Answered: A population of values has a normal…

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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A population of values has a normal distribution with μ=186.8 and σ=70.5. You intend to draw a random sample of size n=41. Please show your answers as numbers accurate to 4 decimal places.

Find the probability that a single randomly selected value is between 197.8 and 198.9.
P(197.8 < X < 198.9) =

Find the probability that a sample of size n=41 is randomly selected with a mean between 197.8 and 198.9.
P(197.8 < x < 198.9) =

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

A z-score represents the position of a data point in terms of its distance from the mean when measured in standard deviation units, i.e, how far the observation is from the mean in units of standard deviation.

The z of a variable (x) is the ratio of the difference of mean $μ$ from the data point (x) by the standard deviation $σ$ , i.e, $z = \frac{x - μ}{σ}$

For x=197.8

$\begin{array}{rcl} z & = & \frac{197.8 - 186.8}{70.5} \\ = & 0.16 \end{array}$

For x=198.9

$\begin{array}{rcl} z & = & \frac{198.9 - 186.8}{70.5} \\ = & 0.17 \end{array}$

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