Quantitative Methods Flashcards
What is statistics ,what’s the goals
How is it applied?
Types of statistics
Give some examples of descriptive statistics
What is statistics
•Attempt to summarize data and to represent all values by a single value that describes a particular characteristics of the set of data
•What is the goal of statistics
•To make inferences or draw conclusions on a statistic value computed
•How do we apply statistics
•Check for the probability of the finding due to chance
•How do we interpret the results of our statistical analysis
•When probability due to chance is lower than a threshold then our observation rules out chance occurrence.
•Test of hypothesis
•Making inference and drawing conclusions
There are two types of statistics:
•Descriptive statistics
•Inferential statistics
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Examples: Descriptive stat
Set of values
Weight of twelve year olds in kg
Mean or average is 26.47
Median =
Mode =
Ranks
Percentiles= 50th percetile =median
SD/ Variance
What is correlation and regression
What is the difference between these two
What are independent and dependent variables
Know the application
https: //byjus.com/maths/correlation-and-regression/
https: //www.g2.com/articles/correlation-vs-regression
The main purpose of correlation, through the lens of correlation analysis, is to allow experimenters to know the association or the absence of a relationship between two variables. When these variables are correlated, you’ll be able to measure the strength of their association.
True or false
the other hand, regression is how one variable affects another, or changes in a variable that trigger changes in another, essentially cause and effect. It implies that the outcome is dependent on one or more variables.
For instance, while correlation can be defined as the relationship between two variables, regression is how they affect each other. An example of this would be how an increase in rainfall would then cause various crops to grow, just like a drought would cause crops to wither or not grow at all.
Regression analysis
Regression analysis helps to determine the functional relationship between two variables (x and y) so that you’re able to estimate the unknown variable to make future projections on events and goals.
The main objective of regression analysis is to estimate the values of a random variable (z) based on the values of your known (or fixed) variables (x and y). Linear regression analysis is considered to be the best fitting line through the data points.
True or false
Know the difference between a correlation graph and a regression graph
Regression establishes how x causes y to change, and the results will change if x and y are swapped. With correlation, x and y are variables that can be interchanged and get the same result.
Correlation is a single statistic, or data point, whereas regression is the entire equation with all of the data points that are represented with a line.
True or false
https://www.thoughtco.com/independent-and-dependent-variable-examples-606828
Correlation measures linear relationship between two continuous or ranked variables
Regression defines that linear relationship by a mathematical model
Correlation Relationship between Weight and HB levels
Independent variable causes an effect on the dependent variable. Example: How long you sleep (independent variable) affects your test score (dependent variable). This makes sense, but: Example: Your test score affects how long you sleep doesn’t make sense
Basis Correlation Regression
Meaning-
Correlation : A statistical measure that defines co-relationship or association of two variables.
Regression:Describes how an independent variable is associated with the dependent variable.
Dependent and Independent variables-correlation:No difference
Regression:Both variables are different.
Usage- correlation:To describe a linear relationship between two variables.
Regression:To fit the best line and estimate one variable based on another variable.
Objective :To find a value expressing the relationship between variables.
Regression:To estimate values of a random variable based on the values of a fixed variable
Correlation Regression
When to use -
Correlation:When summarizing direct relationship between two variables
Regression:To predict or explain numeric response
Able to quantify direction of relationship? Yes
Regression: Yes
Able to quantify strength of relationship? Yes
Regression:Yes
Able to show cause and effect? No regression-Yes
Able to predict and optimize? No regression-Yes
X and Y are interchangeable? Yes
Regression-No
Uses a mathematical equation? No
Regression-y= a + bx
What are the uses of inferential statistics
SE means standard error
What’s the diff between standard error and standard deviation
https://www.scribbr.com/statistics/standard-error/
Inferential statistics have two main uses: making estimates about populations (for example, the mean SAT score of all 11th graders in the US). testing hypotheses to draw conclusions about populations (for example, the relationship between SAT scores and family income).
The deviation of the sample means (in this case, of 25 means) is known as the standard error of the mean to distinguish it from the standard deviation of a single sample. The standard error of the mean is denoted as SE .
The standard error of the mean, or simply standard error, indicates how different the population mean is likely to be from a sample mean. It tells you how much the sample mean would vary if you were to repeat a study using new samples from within a single population.
You can decrease standard error by increasing sample size. Using a large, random sample is the best way to minimize sampling bias.
The standard deviation describes variability within a single sample.
The standard error estimates the variability across multiple samples of a population
The standard deviation is a descriptive statistic that can be calculated from sample data. In contrast, the standard error is an inferential statistic that can only be estimated (unless the real population parameter is known).
What is probability
Explain the addition and multiplication rules of probability
Explain the two major factors of probability
Probability is the estimation of the likelihood or chance of occurrence of an event
A perfect coin has two faces; the head and tail
If it is tossed there is a chance that it may land on the head or tail
Since there are only two outcomes, each toss has only one outcome.
One will have a head or tail. The likelihood of a head toss will be 1 out of the two cases = ½
For a dice with 6 possible outcomes, each throw will give 1/6 outcomes.
Addition and Multiplication rules
If two events A and B are independent, the probability of A and B occurring in succession is written as;
Pr(A∩B) =Pr(A)*Pr(B)
For conditional probability
Pr(A/B) = Pr(A∩B) / Pr(B)
Two events are said to be mutually when they cannot occur together at the same time, H and T of coin cannot occur at the same time
Two major factors
Types of events
Mutually Exclusive : Mutually exclusive events (ME)
Two events are said to be mutually exclusive if they cannot occur at the same time. They are also said to be disjointed or independent
If two events are ME the probability of both occurring together = 0
ME: P(A and B) =0
For ME event the probability of either of them occurring is the SUM of the probabilities occurring.
and
Non-Mutual events (Conditional events)
Types of rules
Addition and
Multiplication rules n
Under addition rule “OR” or Union
What is the equation for ME and Non ME
For ME (mutually exclusive)events of A or B
P(A or B) = P(A)+P(B)
Eg., given P(A) = 0.20 and P(B) = 0.75,
If A and B are ME, the P(A or B) = 0.20+0.75 =0.95
For Non-Mutual exclusive events
In events that aren’t mutually exclusive, there is some overlap.
When P(A) and P(B) are add twice.
To compensate for that double addition, we subtract the intersection
P(A or B) =P(A)+P(B) – P(A and B)
If P(A)=0.20, P(B)=0.70 and P(A and B)=0.15
P(A or B) = 0.2+0.7-0.15 =0.75
Under multiplication rule “and” or intersection rule
What are independent,dependent and conditional events :
Independent, Dependent and Conditional events
Two events are independent if the occurrence of one does not change the probability of the other
Eg, Rolling a 2 on a die and flipping a head on a coin
If events are independent then the probability of them both occurring is the product of probabilities of each occurring
For independent events
P(A and B) = P(A)P(B)
Eg. P(A)=0.20, P(B)=0.70 and A and B are independent, then
P(A and B) = 0.200.70 = 0.14
If the occurrence of one event does affect the probability of the other occurring then the events are dependent
The probability of that B occurring given that A has already occurred is written as P(B│A)
P(A and B)= P(A) * P(B│A)
Eg., P(A)=0.20, P(B)=0.70, P(B│A)=0.40
If A and B are independent events then
P(A and B) =P(A)*P(B)
P(A│B)= P(A)
P(B│A) = P(B)
This is because for independent events probability of one event has no effect on the other event
For conditional probabilities
P(B│A) = P(A and B) / P(A)
Solve this question
For a motion tabled in parliament after debate, the two parties voted in a manner captured in the table below
KBP. NCP. Total
For. 95. 82. 177
Against. 55. 23. 78
Total. 150. 105. 255
What is the probability that some one chosen will belong to KBP
- What is the probability that Among those are against will be from NCP
- What is the probability that some one chosen from NCP will be against
- What is the probability that a person chosen will be both KBP and Against
- What is the probability that randomly selecting a person will be an Against given that the person was an NCP member.
KBP. NCP. Total
For. 95. 82. 177
Against. 55. 23. 78
Total. 150. 105. 255
What is the probability that some one chosen will belong to KBP = 150/255 = 0.588
- What is the probability that Among those are against will be from NCP = 23/78 =0.295
- What is the probability that some one chosen from NCP will be against = 23/105 = 0.219
- What is the probability that a person chosen will be both KBP and Against = (150/255)*(55/78) = 0.414
- What is the probability that randomly selecting a person will be an Against given that the person was an NCP member.
= P(Against │NCP) = P(Against) = 78/255 = 0.306
implies: P(A and B)/P(B) = (78/255*105/255)/ (105/255) = 78/255 = 0.306
State the rules of exponents
Solve these questions
2 to the power 72 to the power 9 =
5 to the power 95to the power 5 =
19 to the power 5*19 to the power 4 /19 to the power 3 =
4 to the power 12/4 to the power 5 =
1/9 to the power 6 =
5 to the power 14/7 to the power 2 =
(7 to the power 3) to the power 5 =
((2 to the power 6) to the power 3) to the power 4 =
[(5 to the power 3) to the power 5 multiplied by (5 to the power 5) to the power 4] / (5 to the power 3) to the power 8 =
If a you invested GHC 10, 000 in a bank and the compound rate is 24% per annum, what will be your account after 5 years.
Exponents
2+2+2+2+2 = 2x5=10
2x2x2x2x2 = 25 = 32; Thus 2 raised to the power 5
This means 2 is the base and 5 the exponent
Eg.
7+7+7+7 = 7x4=28
7x7x7x7= 74=2,401
Product of like bases = atp the power ma to the power n=a to the power (m+n)
example = 2 to the power 32 to the power 5=2 to the power 8
Quotient of like bases= a to the power m/a to the power n = a to the power (m-n) = 3 to the power 6/3 to the power 4 = 3 to the power 2
Power to a power = (a to the power m)to the power n = a to the power mn = (5 to the power 5) to the power 3 = 5 to the power 15
Product to a power = (ab)to the power x = a to the power xb to the Power x = (35) to the power 3=3 to the power 3*5 to the power 3
Quotient to a power= (x to the power 3/y to the power 5) to the power n = x to the power 3n/y to the power 5n = (25/34)2 =210/38
Zero Exponent =a to the power 0 = 1 = 5 to the power 0 = 1
Negative exponents = a to the power -n = 1/a to the power n =
8 to the power -3 = 1/8 to the power 3
1st Rule: To multiply identical bases, just add the exponents
2 to the power 3 x 2 to the power 5 =2 to the power 8
= (2x2x2)*(2x2x2x2x2)=256
3 to the power 4x3 to the power 6 = 3 to the power 10 =59,049
5to the power 3 multiplied by 5 to the power 8 multiplied by 3 to the power 5 multiplied by 3 to the power 8
= 5 to the power 11 *3 to the power 13 = 48,828,125 *1,594323
2nd Rule: To divide identical bases, subtract the exponents
4 to the power 5/4 to the power 2 = 4 to the power (5-2) =4 to the power 3
= 64
3rd Rule: when there are two or more exponents and only one base , multiple the exponents
(5 to the power 3) to the power 6 = 5 to the power 18 = 3.8146 x10 to the power 13
The rate is 24% per anum
So at the end of every month rate is 24/12 = 0.02.
Investing 1 cedi every month will yield 1.02
For 5 years will be 60 months
(1.02) to the power 60 x 10,000 = 32,810.3
What are logarithms
State the properties of logarithms
Logarithm is the inverse operation of exponentiation just as addition and subtraction are inverse operations
53 = 125 : where 5 is the base and 3 the exponent.
Log5125 = 3; meaning that 3 is the power to which 5 needs to be raised to obtain the value of 125
Thus knowing the value and the base at which to work then what is the power at which that base be raised to obtain our value
by=x can be written in log terms as:
Y=log subscript b X
Log subscript b(xy) = log subscript b x+log subscript b y
Log subscript b(x/y) = log subscript b x-log subscript b y
Log subscript b(x to the power n) = nlog subscript b x
Log subscript b x = log subscript a x/log subscript a b
Reciprocal properties
Log subscript b 1 = 0: (log of 1 no matter the base =0)
Log subscript b b = 1: (log of any number to the same base =1
Log subscript b ,b to the power 2 = 2
Log subscript b ,b to the power x = x
b to the power log ,subscript b to the power x = x
Log subscript a ,b = 1/log subscript b , a
Solve Log(2x+2)+log x – log(12) = 0 Log(2x to the power 2+2x) – log(12)=0 Log(2x to the power 2+2x)/12=0 Divide all by 2 (X to the power 2+x)/6 =b to the power 0 =1 X to the power 2+x=6 X to the power 2+x-6=0 (x+3)(x-2) X=-3, x=2 If log 3=a, log7=b, find the value of log63
Log3=a; log7=b
log63=log7+log9
Log9 = log3 to the power 2
Log63=log7+2log3
b+2a = 2a+b
What are quadratic equations
Solve the following problem
3 to the power (x+y) = 27
Any equation that has the form as
ax2+bx+c=0 is referred to as a quadratic equation
The Quadratic Formula:Forax2+bx+c= 0, the values ofxwhich are the solutions of the equation are given by:
x= -b plus or minus square root of b squared - 4ac all divided by 2a
3x+2y=7 equation (2)
27 = 33 ; therefore 3x+y = 33
X+y=3: x=3-y
Substituting into equation 2 we get 3(3-y)+2y=7
9-3y+2y=7 = -y=-2 =y=2
If x=3-y then x=3-2 = 1
Under Venn diagrams and sets
Out of forty students,14are taking English Composition and29are taking Chemistry.
a) If five students are in both classes, how many students are in neither class?b) How many are in either class?c) What is the probability that a randomly-chosen student from this group is taking only the Chemistry class?
Suppose I discovered that my cat had a taste for the adorable little geckoes that live in the bushes and vines in my backyard. In one month, suppose he deposited the following on my carpet: six gray geckoes, twelve geckoes that had dropped their tails in an effort to escape capture, and fifteen geckoes that he’d chewed on a little. Only one of the geckoes was gray, chewed on, and tailless; two were gray and tailless but not chewed on; two were gray and chewed on but not tailless. If there were a total of24geckoes left on my carpet that month, and all of the geckoes were at least one of “gray”, “tailless”, and “chewed on”, how many were tailless and chewed on but not gray?
Solve the question on venn djagram in the slides
Name three dimensions and give examples
State their formulas
Arear
Square, rectangle,
Circumference:
Circle, sector, cone
Volume:
Rectangular Tank, cylinder
Sphere
Cuboid
Areas (space covered by a planeobject)
Area of a square: the square of one of the sides (L to the power 2)
Area of a rectangle: Length x breath (L*B)
Area covered by a circle (πr to the power 2)
Perimeter and Circumference (distance about a plane object)
Perimeter of a rectangle: (2L+2B)
Perimeter of a square: (4L)
Circumference of a circle : 2πr
Volumes of objects (total space occupied by three dimensional objects)
Volume of a cube: (L to the power 3)
Volume of Cylinder (Area of the base of the cylinder x the height) = (πr to the power 2h)
A man planned to tile the floor of his new building. There are three bedrooms each with dimensions 5m by 4.5m. There is a single living room with dimensions 8m by 7.5m. In addition he has two bathrooms and toilet each with dimensions 2.5 by 2m. If a square meter of tiles cost GHC 45.00, what will be the total cost of tiles for his house?
What is a ratio
A unit rate
A unit price
A rate is a ratio that compares two different kinds of numbers, such as miles per hour or dollars per pound.
A unit rate compares a quantity to its unit of measure.
A unit price is a rate comparing the price of an item to its unit of measure.
Ratio? A ratio is a comparison of two numbers or measures that have the same unit. The numbers in a ratio are sometimes called terms.Example: 4 to 3 : This ratio can be expressed as 4 to 3, 4:3, or 4/3Rate?A rate is a ratio between two quantities of different units, likes miles and hours, or dollars and gallons.Example: 60 miles per hour and $1.20 per gallon are rates.What is Unit Rate?When the second number of a rate is 1, then the rate is a unit rate.Example: Speed is a rate. The speed of 60 miles per hour compares 0 miles to 1 hour.Since 1 is the second number in the rate 60 miles per hour = a unit rate.What is a Unit Price?Unit price is the cost per unit. Unit price can be used to help you find the better buy for prices of things.
