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- Let Y₁, Y2, ..., Yn be a random sample from a population with a Gamma distribution with parameters a = 3 and unknown. Find the Minimum Variance Unbiased Estimator for B. Show your work, and explain how you know your estimator is the MVUE.Show that if X₁ and X₂ are standard normal variate with correlation cofficient p between them, then the correlation cofficient X and X2 is given by p².Suppose Y₁, Y2, …‚ Yn is an iid sample from a Poisson(X) distribution, where À > 0. Is Ÿ a consistent estimator of X? Explain.
- Explain at least two differences between the following two point estimations of the population variance. 22 = Σ'1(X, – X)2 n-1 62 _ Σ 1 (X - X)2 =1 = nQ3) a) What is Mean Squared error (MSE) of an estimator ô ? Show that MSE (Ô) = Variance (Ô ) + (Bias ( ))? b) Suppose that X denotes the mean of a random sample of size n from a normal population with mean u and variance of whereas Y denotes the mean of a random sample of size n from a normal population with mean u and variance ož .Let the estimator for µ be a = aX +(1-a)Y i) Show that A is an unbiased estimator of µ. ii) For what value of a would the variance of this estimator be a minimum? c) Let X be a continuous random variable that is uniformly distributed over (0, B). Find the moment estimator of ß.13. The population relationship between household food expenditure, household income and household size is log(FoodExpenditure) = a + B, log(HH _Income)+ B,HH _size +u . Suppose you are interested in estimating this population relationship. In which of the following cases OLS estimator of B and B2 will be biased, assume what's described is the only problem? a) The variance of u is correlated with log(HH_Size). b) The mean of u depends on log(HH_ Size). c) The variance of u is large. d) Both a) and b) will cause biased estimates of the coefficients.
- 1. You plan to use a rod to lay out a square, each side of which is the length of the rod. The length of the rod is μ, which is unknown. You are interested in estimating the area of the square, which is μ^2. Because μ is unknown, you measure it n times, obtaining observations X1, X2,..., Xn. Suppose that each measurement is unbiased for μ with variance σ^2. a. Is X^2 a biased or an unbiased estimate of the area of the square? b. Find an unbiased estimate of the area of the square.2. The average zone of inhibition (in mm) for mouthwash L as tested by the medical technology students has been known to be 9mm. A random sample of 10 mouthwash L was tested and the test yielded an average zone of inhibition of 7.5mm with a variance of 25 mm. Is there enough reason to believe that the anti-bacterial property of the mouthwash has decreased? Test the hypothesis that the average zone of inhibition of the mouthwash is no less than 9mm using 0.05 level of significance. A. State the hypotheses. B. Determine the test statistic to use. C. Determine the level of significance, critical value, and the decision rule. D. Compute the value of the test statistic. E. Make a decision. F. Draw a conclusion.11. 11. In a part of an oil refinery, oil flows through three meters A, B and C which give readings of flow that are subject to errors with zero mean and a common variance. The random variables X,Y and Z denotes the observed readings respectively for these meters while the true (unknown) readings are 01, 02 and 03 respectively with 03 = }(0, + 02). Thus E(X) = 61, E(Y) = 02 E(Z) = 2) = ;(0, + 02)
- Let Y₁, Y2, Y3, Y4, Y5 and X₁, X₂,..., X, be independent and normally distributed random samples from populations with means ₁ = 2 and ₂ = 8 and variances ₁²=5 and 0₂² = k₁ respectively. Suppose that P(X-Y> 10) = 0.02275, find the value of 0₂² = k.Suppose X1, X2,...,X, from a real-valued random variable X with unknown mean u and standard deviation o. What is the estimator of the signal to noise ration u/o.?The table below shows a sample of waiting times (in days) for a heart transplant for two age groups. At α=0.05, can you conclude that the variances of the waiting times differ between the two age groups?