Using a long rod that has length μ, you are going to lay out a square plot in which the length of each side is μ. Thus the area of the plot will be μ2 However, you do not know the value of μ, so you decide to make n independent measurements X1, X2,..., Xn of the length.Assume that each Xi has mean μ (unbiased measurements) and variance σ2. a. Show that X2 is not an unbiased estimator for u?. [Hint: For anyrv Y, E(Y) = V(Y) + [E{Y)]°. Apply this with Y= X.]b. For what value of kis the estimator X2 - ks unbiased for u??[Hint: Compute E( X - ks²).]

If an expected value of sample statistic is equals to the population parameter then that…

Answered: a. Show that X2 is not an unbiased…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter2: Equations And Inequalities

Section2.7: More On Inequalities

Problem 44E

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Using a long rod that has length μ, you are going to lay out a square plot in which the length of each side is μ. Thus the area of the plot will be μ²
However, you do not know the value of μ, so you decide to make n independent measurements X₁, X₂,..., X_n of the length.
Assume that each Xi has mean μ (unbiased measurements) and variance σ².

a. Show that X2 is not an unbiased estimator for u?. [Hint: For any
rv Y, E(Y) = V(Y) + [E{Y)]°. Apply this with Y= X.]
b. For what value of kis the estimator X2 - ks unbiased for u??
[Hint: Compute E( X - ks²).]

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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