Answered: For a sample of n=36, find the…

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

The population mean and standard deviation are given below. Find the indicated probability and determine whether a sample mean in the given range below would be considered unusual. If convenient, use technology to find the probability.

For a sample of

n=36,

find the probability of a sample mean being less than

12,749

or greater than

12,752

when

μ=12,749

and

σ=1.2.

For the given sample, the probability of a sample mean being less than

12,749

or greater than

12,752

nothing.

(Round to four decimal places as needed.)

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

Let $\bar{x}$ denote the sample mean which follows normal distribution. The sample size is 36, mean is 12,749, and standard deviation is 1.2.

The probability of a sample mean being less than 12,749 or greater than 12,752 is,

$\begin{array}{rcl} P (\bar{x} < 12, 749 o r \bar{x} > 12, 752) & = & P (\frac{\bar{x} - μ}{(\frac{σ}{\sqrt{n}})} < \frac{12, 749 - 12, 749}{(\frac{1.2}{\sqrt{36}})} o r \frac{\bar{x} - μ}{(\frac{σ}{\sqrt{n}})} > \frac{12, 752 - 12, 749}{(\frac{1.2}{\sqrt{36}})}) \\ = & P (z < \frac{0}{0.2} o r z > \frac{3}{0.2}) \\ = & P (z < 0 o r z > 15) \\ = & P (z < 0) + P (z > 15) \\ = & P (z < 0) + 1 - P (z < 15) \end{array}$

The probability of z less than 0 can be obtained using the excel formula “=NORM.S.DIST(0,TRUE)”. The probability value is 0.50.

The probability of z less than 15 can be obtained using the excel formula “=NORM.S.DIST(15,TRUE)”. The probability value is 1.

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