Answered: population has a mean equal to 80 and 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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Suppose a population has a mean equal to 80 and a standard deviation equal to 6; also, suppose that the population is not extremely skewed. If one is to draw a random sample of 36 observations from this population, then what is the probability that the sample mean will be larger than 82?

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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Step 1

Population has a mean equal to 80 and a standard deviation equal to 6.

i.e. µ=80 and σ=6, Random sample is drawn of 36 observations.

The probability that sample mean will be larger than 82 can be calculated using central limit theorem.

Central limit theorem states that, X₁, X₂, …, X_nis a random sample from probability distribution with mean(µ) and variance(σ²) then probability distribution of $Z = \frac{X - μ}{σ / \sqrt{n}}$ converges to N(0,1) as n tends to infinity.

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