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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- Suppose that Dr. Grandtete does some measurements and states that there is a correlation (r) of +1.2 between head circumference and 1.Q. What can you conclude from this result? That head circumference and IQ are very strongly and positively related That head circumference and IQ are weakly, but positively, related That there is a mistake because r cannot equal +1.2 That there is a mistake becauser can only be used to measure psychological variables, not physical ones like head circumference.suppose you and a friend each take different random samples of data pairs (x,y) from the same popuation. Assumethe samples are the same size. Based on your samples, you compute r=0.83. Based on her sample, your friend computes r=0.79. Is your friends value for r wrong?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)
- Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal. The table available below shows the prices (in dollars) for a sample of automobile batteries. The prices are classified according to battery type. At α=0.10, is there enough evidence to conclude that at least one mean battery price is different from the others? Complete parts (a) through (e) below.A computer while calculating correlation coefficient between two variables X and Y from 25pairs of observation obtained the following results: n=25, 2X=125, EX2 = 650, Y=100,Y2= 460, 2XY=508. But on subsequent verification it was found that he had copied downtwo pairs as (6,14) and (8,6) while the correct values were (8,12) and (6,8). Obtain the correctvalue of correlation coefficient?5. 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.The following data show the number of class sessions missed during a semester of SOC221 and the final grade for a sample of 8 students selected at random. Number of Sessions Missed Final Grade (x) (y) 0 96 2 88 12 68 6 91 8…Let Y=5X+10 and X be normally distributed with a mean 10 and variance 25. Find the following: I). P(Y<54) II). P(Y268) III). P (525Y567)
- 4, 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 choice7. Let X₁, X2, ..., Xn be a random sample from normal distribution with mean and standard deviation o. (a) Find the maximum likelihood estimators of u and ² μ (b) Show that, the estimator of μ is unbiased but the estimator of o2 is biased (c) Compute the bias of the estimator of o2 and comment on it. Hence, write a function of this estimator that is unbiased, find its variance and show that, it is consistent. (d) Find the Cramer-Rao lower bound variance of o² and hence find the absolute efficiency of the unbiased estimator you proposed in (c).
