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Friday, December 12, 2014

ANALYTIC GEOMETRY TRINAL PART 1



1.  Show that the quadrilateral with vertices (12, 2) ,  (1, 1),  (0, –2)  and (11, –1)  is a
     parallelogram and find its area.

2.  Show that the points (2, 5), (8, 1), (6, –2 )  and  (0 , 2 ) are vertices of  a rectangle
     and find its area.

3.  Show that the points  (–1,  –2),  (5,  4)  and  (–3, 0)  are vertices of a right triangle
     and find its area.

4.  Find the counterclockwise angle from  L1 through (1, 9), (2, 6)  and L2 through (3, 3), (–1, 5).

5.  Find the counterclockwise angle from  L1 through (– 4, 3 ) and ( 6, 7 )  and L2 through
 (9, –5 )  and ( 6, 7 ).

 6. Determine the acute angles of the triangle in problem # 3.   

 7. Find the equation of the locus of a point ( x, y ) which is equidistant from (–2, 0) and (6, 4).
     
 8. Locate the point which is equidistant from (3, 8),  (5, 2)  and  (–3, –4).

 9. Find the area of the quadrilateral formed by the given points ;

               a)  ( 5, 2 ),  ( 4, 3 ) ,  ( 2, 4 ) and  ( -8,  -1 )  
  
               b)  ( 3, 5 ),  ( 9, 5 ),  ( 0, 5 )  and  ( -2,  1).

                                                                                                 

       “Difficulties are insurmountable mountains if you go away from them but they will
                      prove  to be mere molehills if you courageously go after them”
                                                                                - ifraincanhear 2014



SOLID MENSURATION


MATH E 115  TRINAL  EXAMINATION                                                     December  17,  2014

1.  A vegetable bin built in the form of a cube with an edge of 2 m  is divided by a vertical partition which passes through two diagonally opposite edges. Find the lateral surface of either compartment.

2.  How many cubes of maximum edge can be fabricated from a piece of log 28.28  cm  in diameter and 3 m long ? What is the volume of one cube ?

3.  A rectangular lot has an area of  800 square meter. If the length is to be twice its width, determine
     the length and the width of the lot.

4.  a) A circular cylinder has a base area of 324 square centimeter. Determine the radius of this cylinder.  ( b) The circumference of a circle is  113.09733  cm. What is the area of this circle ?

5.  Determine the area of the largest circle which can be cut from a square of edge  8 cm. What is the 
      area of the material wasted ?

6.  a) Determine the dimensions of rectangle if the perimeter is 72 cm  and the area is 308 square
      meters.  ( b ) The diagonal of square is  512 cm, determine the area of this square.

7.  A circular cake  8 cm thick, is divided to make exactly  4  cubes of maximum edge. What is the volume   of one of the cubes ? What is the diameter of the cake ? What is the volume of the cake that is not included in the cubes ?     

8.  A block of wood 10 cm by 12 cm by 3 m  is to be cut into cubes of maximum edge. How many cubes  be cut from the given wooden block ?  What is the volume of one of these cubes. 

9.  A  block of lead having a volume of  6100  cubic centimeter is allowed to melt in a furnace and the  molten metal is poured into a cubical mold  with an edge of 10 cm.  How many cubes can be fabricated in these process ? 

10.  The diagonals of a rhombus are in the ratio of 3 : 5. If the area of the rhombus is  120  square  centimeter,  determine the two diagonals of the rhombus.

11.  The face diagonal  of a cube is  16.97056 cm. Determine the length of the edge, the  volume and the total surface area of the cube.

12.  The diagonal of a cube is  25.98076 cm. Determine the length of the edge, the  volume and the  total surface area of the cube.


E N D 

Saturday, October 11, 2014

STATISTICS


COLLECTION AND PRESENTATION OF  DATA
 Types of data :
1.      Qualitative data are categorical data taking the form of attributes or categories such as sex, course, race, religion, blood type, etc.  This type of data can only be categorized but cannot be quantified.

2.      Quantitative data or numerical data are obtained from measurements like weights, ages, heights, temperatures, score and other measurable quantities.


Measurement is a process to convert qualitative data to quantitative data. By measurement, numbers are used to code objects or items so that they can be treated statistically.

Measurement scales
1.      Nominal measurement is used merely for identification or classification purposes.
Example. A group of students maybe classified according to their courses:
    1 – BSED                              4 – BSBA                   7 – BSIS 
    2 – BEED                              5 – BSIT                     8 – BSA
    3 – BSCS                              6 – BSN                      9 – BSCpE

2.      Ordinal measurement. The ordinal measurement does not only classify items but also give the order or rank of classes, items or objects.
          Example.  a) Ranks as  1st,  2nd,  3rd  in an oratorical contest
                           b) Ranks as  1st, 2nd,  3rd  in a poster making contest

3.      Interval measurement. In the interval measurement, numbers assigned to the items or objects not only identify and rank the objects but also measure the degree of differences between two classes.
Example.  Height, mass, temperature, score
a)      If  Jay has a mass of 48 kg and Claire is  54 kg, the mass difference of 6 kg implies that Claire is 6 kg heavier than Jay.
b)      If water from the faucet is 22 degrees Celcius, and that coming from the well is  27 degrees Celcius, the temperature difference of 5 degrees implies that water from the well is warmer than the water from the faucet.
c)      If  John has a height of  5’ 8” and  Jim stands  5’ 4”, then the difference of 4 inches implies that John is taller than Jim by 4 inches.
4.      Ratio measurement.  For ratio measurement, the ratio of the numbers assigned in the measurement reflects the ratio in the amount of property being measured. Multiplication and division have meaning in ratio measurements.
          Thus, if Rezzie is 18 years old and Stephen is 24 years old, then their ages can expressed              
          in the ratio of  3 : 4.
          If a motorcycle moves 80 km/hr and a car moves with a speed of  100 km/hr, then their 
          speeds can be expressed in the ratio of  4 : 5. 

SAMPLING TECHNIQUES

Sampling techniques are use to ensure the validity of conclusions or inferences from the sample to the population.
 Common types of sampling techniques:
    1.  Simple random sampling
    2.  Stratified random sampling
    3.  Systematic random sampling
    4.  Cluster sampling
    5.  Multi-stage sampling

Samples can either be a Judgment sampleConvenience sample or a Random sample.
   
* Judgment sample is one in which the individual selecting the sample items uses experience as the basis for choosing the items to be included in the sample with the objective to make the sample as representative of the population as possible. An example is an auditor who chooses only few records necessary for sample audit base on the judgment that these records are representative of the records in general. 

* Convenience sample includes the most easily accessible measurement or observations. An example is taking an opinion poll from students passing by the Plaza Madonna entrance gate.

* Random sample is a sampling procedure by which every element in the population has a known and usually equal chance of being chosen for inclusion in the sample.

 *  Simple random sampling.
         A random sample has limited number of individuals chosen from the population. Before the selection is done every individual has an equal chance of being selected in the sample.
        Lottery is the simplest method of random sampling. To illustrate the process, consider a population of 356 CITE students. Obtain a coded or numbered list of all the students. Write the numbers or codes in a small slip of paper, roll the papers and place in a container. Jumble it thoroughly; and without looking at the slip of papers draw, i.e. 160 slips representing 160 students, the desired sample size. The process ensures a representative sample because each item has the same chance of getting into the sample.

        If the population is large, the table of random numbers can be used. This allows us to select the desired sample on a purely chance basis.
1.      Identify the population and determine the sample size.
2.      Make a list of all members of the population and number them consecutively from zero to the last number of the population.
3.      Using the table of random number, point at any number without looking at the table. Since the code numbers is composed of 3 digits, just look at the last three digits of the number pointed out in the table. If the last three digits is a number in the list, then the member corresponding to that number is selected to be included in the sample.
4.      Using this number as reference, read the table horizontally whether to the left or to the right; or vertically up or down or diagonally. Continue recording those numbers found in the population ignoring those numbers which are not found. Discard duplicate numbers and continue until the required sample size is completed.

*  Stratified Random sampling
           - is used to avoid biasedness. This sampling technique is done by dividing the population into categories or strata and drawing the members to be included in the sample at random proportionate to its stratum or subgroup. T select a stratified random sample, the researcher must possess a clear understanding of the problem in order to determine the strata into which the population should be divided.
           Example. a) A college has a population of 2000 students of which 1200 are females and 800 are males. Here, the strata is the male and the female group of students.  After the categories or strata have been determined, a random sample is selected from each stratum which is proportional to its sample size. This avoids the error of selecting too few or too many members from each stratum. A sample of 400 students is required.  To obtain a sample proportional to the given members in each stratum, compute first for the proportion of the sample to the population. Then multiply the result to the number of males and to the number of females.

            400/2000 = 0.20

            ( 0.20 ) * ( 1200 ) =  240 females in the sample
            ( 0.20 ) * ( 800 ) =  160  males in the sample

The last step is to draw the required random sample size of 400 students either by lottery or by the use of table of random numbers from the 240 females and 160 males.

*  Systematic random sampling
          - is a process of selecting every nth element of the population until the desired sample size is obtained. The members or elements maybe arranged alphabetically or in any systematic fashion. To obtain n, divide the population size by the sample size. In the above example, 2000/400 = 5, which implies that every 5th element of the given population is chosen to be included in the sample.
           To know where to start from the population list, use the table of random numbers. Without looking, point at any number in the table and get the last digit. Supposed that the last digit in the number pointed is 3, it means that the 3rd member in the list is included. From the number 3, take every 5th element in the population as belonging to the sample. This means in the list 3, 8, 13, 18, 23, 28, 33, 38, 43, 48, 53, etc. are included in the sample.  

*  Cluster sampling
        Cluster is an intact group possessing common characteristics.
        Cluster sampling is an advantageous procedure if the population is spread out over a wide geographical area. It is also a practical sampling technique used if the complete list of members of the population is not available. An example is a population consisting of 600 teachers in the 30 public schools of Midsayap. To illustrate how the desired sample size of 140 teachers could be selected from the given population,  the steps are as follows :
       1.  Prepare a list of the cluster comprising the population and determine the sample size. The logical cluster is by school. Prepare a list of all the 30 public schools.
       2.  Estimate the average number of members per cluster in the population. Assume that the average number of teachers per school is 20.
       3.  Divide the required sample size by the average number of teacher per cluster to get the number of clusters to be selected. Since, the sample size is 140 and the average teacher per school is 20, the number of clusters shall be :   140/ 20 = 7                                          
       4.  Select the needed number of clusters. By using the table of random numbers, select the 7 public schools from the 30 population schools. Include all the members in the selected clusters. Since there is an average of 20 teachers per school, and 7 clustered school, the sample size of 140 teachers is completed.
                        Summary :
                        Number of public schools                               :  N = 30
                        Average number of teachers per school          :  x = 20
                        Desired sample size                                         :  n = 140
                        Required cluster                                              :  y = n/x = 140/20 = 7
Hence, there are 7 clusters. 

*  Multi stage sampling
             This is a more complex sampling technique which involves the following steps :
   1.  Divide the population into strata.
   2.  Divide each stratum into clusters.
   3.   Draw a sample from each cluster using the simple random sampling technique.
To illustrate, take a random sampling of all first year high schools in the island of Mindanao. Mindanao is divided into 6 regions which forms our stratum. From each stratum, we choose a private and a public high school which again forms a stratum. From these public and private high school, we select first year high school classes forming our clusters. From these clusters, we select first year students from each class by using the simple random sampling technique.


METHODS OF COLLECTING DATA
There are several methods of collecting data. There is however, no best method to obtain the desired information under investigation. The choice of appropriate methods to be used depends on the following factors :
1.      Nature of the problem.
2.      The population under investigation, and
3.      The time and the material factors.

The methods of collecting data are :
1.      The direct interview method
2.      The indirect or questionnaire method.
3.      The registration method, and
4.      Other methods such as;
a)      Observation
b)      Telephone interview
c)      Experiments

*  The direct or interview method
          This is considered as one of the most effective methods of collecting original data. To obtain accurate responses, the maybe conducted by well-trained interviewers. The interviewers maybe of great help to the respondents in answering questions which the respondents could not understand.

Advantages of the interview method
1.      It can give complete information needed in the study.
2.      It can yield precise and consistent information since the interview can immediately clarify any misinterpretation made by the respondents.

Limitations of the interview
      1.  It is more expensive and time consuming.       
      2.  It may yield inaccurate information since the interviewer  can influence the respondent’s    
           through his facial expression, tone of voice or wording the question.
    3. The interviewer may cheat by turning in dishonest responses if their expected or desired responses 
         are  not obtained. 
*  Indirect or Questionnaire method
           - is one of the earliest method of data gathering. It takes time to prepare because questionnaires need to be attractive. It can include illustrations, pictures and sketches. Its content especially the instructions, must be precise, clear, and self-explanatory.

Advantages of the questionnaire method       
1.      It is less expensive since questionnaires can be distributed personally or by mail.
2.      It is less-time-consuming since it can be distributed over a wider geographical area in a shorter time.
3.      It can give confidential responses since the respondents can answer the questionnaire privately.
4.      The answers obtained are free from any influence from the interviewer.

Limitations of the questionnaire method
1.      It cannot be accomplished by illiterates.
2.      It has a high proportion of non-response or non-return.
3.      It tends to yield wrong information since answers cannot be corrected right away.
4.      It tends to give incomplete information.

*   The Registration method
       By registration method, the respondents give information in compliance with certain laws, policies, rules, regulations, decrees or standard practices. Data which can be collected by registration method are as follows : marriage contracts, birth certificates, motor vehicle registration, license of firearms, registration of corporations, real estates, voters, etc.

Other methods :  
 *  Observation method is used to gather data regarding attitudes, behaviour, values, and cultural pattern of the sample under investigation.

*  Telephone interview are employed if the questions to be asked are brief and few. An example is the checks made on listeners to certain radio programs, like asking what programs his radio is turned on to. This method is used to find the most popular Radio or TV programs.

*  Experiment is employed to collect data if the investigator wishes to control the factors affecting  the variables being studied.  An example is when the researcher wants to determine the different factors affecting the academic performance of the students such as IQ methods or approaches used in teaching.

METHODS OF PRESENTING DATA
1.      Textual or Textular method
2.      Tabular method
3.      Semi-tabular method
4.      Graphical method

*  Textual presentation – the data is presented in paragraph form. The reader gets information by reading the gathered data in the paragraph.
*  Tabular presentation – the data is presented in rows and columns. This is more effective way of showing relationships or comparison of numerical data. It gives more precise, systematic and orderly presentation of data in rows and columns. This method makes comparison of figures easy and comprehensible.

*  Semi-tabular presentation – employs both the textual and the tabular methods. This method is used only if there are a few figures to be tabulated. The tables are followed by narrative explanations to make he facts more understandable.

*  Graphical presentation. The use of graphs is the most effective method of presenting statistical results or findings. It gives the relationship of data in pictorial form. Presentation of facts are made attractive and meaningful especially if colors are made and pictures re used, making it easy for important information to be grasped by the reader.

   **  Advantages of Graphical presentation
1.      Graphs enable students, readers and the busy executive to easily understand the essential facts that numerical data intend to convey. Private and government agencies use charts and graphs in their reports.
2.      They can easily attract attention and are more readily understood. It is easier to go through graphs than through quantitative data.
3.      Graphs simplify concepts that would otherwise have been expressed in so many words.

   **  Limitations of graph
      1.  Graphs are not as precise as tables.
      2.  Graphs require more skill and time to prepare.
      3.  Graphs can only be made after the data have been presented in tabular form.

Parts of a Statistical Table
  1.  Table heading – consist of a table number and the title.
  2.  Stub – are found at the left side of the body of the table which are categories or
       classifications.
  3.  Box head – The box head identifies what are contained in the column. Included in the Box
        head are the stub head, master caption and column caption.
  4.  Body – main part of the table. This contains the substance or the figures of one’s data.  

Table number
Title

Stub
Head
Master  Caption
Column Caption
Column Caption
Column Caption
Row Caption



Row Caption



Row Caption
B O D Y
of  the
Table
Row Caption



Row Caption




               Footnote
               Source of data

Example
Table 1.1
College Student Population of XYZ University
S. Y. 2007 – 2012

School Year
BSCS*
BSIT*
BSED*
BEED*
BSBA*
AS*
BSN*
2007 – 2008 
75
150
45
60
290
130
56
2008 – 2009 
70
140
 34
50
285
108
45
2009 – 2010 
50
120
30
45
270
62
34
2010 – 2011 
48
85
28
43
225
48
23
2011 – 2012 
30
75
22
38
213
44
18

 *  Summer enrolment is not included
     Source of data : Registrar’s Office

TYPES OF GRAPHS COMMONLY USE

1.      Linear graph ( Line graph )
2.      Bar graph
3.      Hundred percent chart ( Pie chart or Circle graph )
4.      Pictogram ( Pictograph )
5.      Statistical maps

*  Linear graph is a practical device uniquely suited to portray changes in values effectively over successive periods of time. Variations in the data are indicated by the variation in the movement of linear curves.
   **  Advantages of  linear graph
1.      The curves shows data as a continuous line; hence, it is continuous in its effect.
2.      The wandering line of the curve tells the whole story. At a glance one can see just what the situation is and what is likely to happen.
3.      Its preparation requires less time and skill. 

*  Bar graph – is one of the most common and widely used graphical devices. This consists of bars of equal width either all vertical or all horizontal. The length of each bar represents the frequency of each class. The bars must be of the same width and is arbitrary. The space between bars is about one-half of the width of the bars itself.

*  Pie chart ( circle graph ) – is use to represent quantities that make up a whole. It consists of a circle subdivided into sectors which look like pieces of a pie whose sizes are proportional to the magnitudes or percentages they represent. The pie chart aims to show per cent distribution of a whole into its component parts.

     **  Construction of a Pie chart    
1.      Express each component part as a certain per cent of the whole and multiply the result by 3.6 since 1 %  of the circumference is equal to 3.6 degrees.
2.      Mark off the desired number of degrees on the circumference of the circle using a protractor; connect this points to the center of the circle in order to produce pie-shaped areas which make-up the whole circle.
3.      Label the various segments horizontally whenever possible with the percentages indicated.
4.      Arrange the sector clockwise according to size.
5.      Use cross-hatching, coloring, shading, etc. to increase the effectiveness of a pie chart.
6.      Avoid overloading the chart by showing too many categories; simplicity is a fundamental characteristics of all charts.

*   Pictogram ( pictograph )
        An excellent device for portraying data is by means of pictures or symbols. Its chief purpose is to catch the reader’s attention to convey to him in a vivid manner basic numerical facts. It does not attempt to show details; it simply tries to facilitate comparison of approximate quantities.
        The symbol or picture should suggest the nature of the data being presented. For instance      
              a) corn production can be represented by a picture of a sack of corn;
b) truck importation can be represented by a picture of a truck;
c) population in one’s place can be represented by a person, etc.
    
*  Statistical maps
        One of the best ways to present data is through statistical maps. A  map can be drawn and divided into desired regions. Each region maybe distinguished from the other by use of shades, dots or cross-hatching. A statistical map is always accompanied by a legend which tells the meaning of the lines, colors or other symbols used.