Answer the following questions with TRUE or FALSE, and explain your answer:(a) Let a random sample of independent X1 , . . . Xn be taken from a population with mean µx , variance σ2x . Then, the variance of X bar, (the sample mean) will always be less than or equal to the variance of X.(b) If X and Y have correlation 0.50, then the correlation between 4X and 5Y will not be 0.50.(c) For n dependent random variables X1 , . . . , Xn , drawn from the same population with mean µx ,E[X bar] = µx .(d) If X has a variance of 4 and mean of 2, and a sample mean based on 4 independent Xi wascalculated, the probability that X will be larger than 8 is larger than the probability that X bar islarger than 8.***Note: (Please do not copy answer from Chegg!)

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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Answer the following questions with TRUE or FALSE, and explain your answer:

(a) Let a random sample of independent X₁ , . . . X_n be taken from a population with mean µ_x , variance σ²_x. Then, the variance of X bar, (the sample mean) will always be less than or equal to the variance of X.

(b) If X and Y have correlation 0.50, then the correlation between 4X and 5Y will not be 0.50.

(d) If X has a variance of 4 and mean of 2, and a sample mean based on 4 independent X_i was
calculated, the probability that X will be larger than 8 is larger than the probability that X bar is
larger than 8.

***Note: (Please do not copy answer from Chegg!)

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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Hello! As you have posted 4 different questions, we are answering the first question. In case you require the unanswered question also, kindly re-post them as separate question.

Given information:

Let a random sample of independent X₁ , . . . X_n be taken from a population with mean µ_x , variance σ²_x

From the central limit theorem,

The mean and variance of sample mean is,

$E (\bar{x}) = μ_{x} V (\bar{x}) = \frac{σ_{x}^{2}}{n}, h e r e, n = s a m p l e s i z e .$