Math By: Janelle Maffucci

Normal Distributions and Empirical Rule

Characteristics of normal distribution● One mode● Symmetric-equal amount of data on each side● Asymmetric- approaches but never touches x-axis● Mean=median=mode

The Standard Normal Distribution:A z-score is used to tell how many standard deviation away from the mean. Comparing 2 things that aren't normally comparable.

Example:Jack and Calvin are competing to see who scored higher. The mean score on an IQ Test is 97 with a standard deviation of 30. The Fluid Intelligence Test, similar to an IQ Test has a mean of 55 and standard deviation of 10. If Jack scored a 147 on an IQ test and Calvin scored a 61 on The Fluid Intelligence Test, who scored higher?Jack- Calvin-Z= 147-97/30 Z= 61-55/10Z=50/30 Z= 6/10Z= 1.6 Z= .6Jack Scored Higher!

The Empirical Rule:If data is normally distributed, then 68% of the data will fall between 1 standard deviation of the mean.95% of the data will fall between 2 standard deviation of the mean99.7% of the data will fall between 3 standard deviations of the mean

Example:The normal distributions of height for a Russian female age 18-24 is bell curved with a mean height of 65 inches and a standard deviation of 4 inches.A.) What percent of Russian women 18-24 are between 65 & 70 inches tall?B.) What percent of Russian women 18-24 are less than 65 inches tall?

A.) 34+4.5= 38.538.5%B.) 34+13.5+2.35+.15=5050%

Comparing Data with z-scores

When a set of data values are normally distributed, we can standardize each score by converting it into a z-score.z-scores make it easier to compare data values measured on different scales

If a Z-Score…. Has a value of 0, it is equal to the group mean. Is positive, it is above the group mean. Is negative, it is below the group mean. Is equal to +1, it is 1 Standard Deviation above the mean. Is equal to +2, it is 2 Standard Deviations above the mean. Is equal to -1, it is 1 Standard Deviation below the mean. Is equal to -2, it is 2 Standard Deviations below the mean

Example:A group of friends compares what their candy after they got home from trick or treating. They find that the average number of pieces of candy received is 134, with standard deviation of 9. What is the z-score corresponding to 78 pieces of candy?

● 78-134=-56/9= -6.2222222

The weight of chocolate bars from a Charlie’s Chocolate Factory has a mean of 15 ounces with standard deviation of .3 ounce. What is the z-score corresponding to a weight of 10.16 ounces?

● 10.16-15=-4.84/.3=-16.133

Using normcdf and invnorm

normalcdf cdf = Cumulative This function returns the cumulative probability from zero up to some input value of the random variable x. Technically, it returns the percentage of area under a continuous distribution curve from negative infinity to the x.

invNorm inv = Inverse Normal Probability This function returns the x-value given the probability region to the left of the x-value.

(0 < area < 1 must be true.) The inverse normal probability distribution function will find the precise value at a given percent based upon the mean and standard deviation. Example:

The attendance at NC State Football games is normally distributed, with a mean of 45,000 and a standard deviation of 3,000 when Unc Chapel hill plays.For 80% of the time the attendance should be at least how many?

● 2nd DISTR 3 invnorm(0.20, 45000, 3000) ENTER answer: 42476

1a: Find the probability that a value is between 65 and 80, inclusive. (This is accomplished by finding the probability of the cumulative interval from 65 to 80.)Syntax:normalcdf(lower bound, upper bound, mean, standard deviation)Answer: The probability is 85.361%.

1d: Find the 90th percentile for this distribution.

(Given a probability region to the left of a value (i.e., a percentile), determine the value using invNorm.)Answer: The x-value is 75.767.

Samples/statistics vs populations/parameters

Sample Statistic- A numeric value that characterizes some aspect of the sampleSample Mean- The mean of the samplePopulation Parameter- A numerical value that characterizes some aspect of the populationPopulation Mean- The mean of the population

Ex.Identify the population and the sample A Survey on favorite sports team will be conducted to 300 sophomore students at Heritage High.Population- 300 studentsSample- Sophomore

Sixty high heels were selected from Nordstrom for a fashion show. These 60 heels are referred to as?Parameter

Parallel and Perpendicular LinesGiven the line 2x – 3y = 9 and the point (4, –1), find lines through the point that are:

● parallel to the given line and● perpendicular to it

In other words, they've given me a reference line — 2x – 3y = 9 — that I'll be comparing to, and some point somewhere else on the plane — namely, (4, –1). Then they want me to find the line through (4, –1) that is parallel to (that has the same slope as) 2x – 3y = 9. On top of that, they then want me to find the line through (4, –1) that is perpendicular to (that has a slope that is the negative reciprocal of the slope of) 2x – 3y = 9.Clearly, the first thing is to solve "2x – 3y = 9" for "y=", so that you can find the slope: Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

● 2x – 3y = 9● –3y = –2x + 9d● y = ( 2/3)x – 3

So the reference slope from the reference line is m = 2/3.Since a parallel line has an identical slope, then the parallel line through (4, –1) will have slope m = 2/3. use the point-slope form to find the line:

y – (–1) = ( 2/3 )(x – 4)● y + 1 = ( 2/3 ) x – 8/3

● y = ( 2/3 ) x – 8/3 – 3/3

● y = ( 2/3 ) x – 11/3

This is the parallel line that they asked for.For the perpendicular line, I have to find the perpendicular slope. The reference slope is m = 2/3, and, for the perpendicular slope, I'll flip this slope and change the sign. Then the perpendicular slope is m = – 3/2. So now I can do the point-slope form. Note that the only change from the calculations I just did is that the slope is different now.

● y – (–1) = ( – 3/2 )(x – 4)● y + 1 = ( – 3/2 ) x + 6● y = ( – 3/2 ) x + 5

Then the full solution to this exercise is:● parallel: y = ( 2/3 ) x – 11/3

● perpendicular: y = ( – 3/2 ) x + 5Example:Find the slope of the perpendicular line:

● x+2y=8○ 8-x=2y○ y=2x-2/8○ 2

Write the slope intercept form of the following:● Through (-2,-1), Parallel to y= -1/2x-2

○ y=-1/2x-2

Linear Models

Linear ProgramsGiven the following constraints, maximize and minimize the value of z = –0.4x + 3.2y.

●First I'll solve the fourth and fifth constraints for easier graphing:

●The feasibility region looks like this:

●From the graph, I can see which lines cross to form the corners, so I know which lines to pair up in order to verify the coordinates. I'll start at the "top" of the shaded area and work my way clockwise around the edges:

● y = –x + 7● y = x + 5

● y = –x + 7● x = 5

● x = 5● y = 0

● –x + 7 = x + 5● 2 = 2x● 1 = x

● y = (1) + 5 = 6

● y = –(5) + 7 = 2 ● [nothing to do]

● corner at (1, 6) ● corner at (5, 2) ● corner at (5, 0)

●

●● y = 0

● y = –( 1/2 )x + 2● y = –( 1/2 )x + 2

● x = 0● x = 0

● y = x + 5

● –( 1/2 )x + 2 = 0● 2 = (1/2)x

● 4 = x

● y = –( 1/2 )(0) + 2● y = 0 + 2

● y = 2

● y = (0) + 5 = 5

● corner at (4, 0) ● corner at (0, 2) ● corner at (0, 5)

Now I'll plug each corner point into the optimization equation, z = –0.4x + 3.2y:● (1, 6): z = –0.4(1) + 3.2(6) = –0.4 + 19.2 = 18.8● (5, 2): z = –0.4(5) + 3.2(2) = –2.0 + 6.4 = 4.4● (5, 0): z = –0.4(5) + 3.2(0) = –2.0 + 0.0 = –2.0● (4, 0): z = –0.4(4) + 3.2(0) = –1.6 + 0.0 = –1.6● (0, 2): z = –0.4(0) + 3.2(2) = –0.0 + 6.4 = 6.4● (0, 5): z = –0.4(0) + 3.2(5) = –0.0 + 16.0 = 16.0

Then the maximum is 18.8 at (1, 6) and the minimum is –2 at (5, 0).Example:Janelle is assigned a 3000 word essay for Mr.King’s class. She is an amazing typer and can type 140 words in 4 minutes. At this rate how long will it take for her to complete the paper?Let the number of words typed be w and the time it takes be t.w1 = 120 when t1 = 4 minTo find t2 when w2 = 3000, use ratios as given below:

Cross multiply.

It will take 100 min to complete the paper.You spend $30.00 on snacks for a party. The tootsie rolls cost $3.00 per pound and the swedish fish cost $4.50 per pound. Write an equation in standard form that represents the different amounts (in pounds) of tootsie rolls, T, and Swedish Fish, S, you could buy.If you buy 2 pounds of swedish fish, how many pounds tootsie rolls of can you buy?

The standard–form equation is:3T + 4.5S = 30

Substitute S = 2.3T + 4.5(2) = 303T = 21T = 7So, if you buy 2 pounds of tootsie rolls, you have money left to buy 7 pounds of swedish fish.

Arithmetic sequencing

A Sequence is a set of things (usually numbers) that are in order.

Arithmetic SequenceIn an Arithmetic Sequence the difference between one term and the next is a constant.In other words, we just add the same value each time ... infinitely

Example:

1, 4, 7, 10, 13, 16, 19, 22, 25, ...

This sequence has a difference of 3 between each number.In General we could write an arithmetic sequence like this:

{a, a+d, a+2d, a+3d, ... }where:

● a is the first term, and● d is the difference between the terms (called the "common difference

Example:

1, 4, 7, 10, 13, 16, 19, 22, 25, ...

Has:● a = 1 (the first term)● d = 3 (the "common difference" between terms)

And we get:{a, a+d, a+2d, a+3d, ... }{1, 1+3, 1+2×3, 1+3×3, ... }{1, 4, 7, 10, ... }

RuleWe can write an Arithmetic Sequence as a rule:

xn = a + d(n-1)(We use "n-1" because d is not used in the 1st term).

Example:

Write the Rule, and calculate the 4th term for4, 9, 14, 19, 24, 29,

34, 39, ...This sequence has a difference of 5 between each number.

The values of a and d are:● a = 4 (the first term)● d = 5 (the "common difference")

The Rule can be calculated:xn = a + d(n-1)

= 3 + 5(n-1)= 3 + 5n - 5

= 5n - 2So, the 4th term is:

x4 = 5×4 - 2 = 18

The explicit formula is also sometimes called the closed form. To write the explicit or closed form of an arithmetic sequence, we use

an is the nth term of the sequence. When writing the general expression for an arithmetic sequence, you will not actually find a value for this. It will be part of your formula much in the same way x’s and y’s are part of algebraic equations.

a1 is the first term in the sequence. To find the explicit formula, you will need to be given (or use computations to find out) the first term and use that value in the formula.

n is treated like the variable in a sequence. For example, when writing the general explicit formula, n is the variable and does not take on a value. But if you want to find the 12th term, then n does take on a value and it would be 12.

d is the common difference for the arithmetic sequence. You will either be given this value or be given enough information to compute it. You must substitute a value for d into the formula.

Factoring and solving● Solve (x – 3)(x – 4) = 0.

● Think: If I multiply two things together and the result is zero, what can I say about those two things? I can say that at least one of them must also be zero. That is, the only way to multiply and get zero is to multiply by zero. (This is sometimes called "The Zero Factor Property" or "Rule" or "Principle".)

Warning: You cannot make this statement about any other number! You can only make the conclusion about the factors ("one of them must equal zero") if the product itself equals zero. If the above product of factors had been equal to, say, 4, then we would still have no idea what was the value of either of the factors; we would not have been able (we would not have been mathematically "justified") in making any claim about the values of the factors. Because you can only make the conclusion ("one of the factors must have equalled zero") if the product equals zero, you must always have the equation in the form "(quadratic) equals (zero)" before you can attempt to solve it.

● The Zero Factor Principle tells me that at least one of the factors must be equal to zero. Since at least one of the factors must be zero, I'll set them each equal to zero:

○ x – 3 = 0 or x – 4 = 0● This gives me simple linear equations, and they're easy to solve:

○ x = 3 or x = 4● And this is the solution they're looking for: x = 3, 4

Note that "x = 3, 4" means the same thing as "x = 3 or x = 4"; the only difference is the formatting. The "x = 3, 4" format is more-typically used.One important issue should be mentioned at this point: Just as with linear equations, the solutions to quadratic equations may be verified by plugging them back into the original equation, and making sure that they work, that they result in a true statement. For the above example, we would do the following:

Remember from your translation skills that "difference" means "subtraction". So a difference of squares is something that looks like x2 – 4. That's because 4 = 22, so you really have x2 – 22, a difference of squares. To factor this, do your parentheses, same as usual:

● x2 – 4 = (x )(x )You need factors of –4 that add up to zero, so use –2 and +2:

● x2 – 4 = (x – 2)(x + 2)

Note that we had x2 – 22, and ended up with (x – 2)(x + 2). Differences of squares (something squared minus something else squared) always work this way:

● For a2 – b2, do the parentheses:○ ( )( )

● ...put the first squared thing in front:○ (a )(a )

● ...put the second squared thing in back:○ (a b)(a b)

● ...and alternate the signs in the middles:○ (a – b)(a + b)

Example:Solve.y^2-81=0(y-9)(y+9)

(x/4) = (49/x)(x-14)(x+14)

Vertex form of a parabolaThe vertex form of a parabola's equation is generally expressed as: y = a(x-h)2+k

● (h,k) is the vertex as you can see in the picture below

●

●● If a is positive then the parabola opens upwards like a regular "U".● If a is negative, then the graph opens downwards like an upside down "U".● If |a| < 1, the graph of the parabola widens. This just means that the "U" shape of parabola

stretches out sideways . Explore the way that 'a' works using our interactive parabola grapher.● If |a| > 1, the graph of the graph becomes narrower(The effect is the opposite of |a| < 1).

What is the vertex of the following parabola: y = (x + 3)² + 4The vertex is the point (-3,4)

Example:

Find the parabola's vertex.y=−2x^2+16x−35

y^2+35=−2x^2+16x−35−2x^2+16xy+35=−2(x2−8x y + 35 = -2(x^2 - 8x) y+35=−2(x2−8x)y+35y+35+−32y+3===−2(x2−8x)−2(x2−8x+16)−2(x2−8x+16)y+3=−2(x−4)2\qquad y + 3 = -2(x - 4)^2 y+3=−2(x−4)2

The vertex is (4,−3)

Will the graph of the parabola y = -8x2 + 5x - 100 open upward or downward?Down b/c of -8

Focus and directixThe ellipse and the hyperbola are often defined using two points, each of which is called a focus. The combined distances from these foci is used to create an equation of the ellipse and hyperbola.

A parabola has one focus point. The graph wraps around this focus. The equation of a parabola can be created using a combination of distances from the focus and from a line, called the directrix, to the graph. Examples are shown below, defining a parabola and creating its equation in this manner. Why we do this:One application of a parabola involves lighting. The focus of a parabola has a property which is important is constructing lamps and lighting. An automobile headlight is constructed of a reflective surface which is parabolic. The actual light bulb is placed at the focus. The light from this bulb reflects off the surface in parallel beams which “focus” the headlight beam directly forward from the car. This is because any light source at the focus of a parabola reflects in beams parallel to the axis of symmetry of the parabola.A parabola can be defined as the set of all points such that the distance from a point on the parabola to a focus point is the same as the distance from the same point on the parabola to a fixed line called the directrix. We illustrate this using a focus at the point (0, 1) and a directrix given by the equation y = –1.

On the graph shown below the distances to the point (x, y) from the focus and the directrix line (the dotted line) must be equal. We will now use this information to derive an equation of this parabola.

Since the focus is (0, 1) and a general point on the directrix can be given by (x, -1), the distance formula

would give us .

Expanding this expression yields .Simplifying by subtracting the common terms y2, 1, as well as -2y from both sides givesx2 = 4y.

Solving this equation for y will produce this equation for our parabola

where in this case, .

Examination of the equation tells us that the vertex of this parabola is the origin. The focal distance [from the vertex (0,0) to the focus (0,1)] is 1. Note that the equation could be rewritten as

where d is the focal distance of 1. This general form of the equation for a parabola tells us

that if , then the focal distance d is because . In this case,

the focus is located at and the equation of the directrix is . The vertex is at the origin, (0, 0). Example:FINd the focus point of the equations:Y= 8(X-3)^2 +21/8Y = (x-3)^2 + ¼4(1/32)Y =(x-3)^2 + ¼Focus = 1/32

x= 2y^21/2X= Y^24( ⅛)X=Y^2Focus= 1/8

Quadratic formula

Often, the simplest way to solve "ax2 + bx + c = 0" for the value of x is to factor the quadratic, set each factor equal to zero, and then solve each factor. But sometimes the quadratic is too messy, or it doesn't factor at all, or you just don't feel like factoring. While factoring may not always be successful, the Quadratic Formula can always find the solution.The Quadratic Formula uses the "a", "b", and "c" from "ax2 + bx + c", where "a", "b", and "c" are just numbers; they are the "numerical coefficients" of the quadratic equation they've given you to solve. The Quadratic Formula is derived from the process of completing the square, and is formally stated as:

● For ax2 + bx + c = 0, the value of x is given by:

●

●

For the Quadratic Formula to work, you must have your equation arranged in the form "(quadratic) = 0". Also, the "2a" in the denominator of the Formula is underneath everything above, not just the square root. And it's a "2a" under there, not just a plain "2". Make sure that you are careful not to drop the square root or the "plus/minus" in the middle of your calculations, or I can guarantee that you will forget to "put them back" on your test, and you'll mess yourself up. Remember that "b2" means "the square of ALL of b, including its sign", so don't leave b2 being negative, even if b is negative, because the square of a negative is a positive.In other words, don't be sloppy and don't try to take shortcutsSolve x2 + 3x – 4 = 0

● This quadratic happens to factor:○ x2 + 3x – 4 = (x + 4)(x – 1) = 0

● ...so I already know that the solutions are x = –4 and x = 1. How would my solution look in the Quadratic Formula? Using a = 1, b = 3, and c = –4, my solution looks like this:

○● Then, as expected, the solution is x = –4, x = 1

Example:x^2-x-30=0(x-6)(x+5)=0x=6 x=5

Note: This is not my problem, when i write it out i will modify it but i couldn't type the square roots symbol.

Real numbers

Real Numbers are just numbers like:

1 12.38 −0.8625 3/4 √2 198

In fact:Nearly any number you can think of is a Real Number

Real Numbers include:

Whole Numbers (like 0, 1, 2, 3, 4, etc)

Rational Numbers (like 3/4, 0.125, 0.333..., 1.1, etc )

Irrational Numbers (like π, √3, etc )

Real Numbers can also be positive, negative or zero.

So ... what is NOT a Real Number?

Imaginary Numbers like √−1 (the square root of minus 1)are not Real Numbers

Infinity is not a Real Number

And there are also some special numbers that mathematicians play with that aren't Real Numbers.

Why are they called "Real" Numbers?Because they are not Imaginary Numbers.

The Real Numbers did not have a name before Imaginary Numbers were thought of. They got called "Real" because they were not Imaginary. That is the actual answer!

End Behavior and MultiplicityThe appearance of a graph as it is followed farther and farther in either direction. For polynomials, the end behavior is indicated by drawing the positions of the arms of the graph, which may be pointed up or down. Other graphs may also have end behavior indicated in terms of the arms, or in terms of asymptotes or limits.Polynomial End Behavior:1. If the degree n of a polynomial is even, then the arms of the graph are either both up or both down.2. If the degree n is odd, then one arm of the graph is up and one is down.

3. If the leading coefficient an is positive, the right arm of the graph is up.

4. If the leading coefficient an is negative, the right arm of the graph is down.

small numbers. So, the sign of the leading coefficient is sufficient to predict the end behavior of the function.

Degree Leading Coefficient

End behavior of the function

Graph of the function

Even Positive Example: f (x) = x2

Even Negative Example: f (x) = –x2

Odd Positive Example: f (x) = x3

Odd Negative Example: f (x) = –x3

To predict the end-behavior of a polynomial function, first check whether the function is odd-degree or even-degree function and whether the leading coefficient is positive or negative.Example:Find the end behavior of the function x4 – 4 x3 + 3 x + 25.The degree of the function is even and the leading coefficient is positive. So, the end behavior is:

The graph looks as follows:

A zero has a "multiplicity", which refers to the number of times that its associated factor appears in the polynomial. For instance, the quadratic (x + 3)(x – 2) has the zeroes x = –3 and x = 2, each occuring once. The eleventh-degree polynomial (x + 3)4(x – 2)7 has the same zeroes as did the quadratic, but in this case, the x = –3 solution has multiplicity 4 because the factor (x + 3) occurs four times (the factor is raised to the fourth power) and the x = 2 solution has multiplicity 7 because the factor (x – 2) occurs seven times.The point of multiplicities with respect to graphing is that any factors that occur an even number of time (twice, four times, six times, etc) are squares, so they don't change sign. Squares are always positive. This means that the x-intercept corresponding to an even-multiplicity zero can't cross the x-axis, because the zero can't cause the graph to change sign from positive (above the x-axis) to negative (below the x-axis), or vice versa. The practical upshot is that an even-multiplicity zero makes the graph just barely touch the x-axis, and then turns it back around the way it came. You can see this in the following graphs: Copyright © Elizabeth Stapel 2005-2011 All Rights Reserved

● y = (x + 6)(x – 7) ● y = (x + 6)(x – 7)2

● ●

● x = –6 once● x = 7 once

● x = – 6 once● x = 7 twice

●

■ y = (x + 6)2 (x –

7)

■ y = (x + 6)2(x –

7)2

■ ■

■ x = – 6 twice

■ x = 7 once

■ x = – 6 twice

■ x = 7 twice

All four graphs have the same zeroes, at x = –6 and at x = 7, but the multiplicity of the zero determines whether the graph crosses the x-axis at that zero or if it instead turns back the way it came.Example:Tell the degree and leading coefficient:x^2 + 56Even, Positive

Are these real numbers:-89, 100000, 3.5,78Yes!

Adding and Subtracting Polynomials

A polynomial looks like this:

example of a polynomial

this one has 3 terms

To add polynomials we simply add any like terms together .. so what is a like term?

Like TermsLike Terms are terms whose variables (and their exponents such as the 2 in x2) are the same.In other words, terms that are "like" each other.Note: the coefficients (the numbers you multiply by, such as "5" in 5x) can be different.

Example:

9x x -10000x πx

are all like terms because the variables are all x

Example:

(1/7)xy2 342xy2 66xy2 7xy2/2

are all like terms because the variables are all xy2

Adding PolynomialsTwo Steps:

● Place like terms together● Add the like terms

Example: Add 2x2 + 6x + 5 and 3x2 - 2x - 1

Start with: 2x2 + 6x + 5 + 3x2 - 2x - 1

Place like terms together: 2x2 + 3x2 + 6x - 2x + 5 - 1

Add the like terms: (2+3)x2 + (6-2)x + (5-1)

= 5x2 + 4x + 4

Adding Several PolynomialsWe can add several polynomials together like that.Example: Add (2x2 + 6y + 3xy) , (3x2 - 5xy - x) and (6xy + 5)Line them up in columns and add:2x2 + 6y + 3xy

3x2 - 5xy - x 6xy + 55x2 + 6y + 4xy - x + 5Using columns helps us to match the correct terms together in a complicated sum.

Subtracting PolynomialsTo subtract Polynomials, first reverse the sign of each term we are subtracting (in other words turn "+" into "-", and "-" into "+"), then add as usual.

Types of sampling techniquesSimple Random Sampling: A simple random sample (SRS) of size n is produced by a scheme which ensures that each subgroup of the population of size n has an equal probability of being chosen as the sample.Stratified Random Sampling: Divide the population into "strata". There can be any number of these. Then choose a simple random sample from each stratum. Combine those into the overall sample. That is a stratified random sample. (Example: Church A has 600 women and 400 women as members. One way to get a stratified random sample of size 30 is to take a SRS of 18 women from the 600 women and another SRS of 12 men from the 400 men.)Multi-Stage Sampling: Sometimes the population is too large and scattered for it to be practical to make a list of the entire population from which to draw a SRS. For instance, when the a polling organization samples US voters, they do not do a SRS. Since voter lists are compiled by counties, they might first do a sample of the counties and then sample within the selected counties. This illustrates two stages. In some instances, they might use even more stages. At each stage, they might do a stratified random sample on sex, race, income level, or any other useful variable on which they could get information before sampling.How does one decide which type of sampling to use?The formulas in almost all statistics books assume simple random sampling. Unless you are willing to learn the more complex techniques to analyze the data after it is collected, it is appropriate to use simple random sampling. To learn the appropriate formulas for the more complex sampling schemes, look for a book or course on sampling.Stratified random sampling gives more precise information than simple random sampling for a given sample size. So, if information on all members of the population is available that divides them into strata that seem relevant, stratified sampling will usually be used.

If the population is large and enough resources are available, usually one will use multi-stage sampling. In such situations, usually stratified sampling will be done at some stages.