Probability and Statics

7/29/2019 Probability and Statics

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Probability and Statistics

Week 1

Text Book:

Probability and Statistics for Engineers and Scientist

By Wallpole and Mayer

Probability and Statistics

Statistic (40%) Probability (60%)

Introduction of Statistics Introduction and Terminology of

Probability

Data Organization Joint Probability, Conditional

Probability, Bays Theorem

Data representation Probability distribution Grouping of Data Normal Distribution

Measures of Central Tendency Binomial distribution

Measure of dispersions Exponential Distribution

Measures of Position Random Variables

Estimation and Expectation

Hypothesis Testing

Core Engineering Applications

1. Information Theory and Coding

2. Computer Communication

3. Digital Communication

4. Digital Signal Processing5. Microwave Engineering

6. Radar and Satellite Communication

7. Artificial Intelligence and Robotics8. Decision Support Systems

9. Product Planning and Marketing Survey

Introduction to Statistics:

The use of statistical methods in manufacturing, development of food products, computer

software, pharmaceutical, and many other areas involves the gathering of information orscientific data. Of course, the gathering of data is nothing new. It has been done for well

over a thousand years. Data have been collected, summarized, reported, and stored for

perusal. However, there is a profound distinction between collection of scientificinformation and inferential statistics.


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Statistic:

Statistics is a mathematical science pertaining to the collection, analysis, interpretation orexplanation, and presentation of data. It provides tools for prediction and forecasting

using data and statistical models.

Types of Statistics:

Descriptive statistics summarize the population data by describing what was

observed in the sample numerically or graphically. Numerical descriptors includemean and standard deviation for continuous data types (like heights or weights),

while frequency and percentage are more useful in terms of describing categorical

data (like race).

Inferential statistics uses patterns in the sample data to draw inferences about thepopulation represented, accounting for randomness. These inferences may take

the form of: answering yes/no questions about the data (hypothesis testing)

estimating numerical characteristics of the data (estimation), describingassociations within the data (correlation) and modeling relationships within the

data (regression).

Types of Data

Group Data

Data that has been organized into groups (into a frequency distribution). If you see a tablesimilar to the one below, you will know that you are dealing with grouped data:

Class Frequency0 5 4

6 10 5

11 15 12

16 20 7

Ungrouped Data

Data that has not been organized into groups. Ungrouped data looks like a big list of

numbers.


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Measure of Central Tendency of Ungroup Data

Measures of central tendency are measures of the location of the middle or the center

of a distribution. The definition of "middle" or "center" is purposely left somewhatvague so that the term "central tendency" can refer to a wide variety of measures. The

mean is the most commonly used measure of central tendency. The following

measures of central tendency are discussed in this text:

1. Mean

2. Median3. Mode

4. Trimmed mean

Mean of ungrouped data:

The arithmetic mean is what is commonly called the average: When the word "mean" is

used without a modifier, it can be assumed that it refers to the arithmetic mean. The

mean is the sum of all the entries divided by the number of smaples.

Let x1, x2, x3, ..., xn be n observations then mean is obtained by dividing the sum of n

observations by n.

Example: Find the mean of 4,6,8,6,7,8

Solution:= (4 + 6 + 8 + 6 + 7 + 8)/6

= 39/6

= 6.5

Median of ungrouped data:

If the observations of an ungrouped data are arranged in increasing or decreasing order oftheir magnitude, a value which divides these ordered observations into two equal parts is

called the median of the data. It is denoted by M.


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If the number of observations (n) is an odd integer, then

M = Value of (n+1)/2th

observation in the arrangement of observations in increasingorder.

If the number of observations (n) is an even integer, then

M = (Value of n/2th observation + Value of (n/2 + 1)th observation)/2

Example:

Find the median of the following observations4,6,8,6,7,8,8

Solution:

Observations in the ascending order are :

4, 6, 6, 7, 8, 8, 8

Here, n = 7 is odd.

Median :

M = Value of (n+1)/2

th

observation= Value of (7+1)/2th observation= Value of 4th observation

= 7

Mode of ungrouped data:

An observation occurring most frequently in the data is called mode of the data. It is

denoted by Z.

Example:

Find the mode of the following observations

4,6,8,6,7,8,8

Solution:

In the given data, the observation 8 occurs maximum number of times (3)

Mode= 8


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Trimmed Mean:

A trimmed mean is computed by "trimming away" a certain percent of both the largest

and smallest set of values. For example, the 10% trimmed mean is found by eliminatingthe largest 5% and smallest 5% and computing the average of the remaining values.

trimming the mean can reduce the effects of outlier bias in a sample


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Questions:

1- The following measurements were recorded for the voltage of a capacitor, of acertain period of time.

3.4 2.5 4.8 2.9 3.62.8 3.3 5.6 3.7 2.8

4.4 4.0 5.2 3.0 4.8

Assume that the measurements are a simple random sample.

Required:

(a) What is the sample size for the above sample?(b) Calculate the sample mean for this data.

(c) Calculate the sample median.

(d) Plot the data by way of a dot plot.

(e) Compute the 20% trimmed mean for the above data set.(f) Comment on the bases of given results

Solution: http://hwvalley.blogspot.com/2012/11/probability-statistics-for-engineers.html


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2- Iqra University purchases a machine trainer according to the Lab Engineer RPM

varies for each trial. A random sample of 20 observation is taken and the was

measured. The following are the RPM values:

18.71 21.41 20.72 21.81 19.29 22.43 20.17

23.71 19.44 20.50 18.92 20.33 23.00 22.8519.25 21.77 22.11 19.77 18.04 21.12.

Required:

(a) Calculate the sample mean and median for the above sample

values.

(b) Compute the 10% trimmed mean.(c) Do a dot plot of the given data.

(d) Comment on the bases of given results

Solution: https://docs.google.com/viewer?a=v&q=cache:-

pV53q5cdQ4J:ihome.ust.hk/~liuzhi/tutorial3.pdf+&hl=en&gl=pk&pid=bl&srcid=ADGEESg-BOnqQ8wNnVBBHUFqUenISRz-

zRIA2brtI6rjjD3KmyNC9dwsFCqTotKxgSpDG9sylubjyM1Av7Ni_J8xpKU2zk

Xv7Y8uTtQlQ8KUqdx_xmYNG4PMoLYBEepCZlc6OldaEPVM&sig=AHIEtbSodcT4kKT2ISugYsPqBs13B2ZBrA


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3- In a study conducted by the Department of Mechanical Engineering , the steelrods supplied by two different companies were compared. Ten sample springs

were made out of the steel rods supplied by each company and a measure of

flexibility was recorded for each. The data are as follows:

Company A: 9.3 8.8 6.8 8.7 8.5

6.7 8.0 6.5 9.2 7.0

Company B: 11.0 9.8 9.9 10.2 10.1

9.7 11.0 11.1 10.2 9.6

Required:

(a) Calculate the sample mean and median for the data for the two companies.(b) Plot the data for the two companies on the same line and give your

impression.

(c) Comment of the given result.


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4- The Collector current of a BJT transistor changes with the variation in , if thebiasing technique is simple base bias as the temperature varies the it cause change

is where as Ic = IB . The random sample of Ic is recorded for the two

temperature level in m A.

Required:

(a) Show the Dot plot for both high and low temperature rating

(b) Calculate the sample mean, median and mode for both temperature(c) Comment on the bases of given results


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5- CRC Department of IQRA University has assigned a task to record the Hard Diskstorage capability in GB New Computer Lab 2 equipped with Core2 Quad. 20

sample were recorded as follows.

160 80 160 350 80 160 350 350 180 20 80

350 500 20 350 80 160 80 80 80

Required:

(a) Compute the Mean, Median and Mode of the given data

(b) Comment which central measure would you prefer and why

(c) Do an Dot plot of given data

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Probability and Statics