show that Y is a biased estimator of and compute the bias. b Find a function of Y that is an unbiased estimator of 0. C Find MSE(Y) when Y is used as an estimator of 0.
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- 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.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. 2) Sunnogo that the random vario obcorvation tributionA 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…
- 3. (Sec 5.5) Suppose the average weight of 25-year-old males is 182 pounds with a standard deviation of 40 pounds. Select a random sample of size 45 from among a large group of 25 year old males. Find P(175For a given set of bivariate data, the following result n²100, EX= 5000; EY : 10000; EX2 - 26000; =1010000 and EXY =516000. Ci) Find the predicted value of Y when x = 60. (i) what is the predicted value of the x when Y = 80 (1) Calcutule mie coefficint of Correlation between & and Y. 2ملاعMove to Archive x 0.3e If 1-.7et is the mgf of a well known function. It is Geometric mgf i) Its name is ii) The mean of this distribution = iii) P(x = 4) = iv) The variance of this distribution is5. Find the method of moments estimator of a based on a random sample from a gamma distribution with B=2. XGAM(a,2) www.5. Let X1, X2, . ,Xn be i.i.d. samples distributed according to Geo(q). Consider the sample mean X = 1EX; as an estimator for the mean u = E[X;]. (a) What is the value of u in terms of q? (b) What is the bias of the estimator? Show explicitly by using the linearity of expectation. What is the variance of the estimator? What is its standard error? Write the expressions in terms of q, n. (d) What is the mean square error of the estimator? Write the expression in terms of q, n.6. An optical device is used to detect the passage of cars in a single lane of a downtown street. Because there must be at least half a second between successive cars, it is assumed that the times T; between = 0.50 + Si, where S1, S2, ··. are independent exponential (A) random cars are of the form T; variables. (a) Find the mean and variance of each T;. (b) Let Y, be the time at which the nth car passes the detector. Calculate the mean and variance of Yn. (c) Under what conditions is Yn approximately normally distributed and why? (d) When n = 50 and A = 0.10, calculate the approximate probability that Yn exceeds 500 seconds.3. Table 1 shows the median rent per m² in Berlin, for each of the 12 districts, and for all market segments, for the years 2020 (variable ✗) and 2021 (variable Y). For these data, the following is known: 12 IM i=1 (b) 12 12 12 12 for π; = 124.31, Σy = 127.93, Στ = 1326.6657, Σy = 1408.3461, Σriyi = 1365.7266 i=1 Calculate and y. 2=1 i=1 i=1 Calculate the sample variance for X and Y. Hint: For the sample variance the following formula holds: (c) (d) n Σ? - 2 Σ + n n 1 S = n-1 i=1 i=1 Calculate the sample standard deviation for X and Y. Calculate the covariance and the correlation coefficient. Recall: The formula for the correlation coefficient is suitable SXY TXY = SXSY4, 12. How Call v. 13. Suppose that two raters (Rater A and Rater B) each assign pirysica attractiveness scO = not at all attractive to 10 = extremely attractive) to a set of seven facial photogranhe 21 %3D Pearson's r can be used as an index of interrater reliability or agreement on quantitative ratings. A correlation of +1 would indicate perfect rank-order agreement between rate while an r of 0 would indicate no agreement about judgments of relative attractiveness An r of .8 to .9 is considered desirable when reliability is assessed. For this example, Rater A's score is the X variable and Rater B's score is the Y variable. The ratings are as followe F follows: Photo Rater A Rater B 1 3. 5. 8. 6. 7. 8. 6. 4. 6. 10 6. 7. 5. 4. a. Compute the Pearson correlation between the Rater A and Rater B attractiveness ratings. What is the obtained r value? b. Is your obtained r statistically significant (using a =.05, two tailed)? Are the Rater A and Rater B scores "reliable"? Is there good or…Consider the model Y = B1 + B2X + e, which we estimate using a random sample with 12 observations. Let bj and b2 be the estimators for B1 and B2 and recall Eê, where ê; = Y; – bị – b2X;. Suppose the sample correlation between {X; }"_-1 and {Y;}"_, is 0.5 and E(Y; – Ỹ„)² = 100. What is ổ?? %3D n-2 Hint: (i) for simple regression, the regression R? is equal to the squared sample correlation between X and Y. (ii) R² = 1 – SSE where SSE = Eế and SST SST = O a. 7.5 O b. 8.5 О с. 9 O d. 10 O e. 8 O f. 7 Clear my choiceSEE MORE QUESTIONS
