Probability and Statistics part 2.pdf

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This document provides an overview of topics covered in a probability and statistics course for an FE exam prep, including combinations and permutations, sets, measures of central tendency such as mode, median and mean, measures of dispersion like standard deviation and variance, expected values, probability distributions, and hypothesis testing. Examples are provided for calculating the mode, median, mean and weighted mean of data sets and for determining population variance and standard deviation.

Engineering

Probability and
Statistics
FE Exam Prep Course
© 2018 Professional Publications, Inc.

PROBABILITY AND STATISTICS
• Combinations and Permutations
• Sets
• Probability
• Laws of Probability
• Measures of Central Tendency
• Measures of Dispersion
• Expected Values
• Probability Density Functions
• Probability Distribution Functions
• Probability Distributions
• Sums of Random Variables
• Hypothesis Testing
• Linear Regression
© 2018 Professional Publications, Inc. 35
Lesson Overview

PROBABILITY AND STATISTICS
mode
the value that occurs most frequently in the sample set
median
the point in the sample set at which there are equal numbers of samples above and
below
mean
the sum of all samples in the set divided by the number of samples
© 2018 Professional Publications, Inc. 36
Measures of Central Tendency

PROBABILITY AND STATISTICS
What are the mode, the median, and the
mean of the following data?
61, 62, 63, 63, 64, 64, 66, 66, 68, 68,
68, 68, 68, 69, 69, 69, 69, 70, 70, 70,
70, 71, 71, 71, 73, 74, 74, 75, 76, 79
© 2018 Professional Publications, Inc. 37
Example: Measures of Central Tendency

PROBABILITY AND STATISTICS
weighted arithmetic mean
• used when some data are more
significant than others
• wi = weight assigned to datum Xi
© 2018 Professional Publications, Inc. 39
Measures of Central Tendency

PROBABILITY AND STATISTICS
Three observations, 62, 64, and 72, are
given weights of 4, 3, and 2, respectively.
Most nearly, what is the weighted
arithmetic mean of these data?
(A) 65
(B) 66
(C) 68
(D) 69
© 2018 Professional Publications, Inc. 40
Example: Measures of Central Tendency

PROBABILITY AND STATISTICS
geometric mean of the sample
• the number that, when raised to the
power of the sample size, gives the
product of all samples
• used when data are consecutive
multipliers in other calculations
root-mean-square value
• the square root of the arithmetic
mean of the squares of all samples
© 2018 Professional Publications, Inc. 42
Measures of Central Tendency

PROBABILITY AND STATISTICS
standard deviation, σ
measures amount of dispersion in data
set
• low σ means that data tend to be
gathered close to mean value
• high σ means that data tend to be
spread out over wide range of values
standard deviation of total population
© 2018 Professional Publications, Inc. 43
Measures of Dispersion

PROBABILITY AND STATISTICS
sample standard deviation, s
estimates standard deviation based on a
limited number of samples taken from a
data set
© 2018 Professional Publications, Inc. 44
Measures of Dispersion

PROBABILITY AND STATISTICS
variance
the square of the standard deviation
population variance, σ2
sample variance, s2
© 2018 Professional Publications, Inc. 45
Measures of Dispersion

PROBABILITY AND STATISTICS
What is the population variance and
standard deviation of the following data?
61, 62, 63, 63, 64, 64, 66, 66, 67, 68,
68, 68, 68, 69, 69, 69, 69, 70, 70, 70,
70, 71, 71, 72, 73, 74, 74, 75, 76, 79
© 2018 Professional Publications, Inc. 46
Example: Measures of Dispersion

PROBABILITY AND STATISTICS
expected value
variance
© 2018 Professional Publications, Inc. 48
Expected Values
standard deviation

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2. PROBABILITY AND STATISTICS • Combinations and Permutations • Sets • Probability • Laws of Probability • Measures of Central Tendency • Measures of Dispersion • Expected Values • Probability Density Functions • Probability Distribution Functions • Probability Distributions • Sums of Random Variables • Hypothesis Testing • Linear Regression © 2018 Professional Publications, Inc. 35 Lesson Overview

3. PROBABILITY AND STATISTICS mode the value that occurs most frequently in the sample set median the point in the sample set at which there are equal numbers of samples above and below mean the sum of all samples in the set divided by the number of samples © 2018 Professional Publications, Inc. 36 Measures of Central Tendency

4. PROBABILITY AND STATISTICS What are the mode, the median, and the mean of the following data? 61, 62, 63, 63, 64, 64, 66, 66, 68, 68, 68, 68, 68, 69, 69, 69, 69, 70, 70, 70, 70, 71, 71, 71, 73, 74, 74, 75, 76, 79 © 2018 Professional Publications, Inc. 37 Example: Measures of Central Tendency

5. PROBABILITY AND STATISTICS What are the mode, the median, and the mean of the following data? 61, 62, 63, 63, 64, 64, 66, 66, 68, 68, 68, 68, 68, 69, 69, 69, 69, 70, 70, 70, 70, 71, 71, 71, 73, 74, 74, 75, 76, 79 Solution The mode is the most common value among the data, which is 68. There are 30 numbers. The median is the mean average of the 15th and 16th numbers, which is 69. The mean is found by adding all the numbers and dividing by 30, which gives 68.96. © 2018 Professional Publications, Inc. 38 Example: Measures of Central Tendency

6. PROBABILITY AND STATISTICS weighted arithmetic mean • used when some data are more significant than others • wi = weight assigned to datum Xi © 2018 Professional Publications, Inc. 39 Measures of Central Tendency

7. PROBABILITY AND STATISTICS Three observations, 62, 64, and 72, are given weights of 4, 3, and 2, respectively. Most nearly, what is the weighted arithmetic mean of these data? (A) 65 (B) 66 (C) 68 (D) 69 © 2018 Professional Publications, Inc. 40 Example: Measures of Central Tendency

8. PROBABILITY AND STATISTICS Three observations, 62, 64, and 72, are given weights of 4, 3, and 2, respectively. Most nearly, what is the weighted arithmetic mean of these data? (A) 65 (B) 66 (C) 68 (D) 69 Solution The weighted arithmetic mean is 65. The answer is (A). © 2018 Professional Publications, Inc. 41 Example: Measures of Central Tendency            4 62 3 64 2 72 4 3 2 64.89 65 i i w i w X X w         

9. PROBABILITY AND STATISTICS geometric mean of the sample • the number that, when raised to the power of the sample size, gives the product of all samples • used when data are consecutive multipliers in other calculations root-mean-square value • the square root of the arithmetic mean of the squares of all samples © 2018 Professional Publications, Inc. 42 Measures of Central Tendency

10. PROBABILITY AND STATISTICS standard deviation, σ measures amount of dispersion in data set • low σ means that data tend to be gathered close to mean value • high σ means that data tend to be spread out over wide range of values standard deviation of total population © 2018 Professional Publications, Inc. 43 Measures of Dispersion

11. PROBABILITY AND STATISTICS sample standard deviation, s estimates standard deviation based on a limited number of samples taken from a data set © 2018 Professional Publications, Inc. 44 Measures of Dispersion

13. PROBABILITY AND STATISTICS What is the population variance and standard deviation of the following data? 61, 62, 63, 63, 64, 64, 66, 66, 67, 68, 68, 68, 68, 69, 69, 69, 69, 70, 70, 70, 70, 71, 71, 72, 73, 74, 74, 75, 76, 79 © 2018 Professional Publications, Inc. 46 Example: Measures of Dispersion

14. PROBABILITY AND STATISTICS What is the population variance and standard deviation of the following data? 61, 62, 63, 63, 64, 64, 66, 66, 67, 68, 68, 68, 68, 69, 69, 69, 69, 70, 70, 70, 70, 71, 71, 72, 73, 74, 74, 75, 76, 79 Solution Use the equation for population variance. © 2018 Professional Publications, Inc. 47 Example: Measures of Dispersion   2 2 1 1 N i i X N      

16. PROBABILITY AND STATISTICS • Combinations and Permutations • Sets • Probability • Laws of Probability • Measures of Central Tendency • Measures of Dispersion • Expected Values • Probability Density Functions • Probability Distribution Functions • Probability Distributions • Sums of Random Variables • Hypothesis Testing • Linear Regression © 2018 Professional Publications, Inc. 49 Lesson Overview

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