Transcript Unit 1 Introduction to Statistics

Unit 1 Introduction to Statistics
Math 3
Ms. C. Taylor
Warm-Up
How do I find the mean and
median of a data set?
Introduction to Statistics
How do I understand statistics as a
process for making inferences about
population parameters based on a
random sample from that population?
How do I recognize the purposes of and
difference among statistical data
gathering methods?
What is/are statistics?
Statistics is a way of reasoning, along with a
collection of tools and methods, designed to
help us understand the world.
Statistics are particular calculations made
from data
Population Data or Sample Data?
Population data is used when you are gathering
data from every individual of interest.
Ex: Asking the entire football team a question
Sample data is used when you are gathering data
from some of the individuals of interest.
Ex: Asking only the offensive line a question and
apply it to the entire football team
Population Data or Sample Data?
You want to know the average
GPA of North Forsyth students,
so you ask all of the students in
all of your classes.
Population
Sample
Parameter vs. Statistic
A statistic is a descriptive measure computed
from a sample of data.
A parameter is a descriptive measure computed
from an entire population of data.
Inferential statistics enables you to make an
educated guess about a population parameter
based on a statistic computed from a sample
randomly drawn from that population.
Correlation vs. Causation
 Correlation: The degree to which two or more
measurements on the same group of elements show a
tendency to vary together.
 Causation: The degree to which one causes the other to
happen.
 Correlation does not imply causation
 Example: There is correlation between the number of
people wearing shorts and temperature. More people
wearing shorts doesn’t cause higher temperatures!
Ways to Gather Data
Survey – a questionnaire used to collect interesting
data on a certain topic from a sample of people.
EX: You want to find out how many students in
your class had a summer job.
EX: The government wants to determine average
household income in the United States.
EX: You want to know if tattoos have an influence
on a person’s GPA.
Ways to Gather Data
 Observational Study – we observe individuals and measure
variables of interest but do not attempt to influence the
responses. Observational Studies may show a correlation
between variables, but cannot always guarantee causation.
 EX: A study of child care enrolled 1364 infants in 1991 and
planned to follow them through their sixth year in school. In
2003, the researchers published an article finding that “the
more time children spent in child care from birth to age fourand-a-half, the more adults tended to rate them, both at age
four-and-a-half and at kindergarten, as less likely to get along
with others, as more assertive, as disobedient, and as
aggressive.”
Ways to Gather Data
 Experiment – we deliberately impose some treatment on
(that is, do something to) individuals in order to observe
their responses. Experiments can carry more convincing
evidence of a cause and effect relationship.
 EX: “Take the Pepsi Challenge” – in the 80’s Pepsi had a
huge marketing scheme that had people do a blind taste test
to see which soda they preferred – Pepsi or Coke.
 EX: Does Vitamin C reduce the causes of getting a common
cold?
Warm-Up
Determine if the following is a survey,
experiment, or observational study.
Students are asked about their favorite
music.
Walking in a classroom to see if technology
is efficiently used.
Deciding if Claritin or Allegra is the best
allergy medicine.
Sampling
When conducting a survey, experiment, or
observational study, it is almost impossible to
survey everyone in a population so people use
various sampling methods to gather
information.
One major concern about sampling methods
is whether it is a biased or unbiased method
to gather information.
Sampling Methods
 Random sampling: when everyone in a population has an
equal chance of being chosen in the experiment.
 Stratified sampling: when the population is first divided into
similar categories and the number of members in each
category is determined.
 Systematic sampling: when you determine a method for
which to choose members of the population (assign numbers
to the population and then choose every 5th person to
participate)
 Cluster sampling: when you randomly put the population
into clusters and then choose a cluster randomly and then
randomly choose people in that cluster to participate.
Example if selecting 10 animals from 25
dogs, 15 cats, and 10 rabbits
 Random sampling: when everyone in a population has an equal chance of being
chosen in the experiment.
Randomly selecting 10 from all 50 animals
 Stratified sampling: when the population is first divided into similar categories
and the number of members in each category is determined. Select 5 from 25
dogs, 3 from 15 cats and 2 from the rabbits
 Systematic sampling: when you determine a method for which to choose
members of the population (assign numbers to the population and then choose
every 5th person to participate) Give every animal a random number and then
choose every 5th number
 Cluster sampling: when you randomly put the population into clusters and then
choose a cluster randomly and then randomly choose people in that cluster to
participate.
Randomly put the animals into 2 groups of 25, choose a group,
choose 10 from that selected group.
and then
Biased Questions
 Some questions may use language that people can associate
with emotions:
 How much of your time do you waste on Facebook?
 Do you prefer the wonderful math class or boring Shakespeare
Class?
 Some questions may refer to a majority or supposed
authority:
 Would you agree with the NCAE that teachers should be paid
more for earning their master’s degree?
 Phrased awkwardly:
 Do you disagree with people who oppose the ban on smoking in
public places?
Sampling Bias
 Occurs when one or more sub groups of a population are either over
represented or under represented when conducting a survey or
experiment. It must be random and fair selection
 Voluntary: People voluntarily turn in a survey
 Convenience: Questioner stays in one place
 Exclusion: Only asking certain members
 Under representation: Not getting 1/6 or at least 30 people
 Non-randomness: Calling the first five people on every page of the
phonebook.
 Self-selection: People choose groups
 Lack of double-blinding: Sampler knows which group/product the
person is selecting
Errors in Summarizing Data
 No causation of effect: The cause could be affected
by something other than what is being studied.
(correlation only)
 (Ex: Frog with no legs are deaf)
 No causation of accurate population: Applying your
results to the population incorrectly.
 (Ex: Just because 85% of this class like math, it doesn’t
mean that 85% of all students at this school like math.)
 www.tylervigen.com shows correlations no causations
Resources used:
 "Next: Introduction to Data and Measurement Issues
Surveys and Samples." CK-12 Foundation. N.p., n.d.
Web. 21 Aug. 2013.
 Yates, Daniel S., David S. Moore, and Daren S. Starnes.
The Practice of Statistics: TI-83/84/89 Graphing
Calculator Enhanced. New York: W.H. Freeman, 2008.
Print.
 Greg Fisher – Mount Tabor High School
 Christina Holst – Parkland High School
 Wendy Bartlett – Parkland High School
Warm-Up
What percentage of the data lie within one
standard deviation of the mean?
What percentage of the data lie within two
standard deviations of the mean?
What percentage of the data lie within three
standard deviations of the mean?
Standard Deviation
First, find the mean (average) of the data set
The way to get the variance is to subtract
the mean from EACH value
Square the each of step 2’s results
Find the mean of the squared results
(variance)
Take the square root of the variance
Warm-Up
What is the standard deviation
for the set of data? 2, 4, 3, 6, 5
Probability
 Denoted by P(Event)
P(E) = favorable outcomes
total outcomes
 This method for calculating probabilities is
only appropriate when the outcomes of
the sample space are equally likely.
Experimental Probability
The relative frequency at which
a chance experiment occurs
Flip a fair coin 30 times & get
17 heads i.e. 17/30
Basic Rules of Probability
 Rule 1. Legitimate Values
For any event E,
0 < P(E) < 1
 Rule 2. Sample space
If S is the sample space,
P(S) = 1
Rules Continued
 Rule 3. Complement
For any event E,
P(E) + P(not E) = 1
 Rule 4. Addition
If two events E & F are disjoint, P(E or F) = P(E) +
P(F)
(General) If two events E & F are not disjoint, P(E
or F) = P(E) + P(F) – P(E & F)
Example #1
 A large auto center sells cars made by many
different manufacturers. Three of these are
Honda, Nissan, and Toyota. (Note: these are
not simple events since there are many types
of each brand.) Suppose that P(H) = .25, P(N)
= .18, P(T) = .14.
 Are these disjoint events?
 P(H or N or T) =
 P(not (H or N or T) =
yes
Example #2
 Musical styles other than rock and pop are
becoming more popular. A survey of
college students finds that the probability
they like country music is .40. The
probability that they liked jazz is .30 and
that they liked both is .10. What is the
probability that they like country or jazz?
 P(C or J) =
Independent
 Two events are independent if knowing that
one will occur (or has occurred) does not
change the probability that the other occurs
 A randomly selected student is female - What is the
probability she plays soccer for PWSH?
 Independent
 A randomly selected student is female - What is the
probability she plays football for PWSH?
 Not Independent
Rules Continued
Rule 5. Multiplication
If two events A & B are
independent, 𝑃 𝐴 & 𝐵 = 𝑃 𝐴 × 𝑃(𝐵)
General rule:
𝑃 𝐴&𝐵 =𝑃 𝐴 ×𝑃 𝐵 𝐴
Example #3
A certain brand of light bulbs are
defective five percent of the time. You
randomly pick a package of two such
bulbs off the shelf of a store. What is
the probability that both bulbs are
defective?
Can you assume they are independent?
P(D & D)=
Conditional Probability
A probability that takes into account
a given condition
P 𝐵𝐴 =
𝑃(𝐴∩𝐵)
𝑃(𝐴)
P 𝐵𝐴 =
𝑎𝑛𝑑
𝑔𝑖𝑣𝑒𝑛
Example #4
In a recent study it was found that
the probability that a randomly
selected student is a girl is .51 and
is a girl and plays sports is .10. If
the student is female, what is the
probability that she plays sports?
P 𝑆 𝐹 =
𝑃(𝑆∩𝐹)
𝑃(𝐹)
Warm-Up
What is the probability of the following
events?
Probability that I roll a 4.
Probability that I get a heads.
Probability that I roll a 1.
Probability
Probabilities are written as:
Fractions from 0 to 1
Decimals from 0 to 1
Percents from 0% to 100%
Probability
If an event is certain to happen, then the probability
of the event is 1 or 100%.
If an event will NEVER happen, then the probability
of the event is 0 or 0%.
If an event is just as likely to happen as to not
happen, then the probability of the event is ½, 0.5 or
50%.
Probability
Impossible
Unlikely
Equal Chances
0
0.5
0%
50%
½
Likely
Certain
1
100%
Fundamental Counting Principle
 The fundamental counting principle says
that the number of possible combinations
is the product of the items multiplied
together.
 Example: Ms. Taylor has 12 shirts, 5 pairs
of jeans, and 6 pairs of shoes to choose
from, how many outfits are possible?
 12 * 5 * 6 = 360 outfits
Permutations
Order DOES matter!
Example: I have 245 students that
competed in a mathematics game.
There is a first, second, and third
place winner, how many ways can I
choose the winners?
Combinations
Order DOESN’T matter!
Joe has 54 species of animals and
he wants to pick 5 different ones,
how many combinations are
possible?