Answered: Discuss the effect of following on 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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Discuss the effect of following on the width of a confidence interval.

a. Confidence level

b. Sample size

c. Sample mean

d. Population standard deviation (sigma)

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

The confidence interval can be calculated as:

$C I = X \pm z \times \frac{σ}{\sqrt{n}}$

Here,

$X$ = Sampmean

σ = Population standard deviation

n = Sample size

Answer (a)

The critical value z increases, as the confidence level increases. It can be seen from the above-mentioned formula that the confidence level is directly related to the width of the confidence interval, so if we increase the value of the confidence level, the width of the confidence interval increases. Similarly, the critical value z decreases, as the confidence level decreases. Hence, if we decrease the value of the confidence level, the width of the confidence interval decreases. Hence, confidence level affects the width of the confidence interval.

Answer (b)

It can be seen from the above-mentioned formula that if we decrease the sample size, the width of the confidence interval increases that is a decrease in sample size increase the value of margin of error because the sample size is inversely related to the width of the confidence interval. And, if we increase the sample size, the margin of error decreases; that is, an increase in sample size decreases the width of the confidence interval.

Hence, the sample size affects the width of the confidence interval.

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