Answered: A population of values has a…

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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Question

A population of values has a distribution with μ=233.5μ=233.5 and σ=4.8σ=4.8. You intend to draw a random sample of size n=157n=157.

According to the Central Limit Theorem:

(a) What is the mean of the distribution of sample means?

μ¯x=

(b) What is the standard deviation of the distribution of sample means?
(Report answer accurate to 2 decimal places.)

σ¯x=

(c) In a random sample of n=157, what is the probability that its sample mean is more than 233.4? Round to three decimal places.

(d) In a random sample of n=157, what is the probability that its sample mean is less than 233.5? Give your answer to three decimal places.

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 + ...

Expert Solution

Step 1

Given,

$μ = 233.5 σ = 4.8 n = 157$

Using Central limit theorem,

a). The required mean of the distribution of the sample means is obtained as-

$μ_{x} = 233.5$

b). The required standard deviation of the distribution of sample means is obtained as-

$\begin{array}{rcl} σ_{x} & = & \frac{σ}{\sqrt{n}} \\ = & \frac{4.8}{\sqrt{157}} \\ = & 0.38 \end{array}$

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