The sampling distribution of the mean was defined in the section introducing sampling distributions. This section reviews some important properties of the sampling distribution of the mean introduced in the demonstrations in this chapter.

Mean

The mean of the sampling distribution of the mean is the mean of the population from which the scores were sampled. Therefore, if a population has a mean μ, then the mean of the sampling distribution of the mean is also μ. The symbol μ_M is used to refer to the mean of the sampling distribution of the mean. Therefore, the formula for the mean of the sampling distribution of the mean can be written as:

μ_M = μ

Variance

The variance of the sampling distribution of the mean is computed as follows:

That is, the variance of the sampling distribution of the mean is the population variance divided by N, the sample size (the number of scores used to compute a mean). Thus, the larger the sample size, the smaller the variance of the sampling distribution of the mean.

(optional) This expression can be derived very easily from the variance sum law. Let's begin by computing the variance of the sampling distribution of the sum of three numbers sampled from a population with variance σ². The variance of the sum would be σ² + σ² + σ². For N numbers, the variance would be Nσ². Since the mean is 1/N times the sum, the variance of the sampling distribution of the mean would be 1/N² times the variance of the sum, which equals σ²/N.

The standard error of the mean is the standard deviation of the sampling distribution of the mean. It is therefore the square root of the variance of the sampling distribution of the mean and can be written as:

The standard error is represented by a σ because it is a standard deviation. The subscript (M) indicates that the standard error in question is the standard error of the mean.

Central Limit Theorem

The central limit theorem states that:

Given a population with a finite mean μ and a finite non-zero variance σ², the sampling distribution of the mean approaches a normal distribution with a mean of μ and a variance of σ²/N as N, the sample size, increases.

The expressions for the mean and variance of the sampling distribution of the mean are not new or remarkable. What is remarkable is that regardless of the shape of the parent population, the sampling distribution of the mean approaches a normal distribution as N increases. If you have used the "Central Limit Theorem Demo," you have already seen this for yourself. As a reminder, Figure 1 shows the results of the simulation for N = 2 and N = 10. The parent population was a uniform distribution. You can see that the distribution for N = 2 is far from a normal distribution. Nonetheless, it does show that the scores are denser in the middle than in the tails. For N = 10 the distribution is quite close to a normal distribution. Notice that the means of the two distributions are the same, but that the spread of the distribution for N = 10 is smaller.

Figure 1. A simulation of a sampling distribution. The parent population is uniform. The blue line under "16" indicates that 16 is the mean. The red line extends from the mean plus and minus one standard deviation.

Figure 2 shows how closely the sampling distribution of the mean approximates a normal distribution even when the parent population is very non-normal. If you look closely you can see that the sampling distributions do have a slight positive skew. The larger the sample size, the closer the sampling distribution of the mean would be to a normal distribution.

Figure 2. A simulation of a sampling distribution. The parent population is very non-normal.

Question 1 out of 5.
The population has a mean of 14 and a standard deviation of 3. The sample size of your sampling distribution is N=10. What is the mean of the sampling distribution of the mean?

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Over Population IN EAST AFRICA

POPULATION OF EAST AFRICA

Population Growth

Population Growth rate refers to the rate at which population changes by increasing in size.

It is calculated as;

Birth rate – Death rate divide by 1000 X 100.

Birth rate should always be higher than death rate in order for population to increase.

Population growth rate (PGR) = Birth rate – Death rate X 100
                                                                          1000   

For example, assessing the birth rate of Tanzania to be 48% per annum and its death rate 12% per annum, calculate the population growth rate? 

PGR     =          48 – 12    X 100   =       3.6%
                           1000            

Factors which lead to High Population Growth Rate in East Africa

• Early marriages.

• High fertility rates.

• Polygamy.

• Low levels of education.

• Lack of family planning facilities.

• Excessive birth rates over death rate.

• Declining death rate due to improved medical facilities.

• Rural – urban population.

• Value the parents attach to their children.

• Level of income.
  
  
  
  Over Population
  Over population refers to a situation where a country’s resources are not adequate to provide the necessary means of survival to its people. The resources are over utilised in order for the people to survive.
  Population migration
  In East Africa people are always moving from time to time. This affects the nature of the population distribution. There are a number of factors why people migrate. These include; 
  • Economic reasons
  • Political   reasons
  • Cultural   reasons
  • Social  reasons
  Migration can be for a short time and for a long time. It can be internal and external or international.   
  Reasons for migration
  
  
  • Unemployed people move from the rural to the urban centres in search for employment opportunities.
  • Cultivators migrate in search for good land for farming (agriculture) purpose.
  • Traders move in search for trade opportunities.
  • In northern Uganda people are frequently moving because of political instabilities etc.   
  
  

  Census

  Questions
  1. What is a census?
  2. Have you ever participated in a census?
  3. If so, what questions were you asked?
  4. What is the importance of carrying out a population census?
  5. What are the attitudes of people towards census exercises?
  A Census is the procedure of systematically acquiring and recording information about the members of a given population. It is a regularly occurring and official counting of a particular population.
   10. POPULATION CENSUS
  This is the counting of people in a given area.
  WHY IT IS CARRIED OUT (IMPORTANCE)
  • To establish the number of people in the country in terms of Age, Sex distribution, level of education, standards of living.
  • To enable the government distribute the resources according to the prevailing population in the area. That is, government plans on how to adequately allocate the resources.
  • To find out the relationship between the prevailing population and their conditions of living for example housing facilities, levels of income, levels of employment.
  • To help the government design a sound policy on population; in terms of birth rates, migration and distribution of resources.
  WHAT TO CONSIDER
  • Age
  • Sex
  • Level of education
  • Occupation
  • Standards of living
  • Mode of transport
  • Etc.
  
  11. LIFE EXPECTANCY
  Life expectancy is the average age at which people die. Life expectancy is very low in the developing countries and high in the developed countries.
  • Optimum population.Optimum population refers to a situation where the number of people is in balance with the available resources. This means that;  
  • Optimum conditions can only be maintained if the exploitation of new resources and development of employment opportunities keeps in pace with the increasing rate of population.
  • The size, distribution and structure of the population within a country must be viewed in relation to its natural resources.
  • If the country’s resources are not adequately supporting its population, then people are likely to be poor, dependant and hungry. A higher rate of unemployment is also an indicator.
  Where a country has gone beyond the optimum population level, there is likely to be a  problem of over population.
  
  

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POPULATION OF EAST AFRICA

POPULATION OF EAST AFRICA

Population Density

Population density describes the number of people per square kilometer of land area.

It is calculated as;
      PD                       =          POPULATION (P)
                                                AREA (A)

ACTIVITY 1: - Warm up / Brain Storming

Ask learners to brainstorm on the following terms;

• Population
• Population census
• Population distribution
• Population density
• Under population
• Over population
• Birth rate
• Growth rate
• Death rate

TABLE 1B: POPULATION DENSITY OF EAST AFRICAN COUNTRIES

Country	Total area (km2)	Total Population 2008	Population density
Uganda	244,400	31,367,972	.......................
Kenya	585,000	37,958,838	.......................
Tanzania	943,000	40,212,612	.......................
Total	1772,400	......................	.......................

Source: Population Reference Bureau (2006)
ACTIVITY 2:

In groups of 5 – 8, study the Table 1B above and answer the following questions

• Which of East African country has the;
• Largest population.
• Least population.

• Calculate the population size of East Africa.

• How is high population an advantage to any one country of East Africa.

• What are disadvantages of high population size to any one country of east Africa?

• To which extent may low population be of advantage to your country?

ACTIVITY 3

Given the formula:

            Population Density     =         Total Population
                                                                        Area

• Use the Table 1B above to calculate the population density of
• Uganda
• Kenya
• Tanzania

• Which of the three countries has the
• Lowest population density?
• Highest population density?

• What could be the major cause of the differences in the population densities?

ACTIVITY 4

Study the sketch map below showing the population distribution of East Africa and answer the following questions

  

This would also be considered a group task

• Using the sketch map provided above, which areas of East Africa are:
• Densely populated
• Sparsely populated

• Why do you think that these areas are;
• Densely populated?
• Sparsely populated?

• What challenges do the people living in those areas mentioned in question (b) above face?

• Suppose a population officer, suggest ways in which you would overcome those challenges

ACTIVITY 5

Study Table 2 below showing population figures in Uganda between 1990 –2001 and answer the questions below.

YEAR	POPULATION (000)
-1991	16.000
1992 - 1993	16.500
1994 - 1995	17,000
1996 - 1997	18,000
1998 - 1999	19,500
2000 - 2001	23,000

 (i) Draw a line graph to show the above information.

 (ii) Describe the trend of population growth in Uganda between 1990  to 2001.

 (iii) State the causes for an increase in population between 1990 –2001.

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Instruments used in Surveying....FORM THREE

  Instruments used in Surveying

                                    Instruments Used for Measuring Distance:

1. Chain 

a.  Metric chain

b.  Steel band chain

c.   Günter’s Chain

d.  Revenue Chain

e.  Engineers chain

2. Arrows (chain pins)

3. Tapes

a.  Cloth or linen Tape

b.  Metric Woven Metallic Tape

c.   Metric steel Tape

d.  Invar tape.

e.  Synthetic Tape.

f.    Wooden pegs.

5. Ranging Road.

6. Ranging Poles.

7. Offset Rod.

8. Laths

9. Whites

10. Plumb Bob

1. Chain

The chain is composed of 100 or 150 pieces of galvanized mild steel wire 4mm in diameter called links. The end pf each link are bent into a loop and connected together by means of three oval rings. The ends of the chain are provided with brass handles for dragging the chain on the ground. The length of link is the distance between the centers of the two consecutive middle rings. The end links includes the handles. Metallic tags or indicators are fixed at various distinctive of the chain to facilitate quick reading.

a. Metric surveying chains:

The chains are made in lengths of 20 and 30 meters. To enable the reading of factious of a chain, tallies (tags) are fixed at every five meter length and small brass rings are provided at every meter length. To facilitate holding of the arrows in position with the handle, a groove is cut on the out side surface of the handle. The handle joints are flexible. the tallies used for marking the distances in a metric chain are marked with letters ‘Me’ and ‘m’.

b. Steel Band Chain:

It consists of a ribbon of steel with bras handle at each end. It is 20 or 30long and 16 mm wide. It is wound on an open steel cross or on the metal reel in a closed case. The graduations are etched as meters decimeters, centimeters on one side and 0.2 m links on the other. Brass tallies are fixed at every 5 m length of the band.

c. Günter’s Chain:

It is 66 fit long and is divided into 100 links. Each link is 0.66 ft long. It is very convenient for measuring distance in miles and furlongs. Also for measuring area and when the units of area is an acre

d. Revenue Chain:

It is commonly used for measuring fields in cadastral survey. It is 33 ft long and divided into 16 links. Each link is 2.0625 ft long.

e. Engineer’s chain:

It is 100 ft long and it is divided into 100 links. Each link is 1 ft in a length. Used in all Engineering surveys.

2. Arrows (chain pins):

They are also called as marking or chaining pins and are used to mark the end of chain during the process of chaining. They are made up of good quality hardened and tempered steel wire of 4mm in diameter. The arrows are made 400 mm in length. They are pointed at one end of inserting in to the ground. The other end is in to a ring.

3. Tapes:

a. Cloth or Linen Tape:

Used for taking subsidiary measurements, such as offset. It is very light and handy. It is easily affected by damp. If wet it shrinks. It stretches easily and likely to twist.

b. Metric Woven Metallic Tape:

They are available in 2, 10, 30, and 50 meters. The tape is made of yarn and metal wire. A metal ring is attached to the outer end of tapes. The length of the tape includes the metal ring. At every centimeter a black line 8 to 10 mm in height is drown. Every 5 centimeters is marked with an arrow in black. Every decimeter and meter is marked with a back line extending over the full width of the tape/ the graduation marks at every decimeter and meter are numbered with black and red figures, respectively.

c. Metric Steel Tape:

Tape is available in 1, 2, 10, 30, and 50 meters. The tape is of steel or stainless steel. The outer end is provided with a ring. The length of the tape includes the metal ring. The tape is marked with a line at every five millimeters, centimeters, decimeters, and meter. Every decimeter and meter shall be marked with Hindu Arabic numerals in bold. When the button release devised is pressed, the tape automatically rewind in to the case.

d. Invar Tape:

For highest precision work the invar tape in used. It is made of an alloy of steel and nickel (36%).

It is 6 mm wide and may be obtained in length of 30m and 100m. It is not calibrated through its length but has terminal lines. Each terminal division has ten 1 mm division. It is very expensive.

e. Synthetic Tape:

The tapes are manufactured of glass glass fiber having PVC coating. They are graduated every 10 mm and figured every 100 mm. Meter, figures are shown in red. They are convenient for measuring shorts lengths.

Instruments for marking stations:

1. Wooden Pegs:

These are used to mark the positions. They are made of hard timber and tapered at one end. They are usually, 2.5 cm square and 15 cm long. But in soft ground 40 to 60 cm long and 4 to 5 cm square is suitable. They should be driven in the ground with about 4 cm lengths, projecting above the ground.

2. Ranging rods:

Used for making the positions of stations and for ranging. They are made of seasoned timber of teak, blue pine, sisov or deodar. They are circular or octagonal in cross section of 3 cm diameter. Lower shoe is 15 cm long. They are made in two sizes as 2 meters and 3 meters and are divided in to equal parts each 0.2 m long. They are painted alternatively black and white or red and white. Now a day instead of timber, mild steel hallo pipes are used.

3. Ranging Poles:

Similar to the ranging rods but are heavier, they vary in length from 4 m to 6 m or more. Used in the case of very long lines.

4. Offset Rod:

Similar to the ranging rod, they are usually 3 m long and is divided into parts each 0.2 m length. Top is an provided with an open ring for puling or pushing the chain through a hedge. It has two short narrow vertical slots. It is used for aligning short offsets.

5. Laths:

Useful for ranging long lines, also used over uneven ground where the ranging rod is not visible due to obstructions, they are light, cheap, being white; they are easily visible at a great distance. Unusually 1.0m long

6. Whites:

When the ranging rod is not available or insufficient, whites are used. These are thin strip of bamboo and 40 cm to 1 m in length. One end is sharp and the other end is split for inserting pieces of white papers. They are also useful for temporary marking of counter points.

7. Plumb Bob:

The plumb bob is required when measuring the distance along slopes in order to transfer points to the ground. It is also used for testing the verticality of ranging poles.

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