Answered: The fingerprint of every finger can be classified into one of three basic patterns: a loop, a whorl, or an arch. Forensic experts have determined that about…

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Description: The fingerprint of every finger can be classified into one of three basic patterns: a loop, a whorl, or an arch. Forensic experts have determined that about 65% of all fingers have loops, 30% have whorls, and 5% have arches. Suppose that these perc

MATLAB: An Introduction with Applications

6th Edition

ISBN: 9781119256830

Author: Amos Gilat

Publisher: John Wiley & Sons Inc

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2-Proportion Test

In a 2-proportion test, two separate proportions are measured at the same time, and their similarity is evaluated. It is also known as a 2-proportion z-test, and a comparison of two different proportions is done in this method.

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The fingerprint of every finger can be classified into one of three basic patterns: a loop, a whorl, or an arch. Forensic experts have determined that about 65% of all fingers have loops, 30% have whorls, and 5% have arches. Suppose that these percentages are exactly correct and patterns are independent from one finger to the next. Select a person at random and classify each of their fingerprints on both hands. Let X = the number of fingerprints with a loop.

(a) Calculate and interpret the mean of X.

(b) Calculate and interpret the standard deviation of X.

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

Binomial Distribution :

A binomial experiment is a statistical experiment that has the following properties:

The experiment consists of n repeated trials.
Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
The probability of success, denoted by P, is the same on every trial.
The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.

Notation :

The following notation is helpful, when we talk about binomial probability.

x: The number of successes that result from the binomial experiment.
n: The number of trials in the binomial experiment.
P: The probability of success on an individual trial.
_nC_r: The number of combinations of n things, taken r at a time.

A binomial random variable is the number of successes x in n repeated trials of a binomial experiment. The probability distribution of a binomial random variable is called a binomial distribution.

The binomial distribution has the following properties:

The mean of the distribution (μ_x) is equal to n * P .
The variance (σ²_x) is n * P * ( 1 - P ).
The standard deviation (σ_x) is sqrt[ n * P * ( 1 - P ) ].

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