Short notes on mathematics

Heron’s Formula
Used to find the area of a triangle when all 3 sides are
given.

Heron’s formula
1. Compute s
2. Plug all numbers into the formula
3. Solve(round to the nearest whole number)

Heron’s formula
Solve the triangle and
find the area

Heron’s formula
Find the area when a = 6, b = 4, and c = 3

Heron’s formula
Try on your own

Definition
⚫ Grid – A pattern of horizontal and
vertical lines, usually forming
squares.

Definition
⚫ Coordinate grid – a grid used to
locate a point by its distances from
2 intersecting straight lines.
A
B
C
D
E
1 2 3 4 5
What are the
coordinates
for the house?

Definition
⚫ x axis – a horizontal number
line on a coordinate grid.
1 2 3 4 50 6
x

Hint
⚫ x is the shortest and likes to lie
down horizontally.
1 2 3 4 50 6
x

Definition
⚫ y axis – a vertical number line
on a coordinate grid.
1
2
3
4
5
0
6
y

Hint
⚫ y is the tallest and stands
upright or vertically.
1
2
3
4
5
0
6
y

Definition
⚫ Coordinates – an ordered pair of
numbers that give the location of a
point on a grid. (3, 4)
1
2
3
4
5
0
6
1 2 3 4 50 6
(3,4)

Hint
⚫ The first number is always the x or first
letter in the alphabet. The second number
is always the y the second letter in the
alphabet.
1
3
2
4
5
0
6
1 2 3 4 50 6
(3,4)

How to Plot Ordered Pairs
⚫ Step 1 – Always find the x value first,
moving horizontally either right
(positive) or left (negative).
1
3
2
4
5
0
6
1 2 3 4 50 6
(2, 3)
y
x

How to Plot Ordered Pairs
⚫ Step 2 – Starting from your new position
find the y value by moving vertically either
up (positive) or down (negative).
1
3
2
4
5
0
6
1 2 3 4 50 6
(2, 3)
(2,3)y
x

How to Find Ordered Pairs
⚫ Step 1 – Find how far over horizontally
the point is by counting to the right
(positive) or the left (negative).
1
3
2
4
5
0
6
1 2 3 4 50 6
(5, )
y
x

How to Find Ordered Pairs
⚫ Step 2 – Now count how far vertically
the point is by counting up (positive)
or down (negative).
1
3
2
4
5
0
6
1 2 3 4 50 6
(5,4)
y
x

What is the ordered pair?
1
3
2
4
5
0
6
1 2 3 4 50 6
(3,5)
y
x

1
3
2
4
5
0
6
1 2 3 4 50 6
(2,6)
y
x

1
3
2
4
5
0
6
1 2 3 4 50 6
(4,0)
y
x

1
3
2
4
5
0
6
1 2 3 4 50 6
(0,5)
y
x

1
3
2
4
5
0
6
1 2 3 4 50 6
(1,1)
y
x

Four Quadrants of Coordinate Grid
⚫When the number lines are extended into
the negative number lines you add 3 more
quadrants to the coordinate grid.
-2
0
-1
1
2
-3
3
-2 -1 0 1 2-3 3
y
x

⚫ If the x is negative you move
to the left of the 0.
-2
0
-1
1
2
-3
3
-2 -1 0 1 2-3 3
x = -2
y
x

⚫ If the y is negative you move
down below the zero.
-2
0
-1
1
2
-3
3
-2 -1 0 1 2-3 3
y = -3
y
x

How to Plot in 4 Quadrants
⚫ Step 1 - Plot the x number first
moving to the left when the
number is negative.
-2
0
-1
1
2
-3
3
-2 -1 0 1 2-3 3
(-3, -2)
y
x

⚫ Step 2 - Plot the y number moving
from your new position down 2 when
the number is negative.
-2
0
-1
1
2
-3
3
-2 -1 0 1 2-3 3
(-3, -2)
y
x

⚫ When x is positive and y is
negative, plot the ordered pair
in this manner.
-2
0
-1
1
2
-3
3
-2 -1 0 1 2-3 3
(2, -2)
y
x

⚫ When x is negative and y is
positive, plot the ordered pair in
this manner.
-2
0
-1
1
2
-3
3
-2 -1 0 1 2-3 3
(-2, 2)
y
x

Plot This Ordered Pair
-2
0
-1
1
2
-3
3
-2 -1 0 1 2-3 3
(-3, -3)
y
x

-2
0
-1
1
2
-3
3
-2 -1 0 1 2-3 3
(-1, 2)
y
x

-2
0
-1
1
2
-3
3
-2 -1 0 1 2-3 3
(1, -1)
y
x

-2
0
-1
1
2
-3
3
-2 -1 0 1 2-3 3
(2, -2)
y
x

-2
0
-1
1
2
-3
3
-2 -1 0 1 2-3 3
(-3, -2)
y
x

Credits
Clipart from “Microsoft Clip
Gallery” located on the Internet
at http://cgl.microsoft.com/
clipgallerylive/default.asp

a. Corresponding
b. Alternate Interior
c. Alternate Exterior
d. Consecutive Interior

Warm up – Nov. 1
List all pairs of angles that fit the description.
a. Corresponding
b. Alternate Interior
c. Alternate Exterior
d. Consecutive Interior
1
2
3
4
5
6
7
8
t
HIDDEN MESSAGE
MAN .
BOARD
Free powerpoints at http://www.worldofteaching.com

Corresponding
1
2
3
4
5
6
7
8
t
4 and 2
3 and 1
5 and 7
6 and 8

Alternate Interior
3 and 7
2 and 6
1
2
3
4
5
6
7
8
t

Alternate Exterior
5 and 1
4 and 8
1
2
3
4
5
6
7
8
t

Consecutive Interior
3 and 2
6 and 7 1
2
3
4
5
6
7
8
t

Find all angle measures
1 67 
3
t
113 
180 - 67
2
5
6 7
8
67 
67 
67 
113 
113 
113 

CHAPTER 3
LINES AND ANGLES
Part II: Equations

Alternate Exterior Angles
⚫Name the angle relationship
⚫Are they congruent or supplementary?
⚫Find the value of x
125 
t
5x 
5x = 125
5 5
x = 25


Corresponding Angles
2x + 1
t
151
2x = 150
2 2
x = 75

2x + 1 = 151
- 1 - 1

Consecutive Interior Angles
81
t
7x + 15
supp
7x = 84
7 7
x = 12
7x + 96 = 180
- 96 - 96
7x + 15 + 81 = 180

Alternate Interior Angles
3x
t
2x + 20

20 = x
2x + 20 = 3x
- 2x - 2x

Homework
P. 147 #18 – 26
and
HANDOUT
All HW due THURSDAY
QUIZ ON THURSDAY

Polynomials
Defining Polynomials
Adding Like Terms

• Monomials - a number, a variable, or a product
of a number and one or more variables. 4x,
20x2yw3, -3, a2b3, and 3yz are all
monomials.
• Polynomials – one or more monomials added or
subtracted
• 4x + 6x2, 20xy - 4, and 3a2 - 5a + 4 are all
polynomials.
Vocabulary

Like Terms
Like Terms refers to monomials that have the same
variable(s) but may have different coefficients. The
variables in the terms must have the same powers.
Which terms are like? 3a2b, 4ab2, 3ab, -5ab2
4ab2 and -5ab2 are like.
Even though the others have the same variables, the
exponents are not the same.
3a2b = 3aab, which is different from 4ab2 = 4abb.

Like Terms
Constants are like terms.
Which terms are like? 2x, -3, 5b, 0
-3 and 0 are like.
Which terms are like? 3x, 2x2, 4, x
3x and x are like.
Which terms are like? 2wx, w, 3x, 4xw
2wx and 4xw are like.

Add: (x2 + 3x + 1) + (4x2 +5)
Step 1: Underline like terms:
Step 2: Add the coefficientsof like terms, do not change
the powers of the variables:
Adding Polynomials
(x2 + 3x + 1) + (4x2 +5)
Notice: ‘3x’doesn’t have a like term.
(x2 + 4x2) + 3x + (1 + 5)
5x2 + 3x + 6

Some people prefer to add polynomials by stacking them.
If you choose to do this, be sure to line up the like terms!
Adding Polynomials
(x2 + 3x + 1) + (4x2 +5)
5x2 + 3x + 6
(x2 + 3x + 1)
+ (4x2 +5)
Stack and add these polynomials: (2a2+3ab+4b2) + (7a2+ab+-2b2)
(2a2+3ab+4b2) + (7a2+ab+-2b2)
(2a2 + 3ab + 4b2)
+ (7a2 + ab + -2b2)
9a2 + 4ab + 2b2

Adding Polynomials
1) 3x
3
− 7x( )+ 3x
3
+ 4x( )= 6x3
− 3x
2) 2w
2
+ w − 5( )+ 4w
2
+ 7w+1( )= 6w2
+ 8w − 4
3) 2a
3
+ 3a
2
+ 5a( )+ a
3
+ 4a + 3( )=
3a3
+ 3a2
+ 9a + 3
• Add the following polynomials; you may stack them if
you prefer:

Subtract: (3x2 + 2x + 7) - (x2 + x + 4)
Subtracting Polynomials
Step 1: Change subtraction to addition (Keep-Change-Change.).
Step 2: Underline OR line up the like terms and add.
(3x2 + 2x + 7) + (- x2 + - x + - 4)
(3x2 + 2x + 7)
+ (- x2 + - x + - 4)
2x2 + x + 3

Subtracting Polynomials
1) x
2
− x − 4( )− 3x
2
− 4x +1( )= −2x2
+ 3x − 5
2) 9y
2
− 3y + 1( )− 2y
2
+ y − 9( )= 7y2
− 4y +10
3) 2g
2
+ g − 9( )− g
3
+3g
2
+ 3( )= −g3
− g2
+ g −12
• Subtract the following polynomials by changing to
addition (Keep-Change-Change.), then add:

What is the sum of angles in triangle ADC? D C
BA
We know that
angle DAC+ angle ACD+ angle D = 180
Similarly in triangle ABC,
angle CAB + angle ACB + angle B = 180
Adding 1 and 2 we get ,
angles DAC + ACD + D + CAB + ACB + B =180 + 180 = 360
Also, angles DAC + CAB = angle A and angle ACD + angle ACB = angle C
So, angle A + angle D +angle B + angle C = 360
i.e. THE SUM OF THE ANGLES OF A QUADRILATERAL IS 360.

Cut out a parallelogram from a sheet of paper and cut it along a diagonal.
Now we obtain two triangles.
What can we say about these triangles?
Place one triangle over the other.
Turn one round.
We observe that the two triangles are congruent to each other.
Thus we can say that –
A diagonal of a parallelogram divides it into two congruent triangles.
A
D C
B

A diagonal of a parallelogram divides it into two congruent triangles.
In a parallelogram ,opposite sides are equal.
If each pair of opposite sides of quadrilateral is equal then it is a parallelogram.
In a parallelogram opposite angles are equal.
If in a quadrilateral each pair of opposite angles is equal then it is a
parallelogram.
The diagonals of a parallelogram bisect each other.
If the diagonals of a quadrilateral bisect each other then it is a parallelogram.

We have studied many properties of a parallelogram in this
chapter and we have also verified that if in a quadrilateral any
one of those properties is satisfied, then it becomes a
parallelogram.
There is yet another condition for a quadrilateral to be a
parallelogram.
It is stated as follows:
A QUDRILATERAL IS A PARALLELOGRAM IF A PAIR OF
OPPOSITE SIDES IS EQUAL AND PARALLEL.

A
Q C
P B
D
S R
Example: ABCD is a parallelogram in which P and Q are mid
points of opposite sides AB and CD. If AQ intersects DP at
S and BQ intersects CP at R, show that:
1. APCQ is a parallelogram
2. DPBQ is a parallelogram
3. PSQR is a parallelogram
SOLUTION: 1. In quadrilateral APCQ,
AP is parallel to QC
AP = ½ AB , CQ = ½ CD , AB = CD, AP = CQ
Therefore APCQ is a parallelogram. (theorem 8.8)
2.Similarly quadrilateral DPBQ is a parallelogram because
DQ is parallel to PB and DQ = PB
3. In quadrilateral PSQR
SP is parallel to QR and SQ is parallel to PR.
SO ,PSQR is a parallelogram.

A
F
B
E
C
Draw a triangle and mark the mid points E and F of the two
sides of the triangle.
Join the points E and F.
Measure EF and BC. Measure angle AEF and angle ABC.
OBSERVATION-
EF = ½ BC
And angle AEF = angle ABC
So EF is parallel to BC .
After performing this theorem we arrive at the following theorem :
The line segment joining the mid points of the two sides of a triangle is parallel
to the third side.
The line drawn through the mid points of the two sides of a triangle is parallel
to another side and bisects the third side.

IMPORTANT TERMS –
Sum of the angles of a quadrilateral is 360.
A diagonal of a parallelogram divides it into two
congruent triangles.
In a parallelogram,
Opposite sides are equal , opposite angles are equal , diagonals bisect each
other.
A line drawn through the mid point of a side of a triangle parallel to another
side bisects the third side.
The quadrilateral formed by joining the mid points of the sides of a
quadrilateral in order is a parallelogram.

Q .1- What is the sum of angles in a quadrilateral ?
Q . 2- What are the conditions for a quadrilateral to be a
parallelogram ?
Q . 3- State the converse of mid point theorem ?

Chapter 12
Circles
Tangent Lines
Objectives:
1) To use the relationship between a radius
and a tangent.
2) To use the relationship between 2
tangents from one point.

Circle – Is the set of all points in a plane that
are a given distance from a given point.
Center
P
•Name the circle after its
center.
•Ex: OP
B
Radius – A
segment that
has one endpt
at the center &
the other endpt
on the circle.
•BP
•All Radii are

C D
Diameter – A chord that
contains the center of the
circle.
•Ex: CD
T
A
Chord – A segment
whose endpts lie on
the circle.
•Ex. TA
•

P
B
C D
Tangent Line – A line in the plane
of the circle that intersects the
circle in exactly one point.
Y
Point of Tangency
•Ex: Y
Tangent

⚫Th(11 – 1) If a line is tangent to a circle,
then the line is ⊥ to the radius at the point
of tangency.
OP ⊥ AB
O A
B
P

Ex.1: Find the missing  measure.
⚫mA + mB + mC = 180
⚫x + 90 + 22 = 180
⚫x = 68
A
C B
22°
x°

Ex.2: Find the mx°
⚫DE and EF are tangent to circle C.
110° x°
90
90
C
D
F
E
Top Triangle:
mC + mD + mE = 180
55 + 90 + mE = 180
mE = 35°
Bottom Triangle:
mC + mF + mE = 180
55 + 90 + mE = 180
mE = 35°
55°
mx = 70°

Ex.3: Finding distance between centers
4in
10in
4in
6in
12in
12in
a2 + b2 = c2
62 + 122 = c2
c = 13.4in

Ex.4: Is a line tangent to circle N
⚫Is LM tangent to circle N?
⚫If so then ΔLNM is right
⚫L will be the right angle
N
L
M
a2 + b2 = c2
72 + 242 = 252
49 + 576 = 625
625 = 625
Yes it is tangent!
7
24
25

Ex.5: Solve for x
N
L
M
6 x
8
6
a2 + b2 = c2
62 + 82 = (6+x)2
36 + 64 = 36 + 6x + 6x + x2
100 = x2 + 12x + 36
0 = x2 + 12x - 64
Factor this!
(x – 4) and (x + 16)
x = 4 and x = -16

Inscribed vs Circumscribed
⚫ Inscribed
Inside of the circle
⚫ Circumscribed
Outside of circle
Δ tangent to the circle in 3
places.

⚫Th(11-3) 2 segments tangent to a circle from a
point outside the circle are .
AB  AC
A
B
C

Ex.5: Solve for x, y, & z
10
x
4y
8
z
x = 10
y = 4
z = 8
Find the perimeter of the Δ. = 44

Statistics
From BSCS: Interaction of experiments and
ideas, 2nd Edition. Prentice Hall, 1970 and
Statistics for the Utterly Confused by Lloyd
Jaisingh, McGraw-Hill, 2000

What is statistics?
⚫ a branch of mathematics that provides
techniques to analyze whether or not your
data is significant (meaningful)
⚫ Statistical applications are based on
probability statements
⚫ Nothing is “proved” with statistics
⚫ Statistics are reported
⚫ Statistics report the probability that similar
results would occur if you repeated the
experiment

Statistics deals with numbers
⚫Need to know nature of numbers collected
Continuous variables: type of numbers
associated with measuring or weighing; any
value in a continuous interval of measurement.
⚫Examples:
• Weight of students, height of plants, time to flowering
Discrete variables: type of numbers that are
counted or categorical
⚫Examples:
• Numbers of boys, girls, insects, plants

Can you figure out…
⚫Which type of numbers (discrete or
continuous?)
Numbers of persons preferring Brand X in 5
different towns
The weights of high school seniors
The lengths of oak leaves
The number of seeds germinating
35 tall and 12 dwarf pea plants
Answers: all are discrete except the 2nd and 3rd
examples are continuous.

Populations and Samples
⚫ Population includes all members of a group
Example: all 9th grade students in America
Number of 9th grade students at SR
No absolute number
⚫ Sample
Used to make inferences about large populations
Samples are a selection of the population
Example: 6th period Accelerated Biology
⚫ Why the need for statistics?
Statistics are used to describe sample populations as
estimators of the corresponding population
Many times, finding complete information about a
population is costly and time consuming. We can use
samples to represent a population.

Sample Populations avoiding Bias
⚫Individuals in a sample population
Must be a fair representation of the entire pop.
Therefore sample members must be randomly
selected (to avoid bias)
Example: if you were looking at strength in
students: picking students from the football
team would NOT be random

Is there bias?
⚫ A cage has 1000 rats, you pick the first 20 you can catch
for your experiment
⚫ A public opinion poll is conducted using the telephone
directory
⚫ You are conducting a study of a new diabetes drug; you
advertise for participants in the newspaper and TV
⚫ All are biased: Rats-you grab the slower rats.
Telephone-you call only people with a phone (wealth?)
and people who are listed (responsible?).
Newspaper/TV-you reach only people with newspaper
(wealth/educated?) and TV( wealth?).

Statistical Computations (the Math)
⚫If you are using a sample population
Arithmetic Mean (average)
The mean shows that ½ the members of the
pop fall on either side of an estimated value:
mean
The sum of all the scores
divided by the total number of scores.
http://en.wikipedia.org/wiki/Table_of_mathematical_symbols

Looking at profile of data: Distribution
⚫What is the frequency of distribution,
where are the data points?
Class (height of plants-cm) Number of plants in each
class
0.0-0.9 3
1.0-1.9 10
2.0-2.9 21
3.0-3.9 30
4.0-4.9 20
5.0-5.9 14
6.0-6.9 2
Distribution Chart of Heights of 100 Control Plants
DistributionChart of Heights of 100 ControlPlants

Histogram-Frequency Distribution Charts
0
5
10
15
20
25
30
35
0.0-0.9 1.0-1.9 2.0-2.9 3.0-3.94.0-4.9 5.0-5.9 6.0-6.9
Number of Plants in each Class
Number of plants in each
class
This is called a “normal” curve or a bell curve
This is an “idealized”curve and is theoreticalbased on an infinitenumber
derived from a sample

Mode and Median
⚫Mode: most frequently seen value (if no
numbers repeat then the mode = 0)
⚫Median: the middle number
If you have an odd number of data then the
median is the value in the middle of the set
If you have an even number of data then the
median is the average between the two middle
values in the set.

Variance (s2)
⚫Mathematically expressing the degree of
variation of scores (data) from the mean
⚫A large variance means that the individual
scores (data) of the sample deviate a lot
from the mean.
⚫A small variance indicates the scores
(data) deviate little from the mean

http://www.mnstate.edu/wasson/ed602calcvardevs.htm
Calculating the variance for a whole population
Σ = sum of; X = score, value,
µ = mean, N= total of scores or values
OR use the VAR function in Excel

http://www.mnstate.edu/wasson/ed602calcvardevs.htm
Calculating the variance for a Biased SAMPLE population
Σ = sum of; X = score, value,
n -1 = total of scores or values-1
(often read as “x bar”) is the mean (average value of xi).
Note the sample variance is larger…why?

Heights in Centimeters of Five Randomly Selected Pea Plants Grown at 8-10 °C
Plant Height
(cm)
Deviations from
mean
Squares of
deviation from
mean
(xi) (xi- x) (xi- x)2
A 10 2 4
B 7 -1 1
C 6 -2 4
D 8 0 0
E 9 1 1
Σ xi = 40 Σ (xi- x) = 0 Σ (xi- x)2 = 10
Xi = score or value; X (bar) = mean; Σ = sum of

Variance helps to characterize the data concerning a sample by indicating
the degree to which individual members within the sample vary from the
mean
Finish Calculating the Variance
Σ xi = 40 Σ (xi- x) = 0 Σ (xi- x)2 = 10
There were five plants; n=5; therefore
n-1=4
So 10/4= 2.5

r2 or R2…
is the fraction of the variation in the
values of y that is explained by the least-
squares regression line of y on x.
Class Attendance
Grades
Example: If r2 = 0.61 in the graph to the left, this means
that about 61% of one’s grade is accounted for by the linear
relationship with attendance. The other 39% could be due
to a multitude of factors. Or even more simply you correlate 61% of
your data with attendance or have 61% confidence in the
relationship… the higher the r value the stronger the correlation.

Standard Deviation
⚫ An important statistic that is also used to
measure variation in biased samples.
⚫ S is the symbol for standard deviation
⚫ Calculated by taking the square root of the
variance
⚫ So from the previous example of pea plants:
The square root of 2.5 ; s=1.6
⚫ Which means the measurements vary plus or
minus +/- 1.6 cm from the mean

What does “S” mean?
⚫We can predict the probability of finding a
pea plant at a predicted height… the
probability of finding a pea plant above
12.8 cm or below 3.2 cm is less than 1%
⚫S is a valuable tool because it reveals
predicted limits of finding a particular value

Pea Plant Normal Distribution Curve with Std Dev

The Normal Curve and Standard
Deviation
http://classes.kumc.edu/sah/resources/sensory_processing/images/bell_curve.gif
A normal curve:
Each vertical line
is a unit of
standard deviation
68% of values fall
within +1 or -1 of
the mean
95% of values fall
within +2 & -2
units
Nearly all
members (>99%)
fall within 3 std
dev units

Standard Error of the Sample Means
AKA Standard Error
⚫ The mean, the variance, and the std dev help
estimate characteristics of the population from a
single sample
⚫ So if many samples were taken then the means
of the samples would also form a normal
distribution curve that would be close to the
whole population.
⚫ The larger the samples the closer the means
would be to the actual value
⚫ But that would most likely be impossible to
obtain so use a simple method to compute the
means of all the samples

A Simple Method for estimating standard
error
Standard error is the calculated standard deviation divided by the square root
of the size, or number of the population
Standard error of the means is used to test the reliability of the data
Example… If there are 10 corn plants with a standard deviation of 0.2
Sex = 0.2/ sq root of 10 = 0.2/3.03 = 0.006
0.006 represents one std dev in a sample of 10 plants
If there were 100 plants the standard error would drop to 0.002
Why?
Because when we take larger samples, our sample means get closer
to the true mean value of the population. Thus, the distribution of the
sample means would be less spread out and would have a lower
standard deviation.

Probability Tests
⚫ What to do when you are comparing two
samples to each other and you want to know if
there is a significant difference between both
sample populations
⚫ (example the control and the experimental
setup)
⚫ How do you know there is a difference
⚫ How large is a “difference”?
⚫ How do you know the “difference” was caused
by a treatment and not due to “normal” sampling
variation or sampling bias?

Laws of Probability
⚫ The results of one trial of a chance event do not affect the
results of later trials of the same event. p = 0.5 ( a coin
always has a 50:50 chance of coming up heads)
⚫ The chance that two or more independent events will occur
together is the product of their changes of occurring
separately. (one outcome has nothing to do with the other)
⚫ Example: What’s the likelihood of a 3 coming up on a dice:
six sides to a dice: p = 1/6
⚫ Roll two dice with 3’s p = 1/6 *1/6= 1/36 which means
there’s a 35/36 chance of rolling something else…
⚫ Note probabilities must equal 1.0

Laws of Probability (continued)
⚫ The probability that either of two or more
mutually exclusive events will occur is the sum
of their probabilities (only one can happen at a
time).
⚫ Example: What is the probability of rolling a total
of either 2 or 12?
⚫ Probability of rolling a 2 means a 1 on each of
the dice; therefore p = 1/6*1/6 = 1/36
⚫ Probability of rolling a 12 means a 6 and a 6 on
each of the dice; therefore p = 1/36
⚫ So the likelihood of rolling either is 1/36+1/36 =
2/36 or 1/18

The Use of the Null Hypothesis
⚫ Is the difference in two sample populations due
to chance or a real statistical difference?
⚫ The null hypothesis assumes that there will be
no “difference” or no “change” or no “effect” of
the experimental treatment.
⚫ If treatment A is no better than treatment B then
the null hypothesis is supported.
⚫ If there is a significant difference between A and
B then the null hypothesis is rejected...

T-test or Chi Square? Testing the validity
of the null hypothesis
⚫Use the T-test (also called Student’s T-
test) if using continuous variables from a
normally distributed sample populations
(ex. Height)
⚫Use the Chi Square (X2) if using discrete
variables (if you are evaluating the
differences between experimental data
and expected or hypothetical data)…
Example: genetics experiments, expected
distribution of organisms.

T-test
⚫T-test determines the probability that the
null hypothesis concerning the means of
two small samples is correct
⚫The probability that two samples are
representative of a single population
(supporting null hypothesis) OR two
different populations (rejecting null
hypothesis)

STUDENT’S T TEST
•The student’s t test is a statistical method that is used to see if to sets of
data differ significantly.
•The method assumes that the results follow the normal distribution (also
called student's t-distribution) if the null hypothesis is true.
•This null hypothesis will usually stipulate that there is no significant
difference between the means of the two data sets.
•It is best used to try and determine whether there is a difference between
two independent sample groups. For the test to be applicable, the sample
groups must be completely independent, and it is best used when the
sample size is too small to use more advanced methods.
•Before using this type of test it is essential to plot the sample data from he
two samples and make sure that it has a reasonably normal distribution, or
the student’s t test will not be suitable.
• It is also desirable to randomly assign samples to the groups, wherever
possible.
Read more: http://www.experiment-resources.com/students-t-
test.html#ixzz0Oll72cbi
http://www.experiment-resources.com/students-t-test.html

EXAMPLE
•You might be trying to determine if there is a significant difference in test scores
between two groups of children taught by different methods.
•The null hypothesis might state that there is no significant difference in the mean
test scores of the two sample groups and that any difference down to chance.
The student’s t test can then be used to try and disprove the null hypothesis.
RESTRICTIONS
•The two sample groups being tested must have a reasonably normal distribution.
If the distribution is skewed, then the student’s t test is likely to throw up
misleading results.
•The distribution should have only one mean peak (mode) near the center of the
group.
•If the data does not adhere to the above parameters, then either a large data
sample is needed or, preferably, a more complex form of data analysis should be
used.
test.html#ixzz0OlllZOPZ

RESULTS
•The student’s t test can let you know if there is a significant difference in
the means of the two sample groups and disprove the null hypothesis.
• Like all statistical tests, it cannot prove anything, as there is always a
chance of experimental error occurring.
•But the test can support a hypothesis. However, it is still useful for
measuring small sample populations and determining if there is a
significant difference between the groups.
by Martyn Shuttleworth (2008).
test.html#ixzz0OlmGvVWD

Use t-test to determine whether or not sample population A and B came
from the same or different population
t = x1-x2 / sx1-sx2
x1 (bar x) = mean of A ; x2 (bar x) = mean of B
sx1 = std error of A; sx2 = std error of B
Example: Sample A mean =8
Sample B mean =12
Std error of difference of populations =1
12-8/1 = 4 std deviation units

Comparison of A and B
B’s mean lies outside (less than
1% chance of being the normal
distribution curve of population A
Reject Null Hypothesis

Online calculators:
http://www.physics.csbsju.edu/stats/t-test_bulk_form.html
online calculates for you… and a box plot also
http://www.graphpad.com/quickcalcs/ttest1.cfm

The t statistic to test whether the means are different can be calculated as follows:

Amount of O2 Used by Germinating Seeds of Corn and Pea Plants
mL O2/hour at 25 °C
Reading
Number
Corn Pea
1 0.20 0.25
2 0.24 0.23
3 0.22 0.31
4 0.21 0.27
5 0.25 0.23
6 0.24 0.33
7 0.23 0.25
8 0.20 0.28
9 0.21 0.25
10 0.20 0.30
Total 2.20 2.70
Mean 0.22 0.27
Variance .0028 .0106
Excel file located in AccBio file folder
How to do this all in EXCEL

http://www2.cedarcrest.edu/academic/bio/hale/biostat/session19links/nachocurve2tail.jpg
Ho = null hypothesis if the t value is larger than the chart value (the
yellow regions) then reject the null hypothesis and accept the HA
that there is a difference between the means of the two groups…
there is a significant difference between the treatment group and
the control group.

T table of values (5% = 0.05) For example:
For 10 degrees of
freedom (2N-2)
The chart value to
compare your t value to
is 2.228
If your calculated t value
is between
+2.228 and -2.228
Then accept the null
hypothesis the mean are
similar
If your t value falls
outside
+2.228 and -2.228
(larger than 2.228 or
smaller than -2.228)
Fail to reject the null
hypothesis (accept the
alternative hypothesis)
there is a significant
difference.

So if the mean of the corn = 0.22 and the mean of the peas =0.27
The variance (s2)of the corn is 0.000311 and the peas is .001178.
Each sample population is equal to ten.
Then:
0.22-0.27 / √ (.000311+.001178)/10
-0.05/ √ 0.001489/10
-0.05/ √ .0001489
(ignore negative sign)
t= 4.10
Df = 2N-2 = 2(10) -2=18
Chart value =2.102
Value is higher than t-value… reject the null hypothesis there is a
difference in the means.

The “z” test
-used if your population samples are greater than 30
-Also used for normally distributed populations with continuous variables
-formula: note: “σ” (sigma) is used instead of the letter “s”
z= mean of pop #1 – mean of pop #2/
√ of variance of pop #1/n1 + variance of pop#2/n2
Also note that if you only had the standard deviation you can square that value and
substitute for variance

Z table (sample table with 3 probabilities
α Zα (one tail) Zα/2 (two tails)
0.1 1.28 1.64
0.05 1.645 1.96
0.01 2.33 2.576
Z table use:
α = alpha (the probability of) 10%, 5% and 1 %
Z α: z alpha refers to the normal distribution curve is on one side only of
the curve “one tail” can be left of the mean or right of the mean. Also
your null hypothesis is either expected to be greater or less than your
experimental or alternative hypothesis
Z α/2 = z alpha 2: refers to an experiment where your null hypothesis
predicts no difference between the means of the control or the
experimental hypothesis (no difference expected). Your alternative
hypothesis is looking for a significant difference
Use a one-tail test to show that sample mean A is significantly greater than (or less than)
sample mean B. Use a two-tail test to show a significant difference (either greater than
Or less than) between sample mean A and sample mean B.

Example z-test
⚫ You are looking at two methods of learning
geometry proofs, one teacher uses method 1,
the other teacher uses method 2, they use a test
to compare success.
⚫ Teacher 1; has 75 students; mean =85; stdev=3
⚫ Teacher 2: has 60 students; mean =83; stdev=
= (85-83)/√3^2/75 + 2^2/60
= 2/0.4321 = 4.629

Example continued
Z table (sample table with 3 probabilities)
α Zα (one tail) Zα/2 (two tails)
0.1 1.28 1.64
0.05 1.645 1.96
0.01 2.33 2.576
Z= 4.6291
Ho = null hypothesis would be Method 1 is not betterthan method 2
HA = alternative hypothesis would be that Method 1 is better than method 2
This is a one tailed z test (since the null hypothesis doesn’t predict that there will be no
difference)
So for the probability of 0.05 (5% significance or 95% confidence) that Method one is
not better than method 2 … that chart value = Zα 1.645
So 4.629 is greater than the 1.645 (the null hypothesis states that method 1 would not be better
and the value had to be less than 1.645; it is not less therefore reject the null hypothesis and
indeed method 1 is better

Chi square
⚫Used with discrete values
⚫Phenotypes, choice chambers, etc.
⚫Not used with continuous variables (like
height… use t-test for samples less than
30 and z-test for samples greater than 30)
⚫O= observed values
⚫E= expected values
http://www.jspearson.com/Science/chiSquare.html

http://course1.winona.edu/sberg/Equation/chi-squ2.gif

Interpreting a chi square
⚫ Calculate degrees of freedom
⚫ # of events, trials, phenotypes -1
⚫ Example 2 phenotypes-1 =1
⚫ Generally use the column labeled 0.05 (which
means there is a 95% chance that any
difference between what you expected and what
you observed is within accepted random
chance.
⚫ Any value calculated that is larger means you
reject your null hypothesis and there is a
difference between observed and expect values.

How to use a chi square chart
http://faculty.southwest.tn.edu/jiwilliams/probab2.gif

Interpreting your chi square calculation.
Ask yourself this question:
Is your calculated value less than or equal to the chart value for the
degrees of freedom:
Is 0.931 < or equal to 3.84 (p=0.05, df=1)
If the answer is yes, then there is no significant difference between
your observed and expected values you can accept the null
hypothesis. (ex. Cats show no preference between wet and dry food)
If the answer is no, then there is a significant difference between
your observed and expected value, you can REJECT the null
hypothesis and ACCEPT the alternate hypothesis (ex. Cats prefer
wet food vs. dry food).

What to do if you have more than 250
data points beside panic?
⚫Free downloadable program called “R”
⚫http://www.r-project.org/
⚫Click on CRAN mirror, choose US, choose
UC Berkeley to download program
⚫See link to R powerpoint (use any Excel
file for practice data)
⚫ www.grochbiology.org/R-statisticsPowerPoint.ppt
⚫ If you need help email Ms. Faerber at MV
(mfaerber@mvhigh.net)... Only if you are my student no
outsiders please.

Triangles
Can be classified by
the number of
congruent sides

Scalene Triangle
Has no congruent
sides

Isosceles Triangle
Has at least two
congruent sides

Equilateral Triangle
Has three congruent
sides

Triangles
Can be classified
by the angle
measures

Right Triangle
Has one right angle

Acute Triangle
Has three acute
angles

Obtuse Triangle
Triangle with one
obtuse angle

Triangles
Cut any shape
triangle out of a
sheet of paper

Triangles
Tear off the corners.
Piece them together by
having the corners
touch.

Triangles
The corners form
what type of angle?

Triangles
The sum of the
angles of a triangle
is 180 degrees

Triangles
To find a missing
angle, add up the 2
given angles

Triangles
Subtract the sum
from 180 and you
have the missing
angle

Congruent Triangles
Have the same SIZE
and the same
SHAPE

Congruent Triangles
May be flipped and/or
rotated

Congruent Triangles
BE CAREFUL
WHEN YOU NAME
THE SHAPE

Congruent Triangles
Name congruent
CORRESPONDING
parts
(sides and angles)
IN ORDER

Congruent Triangles
Make sure the letters
are in
CORRESPONDING
order

⚫ ABC is congruent to DEF
A
B
C
D
E
F

One for you
⚫Name the congruent triangles
remembering to name the angles in order
X
Y
D
E
F
Z

Short notes on mathematics

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