Nihar Ranjan Roy

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Nihar Ranjan Roy

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Business Statistics, By Ken Black, Wiley India Edition

Text Book

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In this section we will cover

Uniform Distribution

Normal Distribution

Coverage

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The uniform distribution, sometimes referred to as the rectangular

distribution, is a relatively simple continuous distribution in which the same height, or f (x), is obtained over a range of values.

The following probability density function defines a uniform distribution.

uniform distribution

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Mean and Standard Deviation of Uniform Distribution

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With discrete distributions, the probability function yields the value of the probability.

 For continuous distributions, probabilities are calculated by determining the area over an interval of the function.

With continuous distributions, there is no area under the curve for a single point.

The following equation is used to determine the probabilities of x for a uniform distribution between a and b.

= −

Where −

a≤x₁ ≤x₂ ≤b

Determining Probabilities in a Uniform Distribution

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A production line is set up to manufacture machine braces in lots of five per minute during a shift. When the lots are weighed, variation among the weights is detected, with lot weights ranging from 41 to 47 grams in a uniform

distribution.

Determine the probability that a lot weighs between 42 and 45 grams.

PROBLEM

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A production line is set up to manufacture machine braces in lots of five per minute during a shift. When the lots are weighed, variation among the weights is detected, with lot weights ranging from 41 to 47 grams in a uniform

distribution.

Determine the probability that a lot weighs between 42 and 45 grams.

= = = =0.5000

Solution

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Suppose the amount of time it takes to assemble a plastic module ranges from 27 to 39 seconds and that assembly times are uniformly distributed. Describe the distribution.

1. What is the probability that a given assembly will take between 30 and 35 seconds?

2. Fewer than 30 seconds?

Problem

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Suppose the amount of time it takes to assemble a plastic module ranges from 27 to 39 seconds and that assembly times are uniformly distributed. Describe the distribution.

1. What is the probability that a given assembly will take between 30 and 35 seconds?

2. Fewer than 30 seconds?

Solution

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The average fill volume of a regular can of soft drink is 12 ounces. Suppose the fill volume of these cans ranges from 11.97 to 12.03 ounces and is uniformly distributed.

1. What is the height of this distribution?

2. What is the probability that a randomly selected can contains more than 12.01 ounces of fluid?

3. What is the probability that the fill volume is between 11.98 and 12.01 ounces?

Problem

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a = 11.97 b = 12.03

Height = = 16.667

P(x > 12.01) = = .3333

P(11.98 < x < 12.01) = = .5000

Solution

97 . 11 03

. 12

1 1

 

 a b

97 . 11 03

. 12

01 . 12 03

. 12



97 . 11 03 . 12

98 . 11 01 . 12



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Probably the most widely known and used of all distributions is the normal distribution.

It fits many human characteristics, such as height, weight, length, speed, IQ, scholastic achievement, and years of life expectancy, among others.

 Like their human counterparts, living things in nature, such as trees, animals, insects, and others, have many characteristics that are normally distributed.

Normal Distribution

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The normal distribution is described or characterized by two parameters: the mean,µ , and σ the standard deviation, . The values of µ and σ produce a

normal distribution.

The density function of the normal distribution is

Probability Density Function of the Normal Distribution

= 1

Where 2

µ=mean of x

σ= standard deviation of x

=3.141… and e=2.71828

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Note that every change in a parameter ( µ or σ) determines a different normal distribution.

This characteristic of the normal curve (a family of curves) could make

analysis by the normal distribution tedious because volumes of normal curve tables—one for each different combination of µ and σ—would be required.

Standardized Normal Distribution

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The normal distribution exhibits the following characteristics.

It is a continuous distribution.

It is a symmetrical distribution about its mean.

 It is asymptotic to the horizontal axis.

 It is unimodal.

It is a family of curves.

Area under the curve is 1.

Characteristics of Normal Distribution

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Assume that the test scores of a college entrance exam fits a normal distribution. Furthermore, the mean test score is

72, and the standard deviation is 15.2. What is the

percentage of students scoring 84 or more in the exam?

Problem

= 1

Where 2

µ=mean of x

σ= standard deviation of x

=3.141… and e=2.71828

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Assume that the test scores of a college entrance exam fits a normal distribution. Furthermore, the mean test score is

72, and the standard deviation is 15.2. What is the

percentage of students scoring 84 or more in the exam?

Solution using R

> pnorm(84, mean=72, sd=15.2, lower.tail=FALSE) [1] 0.2149176

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Fortunately, a mechanism was developed by which all normal

distributions can be converted into a single distribution: the z distribution.

This process yields the standardized normal distribution (or curve).

The conversion formula for any x value of a given normal distribution follows.

= −

σ ≠ 0

Z-Score

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A z score is the number of standard deviations that a value, x, is above or below the mean.

If the value of x is less than the mean, the z score is negative;

if the value of x is more than the mean, the z score is positive;

 and if the value of x equals the mean, the associated z score is zero.

z score

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Z Distribution Table

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In a recent year, the mean GMAT score was 494 and the standard deviation was about 100.

What is the probability that a randomly selected score from this administration of the GMAT is between 600 and the mean?

Problem

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In a recent year, the mean GMAT score was 494 and the standard deviation was about 100.

What is the probability that a randomly selected score from this administration of the GMAT is between 600 and the mean?

The z formula yields the number of standard deviations that the x value, 600, is away from the mean.

The probability value in Table for z = 1.06 is .3554.

The shaded portion of the curve at the top of the table indicates that the probability value given always is the probability or area between an x value and the mean

Solution

> pnorm(1.06)-0.5 [1] 0.3554277

> 0.5-pnorm(600, mean=494, sd=100, lower.tail=FALSE) [1] 0.3554277

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What is the probability of obtaining a score greater than 700 on a GMAT test that has a mean of 494 and a standard deviation of 100? Assume GMAT scores are normally distributed.

Problem

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Solution

From Z distribution table we get probability of .4803 for this z score. This value is the probability of randomly drawing a GMAT with a score between the mean and 700. Finding the

probability of getting a score greater than 700, which is the tail of the distribution, requires subtracting the probability value of .4803 from .5000, because each half of the distribution contains .5000 of the area.

The result is .0197.

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Solution

From Z distribution table we get probability of .4803 for this z score. This value is the probability of randomly drawing a GMAT with a score between the mean and 700. Finding the

probability of getting a score greater than 700, which is the tail of the distribution, requires subtracting the probability value of .4803 from .5000, because each half of the distribution contains .5000 of the area.

The result is .0197.

pnorm(700, mean=494, sd=100, lower.tail=FALSE) [1] 0.01969927

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For the same GMAT examination, what is the probability of randomly drawing a score that is 550 or less?

Problem

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For the same GMAT examination, what is the probability of randomly drawing a score that is 550 or less?

Solution

pnorm(550, mean=494, sd=100) [1] 0.7122603