statistic fail

8/2/2019 Statistic Fail

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http://www.mathsisfun.com/data/probability.html

Probability

How Likely

In the real world events can not be predicted with total certainty. The best we can do is

say how likely they are to happen, using the idea of probability.

Tossing a Coin

When a coin is tossed, there are two possible outcomes:

heads (H) or tails (T)

We say that the probability of the coin landing H is 1/2.

Similarly, the probability of the coin landing T is 1/2.

Throwing Dice

When a singledieis thrown, there are six possible outcomes: 1,

2, 3, 4, 5, 6.

The probability of throwing any one of these numbers is 1/6.

Probability

In general:

Probability of an event happening =Number of ways it can happen

Total number of outcomes
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Example: the chances of rolling a "4" with a die

Number of ways it can happen: 1 (there is only 1 face with a "4" on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability =1

6

Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the

probability that a blue marble will be picked?

Number of ways it can happen: 4 (there are 4 blues)

Total number of outcomes: 5 (there are 5 marbles in total)

So the probability =4

= 0.85

Probability Line

You can show probability on aProbability Line:

The probability is always between 0 and 1

Probability is Just a Guide

Probability does not tell us exactly what will happen, it is just a guide

Example: toss a coin 100 times, how many Heads will come up?

Probability says that heads have a 1/2 chance, so we would expect 50 Heads.

But when you actually try it out you might get 48 heads, or 55 heads ... or

anything really, but in most cases it will be a number near 50.
http://www.mathsisfun.com/probability_line.htmlhttp://www.mathsisfun.com/probability_line.htmlhttp://www.mathsisfun.com/probability_line.htmlhttp://www.mathsisfun.com/probability_line.html


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Words

Some words have special meaning in Probability:

Experiment: an action where the result is uncertain.

Tossing a coin, throwing dice, seeing what pizza people choose are all examples of

experiments.

Sample Space: all the possible outcomes of an experiment

Example: choosing a card from a deck

There are 52 cards in a deck (not including Jokers)

So the Sample Space is all 52 possible cards: {Ace of Hearts, 2 of Hearts,

etc... }

The Sample Space is made up of Sample Points:

Sample Point: just one of the possible outcomes

Example: Deck of Cards

the 5 of Clubs is a sample point

the King of Hearts is a sample point

"King" is not a sample point. As there are 4 Kings that is 4 different sample

points.

Event: a single result of an experiment

Example Events:

Getting a Tail when tossing a coin is an event

Rolling a "5" is an event.

An event can include one or more possible outcomes:

Choosing a "King" from a deck of cards (any of the 4 Kings) is an event

Rolling an "even number" (2, 4 or 6) is also an event


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The Sample Space is all possible outcomes.

A Sample Point is just one possible outcome.

And an Event can be one or more of thepossible outcomes.

Hey, let's use those words, so you get used to them:

Example: Alex decide to see how many times a "double" wouldcome up when throwing 2 dice.

Each time Alex throws the 2 dice is an Experiment.

It is an Experiment because the result is uncertain.

The Event Alex is looking for is a "double", where both dice have the same

number. It is made up of these 6 Sample Points:

{1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}

The Sample Space is all possible outcomes (36 Sample Points):

{1,1} {1,2} {1,3} {1,4} ... {6,3} {6,4} {6,5} {6,6}

These are Alex's Results:

Experiment Is it a Double?

{3,4} No

{5,1} No

{2,2} Yes

{6,3} No

... ...

After 100 Experiments, Alex had 19 "double" Events ... is that close to what

you would expect?


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http://rchsbowman.wordpress.com/2008/10/28/statistics-notes-probability-examples-using-the-

addition-rule-and-conditional-probability/

Statistics NotesProbability Examples using the Addition Rule and

Conditional Probability

Filed under: Statistics Tags: Statistics bowman @ 2:40 am

Addition Rule:

If events A and B are mutually exclusive (disjoint), then P(A or B) = P(A) + P(B). Otherwise, P(A or B) =

P(A) + P(B) P(A and B)

Example 1: mutually exclusive

In a group of 101 students 30 are freshmen and 41 are sophomores. Find the probability that a

student picked from this group at random is either a freshman or sophomore.

Note that P(freshman) = 30/101 and P(sophomore) = 41/101. Thus P(freshman or sophomore) =

30/101 + 41/101 = 71/101

Example 2: not mutually exclusive

In a group of 101 students 40 are juniors, 50 are female, and 22 are female juniors. Find the

probability that a student picked from this group at random is either a junior or female.

Note that P(junior) = 40/101 and P(female) = 50/101, and P(junior and female) = 22/101. Thus

P(junior or female) = 40/101 + 50/101 22/101 = 68/101

Not sure why? When we add 40 juniors to 50 females and get a total of 90, we have overcounted. The

22 female juniors were counted twice; 90 minus 22 makes 68 students who are juniors or female.

.

Conditional Probability

Recall that the probability of an event occurring given that another event has already occurred is called

a conditional probability.
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In a card game, suppose a player needs to draw two cards of the same suit in order to win. Of the 52

cards, there are 13 cards in each suit. Suppose first the player draws a heart. Now the player wishes to

draw a second heart. Since one heart has already been chosen, there are now 12 hearts remaining in a

deck of 51 cards. So the conditional probability P(Draw second heart|First card a heart) = 12/51.

.

Suppose an individual applying to a college determines that he has an 80% chance of being accepted,

and he knows that dormitory housing will only be provided for 60% of all of the accepted students.

The chance of the student being accepted and receiving dormitory housing is defined by

P(Accepted and Dormitory Housing) = P(Accepted) P(Dormitory Housing|Accepted) = (0.80)

(0.60) = 0.48

.

To calculate the probability of the intersection of more than two events, the conditional probabilities of

all of the preceding events must be considered. In the case of three events, A, B, and C, the probability

of the intersection P(A and B and C) = P(A) P(B|A) P(C|A and B).

Consider the college applicant who has determined that he has 0.80 probability of acceptance and

that only 60% of the accepted students will receive dormitory housing. Of the accepted students who

receive dormitory housing, 80% will have at least one room mate. The probability of being accepted

and receiving dormitory housing and having no room mates is calculated by:

P(Accepted and Dormitory Housing and No Roommates) = P(Accepted) P(Dormitory

Housing|Accepted) P(No Room mates|Dormitory Housing and Accepted) = (0.80) (0.60) (0.20)

= 0.096. The student has about a 10% chance of receiving a single room at the college. Not

happening.

.

The probability that event B occurs, given that event A has already occurred is

P(B|A) = P(A and B) / P(A)

This formula comes from the general multiplication principle and a little bit of algebra. I showed you

on an earlier post.


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Since we are given that event A has occurred, we have a reduced sample space. Instead of the entire

sample space S, we now have a sample space of A since we know A has occurred. So the old rule

about being the number in the event divided by the number in the sample space still applies. It is the

number in A and B (must be in A since A has occurred) divided by the number in A. If you then divided

numerator and denominator of the right hand side by the number in the sample space S, then you

have the probability of A and B divided by the probability of A.

The question, Do you smoke? was asked of 100 people. Results are shown in the table.

.Yes No Total

Male ..19 41 ..60

Female 12 28 ..40

Total ..31 69 100

What is the probability of a randomly selected individual being a male who smokes?

This is just a joint probability. The number of Male and Smoke divided by the total = 19/100 = 0.19

What is the probability of a randomly selected individual being a male?

This is the total for male divided by the total = 60/100 = 0.60. Since no mention is made of smoking

or not smoking, it includes all the cases.

What is the probability of a randomly selected individual smoking?

Again, since no mention is made of gender, this is a marginal probability, the total who smoke divided

by the total = 31/100 = 0.31.

What is the probability of a randomly selected male smoking?

This time, youre told that you have a male think of stratified sampling. What is the probability that

the male smokes? Well, 19 males smoke out of 60 males, so 19/60 = 0.3167

What is the probability that a randomly selected smoker is male?

This time, youre told that you have a smoker and asked to find the probability that the smoker is alsomale. There are 19 male smokers out of 31 total smokers, so 19/31 = 0.6129


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http://www.analyzemath.com/statistics/introduction_statistics.html

Introduction to Statistics

Statistics is a mathematical science including methods of collecting,organizing and analyzing data in such a way that meaningful conclusions canbe drawn from them. In general, its investigations and analyses fall into twobroad categories called descriptive and inferential statistics.

Descriptive statistics deals with the processing of data without attempting todraw any inferences from it. The data are presented in the form of tables andgraphs. The characteristics of the data are described in simple terms. Eventsthat are dealt with include everyday happenings such as accidents, prices ofgoods, business, incomes, epidemics, sports data, population data.

Inferential statistics is a scientific discipline that uses mathematical tools tomake forecasts and projections by analyzing the given data. This is of use topeople employed in such fields as engineering, economics, biology, the socialsciences, business, agriculture and communications.

Introduction to Population and Sample

A population often consists of a large group of specifically defined elements.For example, the population of a specific country means all the people living

within the boundaries of that country.

Usually, it is not possible or practical to measure data for every element of thepopulation under study. We randomly select a small group of elements fromthe population and call it a sample. Inferences about the population are thenmade on the basis of several samples.

Example 1:A company is thinking about buying 50,000 electric batteries froma manufacturer. It will buy the batteries if no more that 1% of the batteries aredefective. It is not possible to test each battery in the population of 50,000

batteries since it takes time and costs money. Instead, it will select fewsamples of 500 batteries each and test them for defects. The results of thesetests will then be used to estimate the percentage of defective batteries in thepopulation.

Quantitative and Qualitative Data
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Data is quantitative if the observations or measurements made on a givenvariable of a sample or population have numerical values.

Example: height, weight, number of children, blood pressure, current, voltage.

Data is qualitative if words, groups and categories represents theobservations or measurements.

Example: colors, yes-no answers, blood group.

Quantitative data is discrete if the corresponding data values take discretevalues and it is continuous if the data values take continuous values.

Example of discrete data: number of children, number of cars.

Example of continuous data: speed, distance, time, pressure.


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Analysis of varianceFrom Wikipedia, the free encyclopedia

Instatistics,analysis of variance (ANOVA) is a collection ofstatistical models, and their associated

procedures, in which the observedvariancein a particular variable is partitioned into components attributable todifferent sources of variation. In its simplest form, ANOVA provides astatistical testof whether or not

themeansof several groups are all equal, and therefore generalizest-testto more than two groups. Doing

multiple two-sample t-tests would result in an increased chance of committing atype I error. For this reason,

ANOVAs are useful in comparing two, three, or more means.

http://www.physics.csbsju.edu/stats/anova.html

ANOVA:ANalysis Of VAriance between groups

Click here to start ANOVA data entry

ANOVA: How many groups? Size of largest group?

You are about to enter your data for a ANalysis Of VAriance. For this to make sense

you should have several groups of data (at least 3; maximum: 26).

Number of groups:

Each group includes a certain number of data items. (Often all the groups have the

same number of items, but that is not required.) What is the size (i.e., the number of

items) of largest group? (maximum: 99)

Size of largest group:

There is no harm is over estimating the group size: blanks will be ignored. You do

need to correctly enter the number of groups.

Click here for copy & paste data entry

ANOVA: How many groups?
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You are about to enter your data for a ANalysis Of VAriance. For this to make sense

you should have several groups of data (at least 3; maximum: 26).

Number of groups:

Each group includes a certain number of data items (minimum:3, maximum: 1024).(Often all the groups have the same number of items, but that is not required.) You

will be asked to enter this data into the appropriate box. If you have lots of data you

may want to copy the data from a different screen and paste it into the correct box.

You can also just type in the points separated by spaces, tabs, or commas. Boxes left

completely blank will be ignored.

You might guess that the size of maple leaves depends on the location of the trees. For

example, that maple leaves under the shade of tall oaks are smaller than the maple

leaves from trees in the prairie and that maple leaves from trees in median strips of

parking lots are smaller still. To test this hypothesis you collect several (say 7) groups

of 10 maple leaves from different locations. Group A is from under the shade of tall

oaks; group B is from the prairie; group C from median strips of parking lots, etc.

Most likely you would find that the groups are broadly similar, for example, the range

between the smallest and the largest leaves of group A probably includes a large

fraction of the leaves in each group. Of course, in detail, each group is probably

different: has slightly different highs, lows, and hence it is likely that each group has a

different average (mean) size. Can we take this difference in average size as evidence

that the groups in fact are different (and perhaps that location causes that difference)?Note that even if there is nota "real" effect of location on leaf-size (the null

hypothesis), the groups are likely to have different average leaf-sizes. The likely range

of variation of the averages if our location-effect hypothesis is wrong, and the null

hypothesis is correct, is given by the standard deviation of the estimated means:

/N

where is the standard deviation of the size of all the leaves and N (10 in our

example) is the number of leaves in a group. Thus if we treat the collection of the 7

group means as data and find the standard deviation of those means and it is"significantly" larger than the above, we have evidence that the null hypothesis is not

correct and instead location has an effect. This is to say that if some (or several)

group's average leaf-size is "unusually" large or small, it is unlikely to be just

"chance".

The comparison between the actual variation of the group averages and that expected

from the above formula is is expressed in terms of the F ratio:


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F=(found variation of the group averages)/(expected variation of the group averages)

Thus if the null hypothesis is correct we expect F to be about 1, whereas

"large" F indicates a location effect. How big should Fbe before we reject the null

hypothesis? P reports the significance level.

In terms of the details of the ANOVA test, note that the number of degrees of freedom

("d.f.") for the numerator (found variation of group averages) is one less than the

number of groups (6); the number of degrees of freedom for the denominator (so

called "error" or variation within groups or expected variation) is the total number of

leaves minus the total number of groups (63). The F ratio can be computed from the

ratio of the mean sum of squared deviations of each group's mean from the overall

mean [weighted by the size of the group] ("Mean Square" for "between") and the

mean sum of the squared deviations of each item from that item's group mean ("Mean

Square" for "error"). In the previous sentence mean means dividing the total "Sum of

Squares" by the number of degrees of freedom.

Why not just use the t-test?

The t-test tells us if the variation between two groups is "significant". Why not just

do t-tests for all the pairs of locations, thus finding, for example, that leaves from

median strips are significantly smaller than leaves from the prairie, whereas

shade/prairie and shade/median strips are not significantly different. Multiple t-tests

are not the answer because as the number of groups grows, the number of needed pair

comparisons grows quickly. For 7 groups there are 21 pairs. If we test 21 pairs weshould not be surprised to observe things that happen only 5% of the time. Thus in 21

pairings, a P=.05 for one pair cannot be considered significant. ANOVA puts all the

data into one number (F) and gives us oneP for the null hypothesis.


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http://mathworld.wolfram.com/ANOVA.html

ANOVA

"Analysis of Variance." Astatistical testfor heterogeneity ofmeansby analysis of groupvariances. ANOVA isimplemented asANOVA[data] in theMathematicapackage ANOVA` .

To apply the test, assume random sampling of avariate with equalvariances, independent errors, and

anormal distribution. Let be the number ofreplicates(sets of identical observations) within each of factor

levels(treatment groups), and be the th observation withinfactor level . Also assume that the ANOVA is

"balanced" by restricting to be the same for eachfactor level.

Now define the sum of square terms

(1)

(2)

(3)

(4)

(5)

which are the total, treatment, and error sums of squares. Here, is the mean of observations withinfactorlevel , and is the "group" mean (i.e., mean of means). Compute the entries in the following table, obtainingtheP-valuecorresponding to the calculatedF-ratioof the mean squared values

(6)

categoryfreedom

SS mean squared F-ratio

model SSA

error SSE

total SST

If theP-valueis small, reject thenull hypothesisthat allmeansare the same for the different groups.
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