Statistics Final Project Handout

SAMPLING TECHNIQUES AND NORMAL DISTRIBUTION

1 STATISTICS & PROBABILITY

Foreword

This hand out contains various lectures on sampling techniques and normal

distribution as well as problems and exercises to guide you in. We hope that the

contents of this hand out will be much of help to your study of statistics.


PART 1: SAMPLING TECHNIQUES

In reality there is simply not enough; time, energy, money,

labor/man power, equipment, access to suitable sites to

measure every single item or site within the parent

population or whole sampling frame. Therefore an

appropriate sampling strategy is adopted to obtain a

representative, and statistically valid sample of the whole.

WHAT IS SAMPLING?

A shortcut method for investigating a whole population

Data is gathered on a small part of the whole parent population or

sampling frame, and used to inform what the whole picture is like

CONSIDERATIONS IN SAMPLING

Larger sample sizes are more accurate representations of the whole

The sample size chosen is a balance between obtaining a statistically

valid representation, and the time, energy, money, labor, equipment and

access available

A sampling strategy made with the minimum of bias is the most

statistically valid

Most approaches assume that the parent population has a normal

distribution where most items or individuals clustered close to the mean,

with few extremes


WHAT ARE SAMPLING TECHNIQUES?

Sampling techniques are classified as either probability or

nonprobability. In probability samples, each member of the

population has a known non-zero probability of being selected. Probability

methods include random sampling, systematic sampling, and stratified

sampling. In nonprobability sampling, members are selected from the

population in some nonrandom manner. In this part we will be focusing on the

probability types of sampling techniques.

THREE MAIN TYPES OF SAMPLING STRATEGY:

Random

Systematic

Stratified

Within these types, you may then decide on a; point, line, area method.

RANDOM SAMPLING

Least biased of all sampling techniques, there is no

subjectivity - each member of the total population has an

equal chance of being selected

Can be obtained using random number tables

Microsoft Excel has a function to produce random number


GENERATING RANDOM NUMBERS USING MICROSOFT EXCEL

The generation random numbers can be done through the use of Microsoft excel

using the RAND () function.

Simply type the function into a cell or in the formula bar and it will produce a

random number in that cell.

ILLUSTRATION 1.0


ADDITIONAL KNOWLEDGE

If you want to use RAND to generate a random number but don’t want to

change the numbers every time the cell is calculated, enter the function

in the formula bar and press F9 to change the formula into a random

number.

You can modify the formula to obtain whatever range you wish, for

example if you wanted random numbers from one to 250, you could enter

the following formula: =INT(250*RAND())+1. Where INT eliminates the

digits after the decimal, 250* creates the range to be covered, and +1 sets

the lowest number in the range. (See illustration 1.1).

Paired numbers could also be obtained using; =INT(9000*RAND())+1000

ILLUSTRATION 1.1


PRACTICE WHAT YOU LEARNED

Using the RAND function in Microsoft excel generate a random number between the following values:

1. 0 and 90 2. 0 and 20 3. 30 and 200

4. 10 and 30 5. 100 and 900

METHODOLOGIES

Random point sampling

A grid is drawn over a map of the study area

Random number tables are used to obtain coordinates/grid references for

the points

Sampling takes place as feasibly close to these points as possible

Random line sampling

Pairs of coordinates or grid references are obtained using random number

tables, and marked on a map of the study area

These are joined to form lines to be sampled

Random area sampling

Random number tables generate coordinates or grid references which are

used to mark the bottom left (south west) corner of quadrats or grid

squares to be sampled


ILLUSTRATION 1.3

A random number grid showing methods of generating random numbers, lines and areas

ADVANTAGES AND DISADVANTAGES OF RANDOM SAMPLING

Advantages:

Can be used with large sample populations

Avoids bias

Disadvantages:

Can lead to poor representation of the overall parent population or area

if large areas are not hit by the random numbers generated. This is made

worse if the study area is very large

There may be practical constraints in terms of time available and access

to certain parts of the study area

http://www.rgs.org/NR/rdonlyres/950655FF-290A-48BC-86BD-24CD99C34D13/0/FW_Techs_sample1Big.jpg


SYTEMATIC SAMPLING

Samples are chosen in a systematic, or regular way.

They are evenly/regularly distributed in a spatial context, for example

every two meters along a transect line

They can be at equal/regular intervals in a temporal context, for example

every half hour or at set times of the day

They can be regularly numbered, for example every 10th house or person

METHODOLOGIES

A. Systematic point sampling

A grid can be used and the points can be at the intersections of the grid

lines (A), or in the middle of each grid square (B). Sampling is done at the

nearest feasible place. Along a transect line, sampling points for

vegetation/pebble data collection could be identified systematically, for

example every two meters or every 10th pebble.

B. Systematic line sampling

The eastings or northings of the grid on a map can be used to identify

transect lines (C and D) alternatively, along a beach it could be decided

that a transect up the beach will be conducted every 20 meters along the

length of the beach.

C. Systematic area sampling

A ‘pattern' of grid squares to be sampled can be identified using a map of

the study area, for example every second/third grid square down or

across the area (E) - the south west corner will then mark the corner of a

quadrat. Patterns can be any shape or direction as long as they are

regular (F).


ILLUSTRATION 1.4

ADVANTAGES AND DISADVANTAGES OF SYSTEMATIC SAMPLING

Advantages:

It is more straight-forward than random sampling

A grid doesn't necessarily have to be used, sampling just has to be at

uniform intervals

A good coverage of the study area can be more easily achieved than using

random sampling

Disadvantages:

It is more biased, as not all members or points have an equal chance of

being selected

It may therefore lead to over or under representation of a particular

pattern


STRATIFIED SAMPLING

This method is used when the parent population or sampling

frame is made up of sub-sets of known size. These sub-sets make up different

proportions of the total, and therefore sampling should be stratified to ensure

that results are proportional and representative of the whole.

TYPES OF STRATIFIED SAMPLING

A. Stratified systematic sampling

The population can be divided into known groups, and each group sampled

using a systematic approach. The number sampled in each group should be in

proportion to its known size in the parent population.

For example: the make-up of different social groups in the population of a town

can be obtained, and then the number of questionnaires carried out in different

parts of the town can be stratified in line with this information. A systematic

approach can still be used by asking every fifth person.

B. Stratified random sampling

A wide range of data and fieldwork situations can lend themselves to this

approach - wherever there are two study areas being compared, for example

two woodlands, river catchments, rock types or a population with sub-sets of

known size, for example woodland with distinctly different habitats.

Random point, line or area techniques can be used as long as the number of

measurements taken is in proportion to the size of the whole.


For example: if an area of woodland was the study site, there would likely be

different types of habitat (sub-sets) within it. Random sampling may altogether

‘miss' one or more of these.

Stratified sampling would take into account the proportional area of each

habitat type within the woodland and then each could be sampled accordingly;

if 20 samples were to be taken in the woodland as a whole, and it was found

that a shrubby clearing accounted for 10% of the total area, two samples would

need to be taken within the clearing. The sample points could still be identified

randomly (A) or systematically (B) within each separate area of woodland.

C. Proportionate Stratified Random Sampling

In proportional stratified random sampling, the size of each strata is

proportionate to the population size of the strata when looked at across the

entire population. This means that each stratum has the same sampling

fraction.

For example, let’s say you have four strata with population sizes of 200, 400,

600, and 800. If you choose a sampling fraction of ½, this means you must

randomly sample 100, 200, 300, and 400 subjects from each stratum

respectively. The same sampling fraction is used for each stratum regardless of

the differences in population size of the strata.

D. Disproportionate Stratified Random Sample

In disproportionate stratified random sampling, the different strata do not have

the same sampling fractions as each other. For instance, if your four strata

contain 200, 400, 600, and 800 people, you may choose to have different

sampling fractions for each stratum. Perhaps the first strata with 200 people

has a sampling fraction of ½, resulting in 100 people selected for the sample,

while the last strata with 800 people has a sampling fraction of ¼, resulting in

200 people selected for the sample.


The precision of using disproportionate stratified random sampling is highly

dependent on the sampling fractions chosen and used by the researcher. Here,

the researcher must be very careful and know exactly what he or she is doing.

Mistakes made in choosing and using sampling fractions could result in a

stratum that is overrepresented or underrepresented, resulting in skewed

results.

ILLUSTRATION 1.5

A diagram highlighting the benefits of using stratified random sampling and

stratified systemic sampling within certain fieldwork sites.

http://www.rgs.org/NR/rdonlyres/B1590E3F-99C1-470B-A6C2-C6658A4B2DCC/0/FW_Techs_sample3Big.jpg


ADVANTAGES AND DISADVANTAGES OF STRATIFIED SAMPLING

Advantages:

It can be used with random or systematic sampling, and with point, line

or area techniques

If the proportions of the sub-sets are known, it can generate results which

are more representative of the whole population

It is very flexible and applicable to many geographical enquiries

Correlations and comparisons can be made between sub-sets

Disadvantages:

The proportions of the sub-sets must be known and accurate if it is to

work properly

It can be hard to stratify questionnaire data collection, accurate up to

date population data may not be available and it may be hard to identify

people's age or social background effectively


NONPROBABILITY SAMPLING

The difference between nonprobability and probability sampling is that

nonprobability sampling does not involve random selection and probability

sampling does. Does that mean that nonprobability samples aren't

representative of the population? Not necessarily. But it does mean that

nonprobability samples cannot depend upon the rationale of probability theory.

At least with a probabilistic sample, we know the odds or probability that we

have represented the population well. We are able to estimate confidence

intervals for the statistic.

With nonprobability samples, we may or may not represent the population well,

and it will often be hard for us to know how well we've done so. In general,

researchers prefer probabilistic or random sampling methods over non-

probabilistic ones, and consider them to be more accurate and rigorous.

However, in applied social research there may be circumstances where it is not

feasible, practical or theoretically sensible to do random sampling. Here, we

consider a wide range of non-probabilistic alternatives.

ACCIDENTAL, HAPHAZARD OR CONVENIENCE SAMPLING

One of the most common methods of sampling goes under the various titles

listed here. I would include in this category the traditional "man on the street"

(of course, now it's probably the "person on the street") interviews conducted

frequently by television news programs to get a quick (although non

representative) reading of public opinion.


In many research contexts, researchers sample simply by asking for volunteers.

Clearly, the problem with all of these types of samples is that there is no evidence

that they are representatives of the population the researchers interested in

generalizing to, and in many cases would clearly suspect that they are not.

PURPOSIVE SAMPLING

In purposive sampling, we sample with a purpose in mind. We

usually would have one or more specific predefined groups we are seeking. For

instance, have you ever run into people in a mall or on the street who are

carrying a clipboard and who are stopping various people and asking if they

could interview them? Most likely they are conducting a purposive sample (and

most likely they are engaged in market research). They might be looking for

Caucasian females between 30-40 years old. They size up the people passing

by and anyone who looks to be in that category they stop to ask if they will

participate. One of the first things they're likely to do is verify that the

respondent does in fact meet the criteria for being in the sample. Purposive

sampling can be very useful for situations where you need to reach a targeted

sample quickly and where sampling for proportionality is not the primary

concern. With a purposive sample, you are likely to get the opinions of your

target population, but you are also likely to overweight subgroups in your

population that are more readily accessible.


HOW MUCH HAVE YOU LEARNED?

True or False

1. In reality there is enough time and resources that we don’t need to

adopt an efficient sampling technique

2. In probability sampling techniques there is a zero chance of being

selected.

3. A non-probability sampling technique represents a population

perfectly

4. There is no subjectivity in a random sampling

5. Stratified sampling ensures that results are proportional and

representative of the whole.

Answer here

1.

2.

3.

4.

5.


PART 2: THE NORMAL DISTRIBUTION

Data can be "distributed" (spread out) in different ways. It can

be spread out more on the left ... or more on the right or it can

be all jumbled up but there are many cases where the data tends to be around

a central value with no bias left or right and that’s where normal distribution

comes in.

THE "BELL CURVE" IS THE NORMAL DISTRIBUTION CURVE.

Many things closely follow a Normal Distribution:

• Heights of people

• Size of things produced by machines

• Errors in measurements

• Blood pressure

• Marks on a test

We say the data is "normally distributed"….

The Normal Distribution has:

• mean = median = mode about the center

• Symmetry about the center

• 50% of values less than the mean and 50%

greater than the mean


THE STANDARD DEVIATION

The standard deviation is a measure of how spread out

numbers are

When you calculate the standard deviation of your data, you will find that

(generally):

68% of the values are within

1 standard deviation of the mean

95% are within 2 standard deviations

99.7% are within 3 standard

deviations


STANDARD SCORES

The number of standard deviations from the mean is also

called the "Standard Score", "sigma", "z-score", or “the z-value”

FINDING THE AREA OF A STANDARD NORMAL CURVE

Determining the area of a standard normal curve is a crucial for many problems

involving a normal curve. To find the area, you need a Z-table it is a table

containing the areas under the standard normal curve between 0 and a

specified number Z

ILLUSTRATION 2.1

The shaded part is the area under standard normal curve between 0 and z


USING THE Z TABLE

The table below is called a Z distribution table this is where you will look for

the area between zero and a specified Z number.

ILLUSTRATION 2.2

To determine the area under the standard normal curve for example, between 0 and 0.62, First go down the left-hand column, labeled z to 0.6 , then go across that row until column 0.02, there you will find the number in the table 0.2324 and that is the area of the normal curve between 0 and 0.62

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359

0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753

0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141

0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517

0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879

0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224

0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.254

Tip: In case of negative z scores, apply the fact that the normal curve is symmetric

about 0, by symmetry the area between a positive z score and 0 is the same with its

negative counterpart. In the above example if the given Z score was –0.62 its area

would still be the same with 0.62, note that while Z scores may be negative areas must

be positive.

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359

0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753

0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141

0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517

0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879

0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224

0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.254


MORE ON FINDING THE AREA….

To find the area of a normal curve to the right of a positive value

simply remember that a standard normal curve is symmetric about 0

it follows that the area under the normal curve to the right of z = 0 is 0.5

ILLUSTRATION 2.3

Find the area under the normal curve to the right of z= 0.72

I

The area to the right of 0 is 0.5

Area to the right of 0.72=0.2358

III

II

Area between 0 and 0.72 = 0.2642 0.72

0.72

The area between 0 and 0.72 is 0.2642, since the total area to the right of z = 0 is 0.5

to get the area to the right of 0.72 therefore, 0.5-0.2642 = 0.2358

To find the area between two positive z values, for example 1.02 and 2.1 first get

the individual areas between 0 and 1.02, between 0 and 2.1, the required area

is the difference of the area between 0 and 2.1 and 0 between 1.02


ILLUSTRATION 2.4

Find the area between two positive z values, 1.02 and 2.1

1.02 2.1

Find the area between 0 and 1.02 Find the area between 0 and 2.1

which is 0.3461 which is 0.4821

1.02 2.1

To get the required area subtract the area between 0 and 2.1 with the area between 0 and 1.02 hence:

0.4821-0.3461 = 0.136

Tip: Always area larger minus area smaller.

For areas between a negative and positive value for example, -1.5 and 0.21,

simply add the area between 0 and -1.5 with the area between 0 and 0.21.


HOW MUCH HAVE YOU LEARNED?

I. True or False

1. In a normal curve the data distribution is concentrated on the right or left.

2. In a normal curve the mean, median and mode are not equal.

3. A normal distribution curve symmetrical about the mean.

4. The standard deviation is the measure how the numbers are close to the

mean.

5. You need a Chi-square table to find the area in a normal distribution curve.

6. To get the required area between two positive z values always add the area

larger to the area smaller.

Answer Here:

1.

2.

3.

4.

5.

6.



Find the area of the ff.

1. Between 0 and 1.98










11. Between 0 and -1


13. Between 0 and -2.4



16. Between 0 and – 1.2

17. Between 0 and – 1.64




21. To the right of 2.5

22. To the right of 1.92

23. To the left of -1.42

24. To the left of -2.0

25. Between 1.0 and 2.2

26. Between 0.99 and 2

27. Between 1.21 and 2.8

28. Between -0.92 and 0. 91

29. Between -1 and 1

30. Between -2 and 2.6


FINDING THE Z VALUE

Now we are done with finding the area in a normal curve using

z values but what if we are given the reverse? We are to find

the z value using a given area?

To find the value of the unknown Z score using a given area between it and 0,

simply look it up in the Z distribution table the Z value that corresponds to it,

let us say you have a given area of 0.3212 when you look it up in the table you

can see that the value it corresponds to is 0.92, it’s very similar in finding the

area albeit the reverse way.

When you are given an area to the right or left of an unknown Z score, simply

subtract the area given to 0.5 since we follow the assumption that each half of

the curve has an area of 0.5, then find the resulting difference in the Z table.

KEEP PRACTICING

Find the z value that corresponds to the given area.

0.4938 0.4956


0.3023 0.0281

STANDARDIZING A NORMAL CURVE

We have already seen that the area in a normally distributed

curve can be determined with the use of a z table. However, as we can notice

the z table is only from 0 to 3.0 but in reality, there can be a normal distribution

with the values of the mean and standard deviation different from 0 and 1 for

example 20, 36, 50, in this case the normal distribution must be standardized

so we can find the areas with relative ease since it is within the range of the z

distribution table.

For a normal variable x, to be converted to a z value by using the formula:

𝑧 =x−μ

𝜎

Where µ is the mean and 𝛔 is the standard deviation, x is your normal variable.


ILLUSTRATION 2.5

Using the formula convert 20 to a z value with a mean of 25 and standard deviation of 2

𝑧 =x − μ

𝜎

Hence:

𝑧 =20 − 25

2

Z = -2.5


Using the formula convert the following x values for a normal distribution to z values. Mean = 30; Standard deviation = 3

1. 26

2. 25

3. 19

4. 20

5. 31

6. 37

7. 32

8. 40

9. 26

10. 22


To find the area between two x values for a normal distribution, convert first the

values of x to their corresponding z values, then find the area just like we did in

previous exercises.

KEEP PRACTICING

Find the area under the normal curve with parameters of mean 25 and

standard deviation of 2.25

1. Between 25 and 30




5. Between 21and 26







ADDITIONAL THOUGHT….

Now we know how to standardize a normal curve by converting our x values to

z scores, but what about if we are given the reciprocal task? In that case we

need to simply use this formula.

𝑥 = z𝜎 + μ

THE NORMAL DISTRIBUTION CURVE AS PROBABILITY DISTRIBUTION CURVE

We have already learned how to find the area, z values and

standardization. Now we will move on to the application of the normal

distribution curve as a probability distribution curve for normally distributed

variables. Here the area under the curve corresponds to a probability or the

chance of being randomly chosen. Let us say we want to know the probability

of the values between 0 and 2 being chosen in a random draw, in this case the

area between 0 and 2 represents the probability of any values being chosen

between 0 and 2, and that would be 0.4772 or 47.72% probability.

For probabilities however, a special notation is used. For example if we are to

find the probability of any value between 0 and 2 we write it as P (0<z<2).



Write the following in probability notation

1. The probability between -2 and 0

2. The probability between 1 and 2.5

3. The probability between 2 and 3

4. The probability between -0.21 and 0

5. The probability between -1.89 and 1.5

6. The probability between 1.9 and 3

7. The probability between 2.75 and 1.9

8. The probability between 1.9 and 2.9

9. The probability between 0 and 2

10. The probability between 0 and 2.89

Now that we have familiarized ourselves with the probability

notation let us now move to solving probability problems by

finding the area, the process is just the same as the earlier

illustrations the only difference is that the area will not just an ordinary area

anymore, it represents probability and the givens are in probability notation.


ILLUSTRATION 2.5

Find the probability using the standard normal distribution.

1. P (0<z<1.21)

This means that we need to find the probability between 0 and 1.21 under the

normal curve now, using the z table look up the area between 0 and 1.21 and

that would be 0.3869 or 38.69%

1.21

2. P (-2<z<1.21) here we need to find the probability between -2 and 1.21,

Remember that we mentioned earlier that in finding the area between a

negative and positive z values that we first need to find the area between

-2 and 0, second, find the area between 0 and 1.21 the lastly add the two

areas.

+ =

-2 0 0 1.21 -2 0 1.21

0.4772 0.3869 0.8641 or 86.41%


KEEP PRACTICING

Find the probabilities for each using the standard normal distribution.

1. P (-1.2<z<0.29)

2. P (0<z<1.2)

3. P (0<z<2)

4. P (0<z<1.31)

5. P (-1.89<z<0)


APPENDIX

REFERENCES/SOURCES:

MSA Statistics and Probability

Chapter 7 Normal Distribution Curve

Authors: Merlie S. Alferez and Ma. Cecilia Duro

Statistics and Probability Sampling Techniques http://www.rgs.org/OurWork/Schools/Fieldwork+and+local+learning/Fieldw

ork+techniques/Sampling+techniques.htm

Normal Distribution Curve

http://www.mathsisfun.com/data/standard-normal-distribution.html

http://www.google.com.ph/imgres?q=normal+distribution+curve&hl=fil&sa=X&biw=1280&bih=935&tbm=isch&tbnid=Uup00D7vCzMS5M:&imgrefurl=http:

//www.regentsprep.org/Regents/math/algtrig/ATS2/NormalLesson.htm&docid=H3dht9FFLdiDXM&imgurl=http://www.regentsprep.org/Regents/math/algtrig/ATS2/normal67.gif&w=679&h=357&ei=ip5FUcjsHYbqrAfa3YCoBA&zoo

m=1&ved=0CFUQhBwwAw&ved=1t:3588,r:3,s:0,i:85&iact=rc&dur=396&page=1&tbnh=163&tbnw=310&start=0&ndsp=18&tx=213&ty=102

http://www.rgs.org/OurWork/Schools/Fieldwork+and+local+learning/Fieldwork+techniques/Sampling+techniques.htm

http://www.rgs.org/OurWork/Schools/Fieldwork+and+local+learning/Fieldwork+techniques/Sampling+techniques.htm

http://www.mathsisfun.com/data/standard-normal-distribution.html

http://www.google.com.ph/imgres?q=normal+distribution+curve&hl=fil&sa=X&biw=1280&bih=935&tbm=isch&tbnid=Uup00D7vCzMS5M:&imgrefurl=http://www.regentsprep.org/Regents/math/algtrig/ATS2/NormalLesson.htm&docid=H3dht9FFLdiDXM&imgurl=http://www.regentsprep.org/Regents/math/algtrig/ATS2/normal67.gif&w=679&h=357&ei=ip5FUcjsHYbqrAfa3YCoBA&zoom=1&ved=0CFUQhBwwAw&ved=1t:3588,r:3,s:0,i:85&iact=rc&dur=396&page=1&tbnh=163&tbnw=310&start=0&ndsp=18&tx=213&ty=102








ANSWER KEY

How much have you learned? (Page 16)

1. False

2. False

3. False

4. True

5. True

How much have you learned? (Page 23)

1. False

2. False

3. True

4. True

5. False

6. False


Practice What You Learned (Page 24)

1. between 0 and 1.98

Answer: 0.4761


Answer: 0.4918


Answer: 0.4821


Answer: 0.0832


Answer: 0.2486


Answer: 0.4066


Answer: 0.4898


Answer: 0.2324


Answer: 0.4985


Answer: 0.4987

11. between 0 and -1

Answer: 0.3643

12. between 0 and -2

Answer: 0.4772

13. between 0 and -2.4

Answer: 0.4927


Answer: 0.1255


Answer: 0.4525

16. between 0 and – 1

Answer: 0.3849

17. between 0 and – 1.64

Answer: 0.4495


Answer: 0.4925

19. between 0 and –3

Answer: 0.4987


Answer: 0.4726

21. to the right of 2.5

Answer: 0.5-0.4938= 0.0062

22. to the right of 1.92

Answer: 0.5-0.4726= 0.0274

23. to the left of -1.42

Answer: 0.5-0.4222 = 0.0778

24. to the left of -2.0

Answer: 0.5-0.4772= 0.0228

25. between 1.0 and 2.2

Answer: 0.3413-0.4861= 0.1448

26. between 0.99 and 2

Answer: 0.3389-0.4772= 0.1383

27. between 1.21 and 2.8

Answer: 0.3869-0.4974= 0.1105

28. between -0.92 and 0. 91

Answer: 0.3212+0.3186 = 0.6398

29. Between -1 and 1

Answer: 0.3849+0.3849 = 0.6826

30. Between -2 and 2.6

Answer: 0.9725


Keep Practicing (Page 25)

Z= 2.5

Z= -2.62

Z= 0.85

Solution:

0.5- 0.0281 = 0.4719 (Z value=1.91)

Z= 1.91


Practice What You Learned

(Page 27)

1. 26= (26-30)/3= -1.33

2. 25 = (25-30)/3= -1.67

3. 19 = (19-30)/3= -3.67

4. 20 = (20-30)/3 = -3.33

5. 31 = (31-30)/3 = 0.33

6. 37= (37 – 30)/3= 2.33

7. 32= (32-30)/3= 0.67

8. 40= (40-30)/3= 3.33

9. 26= (26-30)/3= -1.33

10. 22= (22-30)/3= -2.6


1. between 25 and 30

z= (25-25)/2.25= 0

z= (30-25)/2.25= 2.22

Area= 0.4868


z= (25-25)/2.25= 0

z= (20-25)/2.25= -2.22

Area: 0.4868


z= (25-25)/2.25= 0

z= (26-25)/2.25= 0.44

Area: 0.1700


z= (25-25)/2.25= 0

z= (28-25)/2.25= 1.33

Area= 0.4082

5. Between 21and 26

z= (21-25)/2.25= -1.78 = 0.4625

z= (26-25)/2.25= 0.44 = 0.1700

Area: 0.4625+0.1700 =0.6325


z= (22-25)/2.25 z= (27-25)/2.25

z= -1.33 z= 0.89

Area: 0.4082 + 0.3133 = 0.7215

Area: 0.7215


z= (23-25)/2.25 z= (20-25)/2.25

z= -0.89 z= -2.22

Area: 0.4868 – 0.3133 = 0.1735

Area: 0.1735


Z= (27-25)/2.25 z= (30-25)/2.25

Z= 0.89 z= 2.22

Area: 0.4868-0.3133 = 0.1735

Area: 0.1735



Z= (28-25)/2.25 z= (21-25)/2.25

z= 1.33 z= -1.78

Area: 0.4625 + 0.4082 = 0.8707

Area: 0.8707


= (20-25)/2.25 z= (26-25)/2.25

z= -2.22 z= 0.44

Area: 0.4868 + 0.1700 = 0.6586

Area: 0.6586


Practice What You Learned (Page 30)

1. The probability between -2 and 0 = P (-2<z<0)

2. The probability between 1 and 2.5 =P (1<z<2.5)

3. The probability between 2 and 3 =P (2<z<3)

4. The probability between -0.21 and 0= P (-0.21<z<0)

5. The probability between -1.89 and 1.5= P (-1.89<z<1.5)

6. The probability between 1.9 and 3 =P (1.9<z<3)

7. The probability between 2.75 and 1.9= P (1.9<z<2.75)

8. The probability between 1.9 and 2.9 =P (1.9<z<2.9)

9. The probability between 0 and 2= P (0<z<2)

10. The probability between 0 and 2.89 =P (0<z<2.89)


1. P (-1.2<z<0.29)

Solution: Find the respective areas first then add.

Areas: -1.2= 0.3849 0.29= 0.1141

Area1 + Area 2 = 0.3849+0.1141 = 0.499 => 49.9% Probability

2. P (0<z<1.2)

Solution: Find the area between 0 and 1.2

Area= 0.3849=>38.49% Probability

3. P (0<z<2)

Solution: Find the area between 0 and 2

Area=0.4772=> 47.72% Probability


4. P (0<z<1.31)

Solution: Find the are between 0 and 1.31

Area= 0.4049=> 40.49% Probability

5. P (-1.89<z<0)

Solution: Find the area between 0 and -1.89

Area= 0.4706 =>47.06% Probability

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Statistics Final Project Handout