Chapter 1 Introduction to Statistics.pdf

7/26/2019 Chapter 1 Introduction to Statistics.pdf

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CHAPTER 1: INTRODUCTION TO

STATISTICS

- Qualitative data and

Quantitative data

- Measures of central tendencyand dispersion

BEKA 2453 Statistics & Numerical Methods


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WHAT IS STATISTIC???

Your company has created a newdrug that may cure arthritis. Howwould you conduct a test toconfirm the drug's effectiveness?

You want to conduct a poll onwhether your school should use itsfunding to build a new athleticcomplex or a new library. Howmany people do you have topoll? How do you ensure thatyour poll is free of bias? How doyou interpret your results?

Statistics is the mathematical science involved in the application of quantitative

principles to the collection, analysis, and presentation of numerical data. The

practice of statistics utilizes data from some population in order to describe it

meaningfully, to draw conclusions from it, and make informed decisions.

BENG 2142 Statistics


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WHAT DO ENGINEERS DO???

An engineer is someone who solves problems of interest tosociety with the efficient application of scientific principlesby:

Refining existing products

Designing new products or processes



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THE CREATIVE PROCESS

THE ENGINEERING PROCESS


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STATISTICS SUPPORTS THE CREATIVE PROCESS



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1.1 Qualitative data and Quantitative data

1.1.1 Introduction

Statistics - the science of collecting, organizing,

summarizing and analyzing information in order to drawconclusions.

Two types of statistics

Descriptive statistics

Inferential Statistics


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Descriptive statistics

consists of organizing and summarizing theinformation collected. Descriptive statistics

describes the information collected through

numerical measurements, charts, graphs and

tables.

Inferential Statistics

generalize results obtained from a sample to

the population and measure their reliability.



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BEKA 2453 Statistics & Numerical Methods

1.1.2 Basic Terms

Population - consists of all items or elements ofinterest for a particular decision or

investigation.

Example (all FKE students in the UTeM. )

Sample is a certain number of elements thathave been chosen from a population. Sample is

a subset of population.

Example: (a list of students of 2BEKG would be

a sample from the population of all FKE students

in the UTeM.)


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POPULATION VS. SAMPLE



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1.1.2 Basic Terms (cont.)

Random sample is a sample drawn in such a

way that each element of the population has a

chance of being selected.

Simple random sample implies that anyparticular sample of a specified sample size

has the same chance of being selected as any

other sample.

Element / member is a specific subject orindividual about which the information is

collected.



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1.1.2 Basic Terms (cont.) Variable is a characteristic of the individual within the

sample or population.

Observation/Measurement is the value of a variable

for an element

Data set is a collection of values of one or more

variables. Grouped data set is a collection of data which are

grouped in classes.

Population parameter is a descriptive measure

computed from a population data. Sample statistic is a descriptive measure computed

from a sample data.

Outliers / Extreme Values are values that are very

small or very large relative to the majority of the

values in a data set.



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Outlier



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PRACTICE PROBLEM

A random sample of 30 middle school students average 1.8 hoursspent on homework each night. It is believed that middle schoolstudents spend 2 hours each night on homework. Identify the sample,

the population, the sample statistic, and the population parameter.

The sample is the 30 middle school students who are randomly selected. The

population is all middle school students.

The sample statistic is x = 1.8 hours, and the population parameter is = 2 hours.

Remember that sample statistics are values that represent a sample, whilepopulation parameters are values that represent a population.



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1.1.3 Variables

Qualitative variables allow for classification of

individuals based some attribute or

characteristics

Example: the gender of new born babies;the marital status of people, types of cars.

Quantitative variables provide numerical

measures of individuals. (countable).

Example: The weight of children; the

numbers of cars owned.



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1.1.3 Variables (cont.)

Quantitative variables can be further classified

into two groups:

(a) Discrete Variables.

finite / countable number of possible values. Example:

The number of heads obtained by flipping

a coin five times.

The number of cars that arrive at KFCsdrive-through between 1.00 p.m to 2.00

p.m.



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1.1.3 Variables (cont.)

(b) Continuous Variables.

infinite number of possible values that are not

countable. They are obtained by measuring;

include fractions and decimals. Example

Time spent studying for your first statistics

exam.

The height of volleyball players.



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PRACTICE PROBLEM

Determine whether thefollowing variables arequalitative or quantitative.

1. Postal Code

2. Salary

3. PTPTN allowances

4. Gender

5. Marital Status

Determine whether the followingvariables are discrete or continuous.

1. Heights of 2BEKG students in FKE.

2. Number of books that have beenborrowed by FKE students each dayfrom library.

3. Number of 2BEKG studentsattended Statistic class everyTuesday.

4. The time taken for 2BEKG studentsto get to class at 8oclock in themorning.


NG 2 42 S


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1.1.4 Graphical Methods

Qualitative data can be displayed by using Bar graph

Pie chart

Example:


BENG 2142 S i i


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Solution:

Bar graph: Pie chart:

0

50

100

150

200

250

300

A B C D

Frequen

cy

Rating

9%

65%

23%

3%

A

B

C

D


BENG 2142 S i i


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1.1.4 Graphical Methods

Graph grouped data can be classified to:

Histogram

Polygon

Graphing for cumulative frequency distribution used

Ogive (cumulative histograms)


BENG 2142 St ti ti


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Example:


BENG 2142 St ti ti


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Solution:

(a) & (b)




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Solution:

(c) Histogram Polygon




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Solution:

(d)



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PRACTICE PROBLEM

The following scores represent the final examination grade for Statisticsubject:

23 60 79 32 57 74 52 70 82 36 80 77 81 95 41 65 92 85 55 76 5210 64 75 78 25 80 98 81 67 41 71 83 54 64 72 88 62 74 43 60 78

89 76 84 48 84 90 15 79 34 67 17 82 69 74 63 80 85 61

a) Construct a frequency distribution table with the class width is 10.

b) Determine the class boundaries and class midpoints.

c) Calculate the relative frequencies and percentages for all classes.d) Construct a frequency histogram for the data

e) Prepare the cumulative frequency distribution table.

f) Construct an ogive for cumulative frequency.


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MARKS No. of

Student

Relative

frequence

Percentage Class

boundaries

No. of Student Class

midpoints

1-10 1 0.017 2 0.5-10.5 1 5.5

11-20 2 0.033 3 10.5-20.5 2 15.5

21-30 2 0.033 3 20.5-30.5 2 25.5

31-40 3 0.050 5 30.5-40.5 3 35.5

41-50 4 0.067 7 40.5-50.5 4 45.5

51-60 7 0.117 12 50.5-60.5 7 55.5

61-70 10 0.167 17 60.5-70.5 10 65.5

71-80 16 0.267 27 70.5-80.5 16 75.5

81-90 12 0.200 20 80.5-90.5 12 85.5

91-100 3 0.050 5 90.5-100.5 3 95.5

Total 60 1.000 100

0

2

4

6

8

10

12

14

16

18

1-10 11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90 91-100

No.of

Students

Marks

BAR GRAPH

HOW ABOUT

HISTOGRAM (NO

GAP)?



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1.2 Measures of central tendency and

dispersion

Measures of central tendacy

- Indicates the central point around which observations tend to

cluster- Mean, Mode, Median

Measures of dispersion

- The measures that help us know about the spread of a data set

- Range, Variance, Standard deviation




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1.2 Measures of central tendency and

dispersion1.2.1 Numerical Measures

Skewness:

Measures of Central

Tendency

Measures of Dispersion

Mean RangeMedian Variance

Mode Standard deviation

Skewness




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1.2.2 Measures of Central Tendency

Mean is the arithmetic mean or average.

Median of a variable is the value that lies in the

middle of the data when arranged in ascending

order. Mode of a variable is the most frequent observation

of the variable that occurs in the data set.




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RELATIONSHIP BETWEEN MEAN, MEDIAN

AND MODE


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MEAN

POPULATION

Mean

+++ where N is the number of observations in the population

SAMPLE

Mean

+++

where n is the number of observations in the sample


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MEDIAN

Steps in Computing the Median of a Data Set

Step 1: Arrange the data in ascending order.

Step 2: Determine the number of observations, n .

Step 3: Determine the observation in the middle of the data set.

If the number of observations is odd, then the median is the datavalue that is exactly in the middle of the data set. That is, the median

is the observation that lies in the +2 position. If the number of observations is even, then the median is the mean ofthe two middle observations in the data set. That is, the median is the

mean of the data values that lie in the2 and

2 1 positions.


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MODE

To compute the mode, tally the number of observations that occur for

each data value. The data that occurs most often is the mode. A set of

data can have no mode, one mode or more than one mode. If there

is no observation that occurs with the most frequency, we say the

data has no mode.


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EXAMPLE

The following data represent the monthly phone bill for six randomly

selected months (in RM).

35.34 42.09 39.43 38.93 43.39 49.26

Calculate the mean, median and mode for the monthly phone bill.

Solutions:


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1.2.3 Measures of Dispersion

Range of a variable is the difference between

the largest data value and the smallest data

value

Variance is based upon the difference betweeneach observation and the mean; that is, it is

based upon the deviation about the mean

Standard deviation tells us how closely the valuesof a data set are clustered around the mean.



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POPULATION

Mean

2 =

Variance

2 1 = 2 =

2

Standard deviation

1

=

2

= 2



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SAMPLE

Mean

2 =

Variance

2 1 1 = 2 = 2

Standard deviation

1 1 =

2

= 2


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EXAMPLE

The following data represent the monthly phone bill for six randomly

selected months (in RM).

35.34 42.09 39.43 38.93 43.39 49.26

Compute the range, sample variance and sample standard deviation.

Range = Largest Data ValueSmallest Data Value.

= 49.2635.34

=13.92


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Sample Variance, s2


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Sample standard deviation, s


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PRACTICE PROBLEM

An engineer is interested in testing the bias in a pH meter. Data are

collected on the meter by measuring the pH of a neutral substance(pH=7.0). A sample of size 10 is taken with results given by

7.07 7.00 7.10 6.97 7.00 7.03 7.01 7.01 6.98 7.08

Compute the range, sample variance and sample standard deviation.


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MEASURES OF CENTRAL TENDENCY AND

DISPERSION FOR GROUPED DATAMEASUREMENT POPULATION SAMPLE

MEAN

VARIANCE

STANDARD

DEVIATION


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EXAMPLEThe following data give the monthly expenditures (in hundred RM) on

food for 30 households randomly selected from the households who

incurred such expenses.

4.57 3.95 6.95 3.80 1.50 3.99 7.84 5.05 8.00 14.75 9.33 1.055.08 7.00 9.60 18.99 9.15 11.32 4.75 9.95 3.63 1.99 1.39 13.0919.31 11.15 7.73 12.00 7.58 16.35

Find the sample mean for the monthly expenditures on food for 30households


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Sample Variance Sample standard deviation



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1.2.4 Reasons to sampling

To construct the whole population would be timeconsuming.

The cost of studying the all items in population

may be prohibitive. The physical impossibility of checking all items in

the population.

The destructive nature of some tests

The sample results are adequate


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END OF CHAPTER 1

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Chapter 1 Introduction to Statistics.pdf