Analysis of Variance

1-way ANOVA

-          Analysis of variance is used to compare 3 0r more means w/c contains only 1 variable.

2-way ANOVA

-          ANOVA that involves 2 variables.

Reasons why the T-test should not be used on 3 or more populations

1.       When 1 is comparing 2 means at a time, the means of the rest of the study are ignored. W/ F-test, all the means are tested simultaneously.

2.       When 1 is comparing 2 means at a time, the probability of rejecting the null hypothesis when it is true increased, since more T-tests are conducted, the greater is the likelihood of getting significant differences by chance alone.

3.       The more means are to compare, 3 the mote T-tests are needed.

Assumptions for the F-test in comparing three of more means

1.       The populations in w/c the samples were obtained must be normally distributed or approximately normally distributed.

2.       The sample must be independent for each.

3.       Te variances of the population must be equal.

With the f-test, 2 different estimates of the population variances are made. The 1^st estimate is called the between group variance or mean square of the between group that involves finding the variance of the means. The 2^nd estimate, the w/in group variance or mean square of the w/in group. This is made by computing all the variances using the data and is not affected by the difference of the means.

No difference in the means.

·         The between group variance estimate is approximately equal to the w/in group variance.

·         F-test value will be approximately equal to 1.

·         The null hypothesis will not be rejected.

Means differ significantly.

·         The between group variance is much larger than the w/in group variance.

·         F-test will be significantly greater then 1.

·         The null hypothesis will be rejected.

Test for Independence

Test Using Contingency Tables

When data can be tabulated in the table from in terms of frewuencies, several hypothesis can be tested by using the chi-square test. Two such tests are the independence of variables test and the homogenety of proportions test.

·         The test of independence is used to determine wether 2 variables are independent of or related to each other when a single sample is selected.

Steps:

·         State the hypotheses

·         Find the degree of freedom

·         Find the expected value

·         Find the test value

·         Find the critical value

·         Make a decision

·         Summarize

Test for homogenety of proportions

·         It is used ti determine wether the proportions for variables are equal when several samples are selected from different populations.

Steps:

·         State the hypotheses

·         Find the critical value

·         Find the test value

·         Make a decision

·         Summarize

                                              Mobile/Smartphone statistics

• Mobile now accounts for 10% of internet usage worldwide (this has more than doubled over last 18months)
• 1.08 of the world’s 4 billion mobile phones are smartphones
• Apple and Android represent more than 75% of the smartphone market
• 7.96% of all web traffic in the U.S. is mobile traffic. That number skyrockets to 14.85% in Africa, and 17.84% in Asia — up 192.5% since 2010
• 29% of mobile users are open to scanning a mobile tag to get coupons
• 39% of instances where a consumer walks out of a store without buying were influenced by smartphones
• 91% of mobile internet access is for social activities, versus just 79% on desktops 
• Over 1/3 of Facebook’s users access Facebook Mobile; 50% of Twitter’s users use Twitter Mobile
• QR code scans increased 300% in 2011 compared to 2010
• The average tablet user spends 13.9 hours per week with the device
• 73% of smartphone owners access social networks through apps at least once per day 
• There was 103% growth in website traffic from smartphones from 2011-2012
• US consumers spend almost 1 in every 10 ecommerce dollars using a mobile device 
• There are currently 6 Billion mobile subscribers worldwide
• This equals 87% of the world’s population
• China and India account for 30% of this growth
• There are over 1.2 Billion people accessing the web from their mobiles
• Over 300,000 apps have been developed in the past 3 year
• Google earns 2.5 Billion in mobile ad revenue annually

J.Santillan

Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA) is a collection of statistical models, and their associated procedures, in which the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether or not the means of several groups are all equal, and therefore generalizes t-test to more than two groups. Doing multiple two-sample t-tests would result in an increased chance of committing a type I error. For this reason, ANOVAs are useful in comparing two, three, or more means.




POSTED BY: VON ERJUN SABUSAP

Difference Between Means

Hypothesis Testing of the Difference Between Two Means
Do employees perform better at work with music playing.  The music was turned on during the working hours of a business with 45 employees.  There productivity level averaged 5.2 with a standard deviation of 2.4.  On a different day the music was turned off and there were 40 workers.  The workers' productivity level averaged 4.8 with a standard deviation of 1.2.  What can we conclude at the .05 level?
Solution
We first develop the hypotheses
        H₀:  m₁ - m₂  =  0       
        H₁:  m₁ - m₂  >  0
Next we need to find the standard deviation.  Recall from before, we had that the mean of the difference is 
        m_x  =  m₁ - m₂ 
and the standard deviation is 

 s_x  =      

We can substitute the sample means and sample standard deviations for a point estimate of the population means and standard deviations.  We have

        
and 
Now we can calculate the t-score.  We have
                    0.4
        t  =                       =  0.988
                   0.405
To calculate the degrees of freedom, we can take the smaller of the two numbers n₁ - 1 and n₂ - 1.  So in this example we use 39 degrees of freedom.  The t-table gives a value of 1.690 for the t_.95 value.  Notice that 0.988 is still smaller than 1.690 and the result is the same.  Since the t-score is smaller than 1.690, we fail to reject the null hypothesis and state that there is insufficient evidence to make a conclusion about employees performing better at work with music playing. 




POSTED BY: VON ERJUN SABUSAP

                                       ANOVA for simple linear regression

• Total sum of squared deviations is divided into model (regression) and error
(residual) sums of squares

• Their ratio is the coefficient of determination R2

• These are each divided by their degrees of freedom to obtain the mean SS

• Their ratio is distributed as F and can be tested for significance



J.Santillan

                                         Analysis of Variance (ANOVA)

• Partition the total variance in a population into the model and residual

• If the model has more than one term, also partition the model variance into
components due to each term

• Can be applied to any linear additive design specified by a model

• Each component can be tested for signficance vs. the null hypothesis that it
does not contribute to the model fit



J.Santillan

STATISTICAL FACTS ABOUT TEENAGERS

Every 2 hours a youth is murdered.

In the next twenty four hours, 1,439 Teens will attempt suicide. 

A 1995 study by Children Now and Kaiser Permanente found 40% of teen women know someone who was in an abusive relationship. 

In the next twenty four hours, 2,795 Teenage girls will become pregnant.

Every 4 minutes a youth is arrested for an alcohol related crime.

In the next twenty four hours, 3,506 Teens will run away.

Every 7 minutes a youth is arrested for a drug crime.

G.K. Elio

Analysis of Variance

   The z and t tests should not be used when three orb more means are compared, instead, F-test can be used to compare three or more means. This technique is called analysis of variance or ANOVA .For three groups, the F-test can only show whether or not a difference exists among the three means. It cannot reveal where the difference lies. If the F test indicates that there is a difference among the means, other statistical test are used to find where the difference exists. The most commonly used tests are the Scheffe test and the Tukey test.

One-Way Analysis of Variance
- Analysis of variance used to compare three or more means which contains only one variable.

Two-Way Analysis of Variance
-ANOVA that involves two variables.

Reasons why the t test should not be used on three or more populations:
1. When one is comparing two means at a time, the rest of the means under study are ignored. With the f test, all the means are compared simultaneously.
2. When one is comparing two means at a time, the probability of rejecting the null hypothesis when it is true increased, since more t test are conducted, the greater is the likelihood of getting significant differences by chance alone.
3. The more means there are to compare, 3 the more t tests are needed.

Assumptions for the F test for comparing Three or more means
1. The populations from which the samples were obtained must be normally or approximately normally distributed.
2. The samples must be independent for each.
3. The variances of the population must be equal.


J.Santillan

Procedure in Finding F test Value for the Analysis of Variance

Step 1: Find the mean and variance for each sample.

Step 2: Find the grand mean.

Step 3: Find between- group variance.

Step 4: Find within- group variance.

Step 5: Find the F test value

The degrees of freedom are:
       d.f. N= k-1   where k is the number of groups, and
       d.f D= N-k   where N is the sum of the sample sizes of the groups



J.Santillan

Test Using Contingency Tables

When data can be tabulated in the table from in terms of frewuencies, several hypothesis can be tested by using the chi-square test. Two such tests are the independence of variables test and the homogenety of proportions test.

Test for Independence

·         The test of independence is used to determine whether 2 variables are independent of or related to each other when a single sample is selected.

Steps:

·      1.   State the hypotheses

·      2.  Find the degree of freedom

·     3.   Find the expected value

·     4.   Find the test value

·    5.     Find the critical value

·    6.     Make a decision

·     7.   Summarize

Test for homogenety of proportions

·         It is used ti determine whether the proportions for variables are equal when several samples are selected from different populations.

Steps:

·        1.  State the hypotheses

·         2. Find the critical value

·      3.   Find the test value

·       4.   Make a decision

·       5.   Summarize

C.Ordoyo

Majority of establishments is engaged in Water collection, treatment and supply

           The 2010 Annual Survey of Philippine Business and Industry (ASPBI) covered a total of 255 Water supply; Sewerage, Waste Management and Remediation Activities establishments with total employment of 20 and over.
            Among industries, water collection, treatment and supply recorded the highest number of establishments at 232 or 91.0 percent of the total. Waste collection followed a far second with 11 establishments (4.3%). Waste treatment and disposal ranked third with 7 establishments (2.7%) while materials recovery had the lowest number with 5 establishments (2.0%). Figure 1 illustrates the percentage distribution of Water Supply; Sewerage, Waste Management and Remediation Activities establishments by industry group in 2010.

           At the regional level, Central Luzon accounted for the highest number of establishments at 52 or 20.4 percent of the total. This was followed by CALABARZON with 45 establishments (17.6%) and Western Visayas with 20 establishments or 7.8 percent.


 J.Santillan

Statistical Facts About

Google+

                   It is expected that Google’s new social network Google+ will hit 400 million users by the end of 2012. It needs to be kept in mind that Google+ is more about being “core” to Google’s social web strategy as it aims to embed social signals into its whole web and search strategy as it aims to continue to be relevant in an increasingly social and mobile web.

Google makes it mandatory to join Google+ when you register for a Gmail account which really amounts to it being a forced membership. This non organic strategy means that its engagement levels are extremely low at 3 minutes per month compared to Facebook at 405 minutes, in fact they are less than MySpace.

It must be also kept in mind that Google+ is the glue that connects Google’s other social and web properties such as YouTube, Android and of course search.

Some Google+ Facts and Figures

• It was launched on June 28, 2011
• Google+ reached 10 million users by July 14, 2011
• 67% of Google+ users are male
• Google “+1″ button is served more than 5 billion times daily
• It is gaining 625,000 users per day
• In less than one day the Google+ iPhone app became the most popular free application in the Apple App store

J.Santillan

Correlation and Regression

Relationship

1.Simple

-maybe positive

-maybe negative

2.Multiple

Regression- making prediction

*The more stronger the regression the more accurate.

Scatter plot

(x) Independent variable- being controlled

(y) Dependent variable- affected by other variable

J.Santillan

YouTube

Communication and engagement is much more than talking or writing and the popularity of  YouTube is evidence of that. The availability of cost effective high speed internet access is making it easy for people to express themselves via video. Brands have seized on its power to be a viral media that augments traditional advertising media such as TV. YouTube’s advantage is it is always available and searchable. (it is the worlds second largest search engine after Google.

This capability has been displayed by the success of the Old Spice  marketing campaign among many others.

Some facts and figures:

• 3rd most visited website according to Alexa
• 2 billion views per day
• It handles 10% of the internet’s traffic
• Average YouTube user spends 900 seconds per day
• 44% of YouTube’s users are aged between 12 and 34
• Over 829,000 videos are uploaded every day
• Average video duration is 2 minutes 46 seconds

J.Santillan

FACEBOOK STATISTICS

With over 500 million users, Facebook is now used by 1 in every 13 people on earth, with over 250 million of them (over 50%) who log in every day. The average user still has about 130 friends, but that should expand in 2011.

48% of 18-34 year old check Facebook when they wake up, with 28% doing so before even getting out of bed. The 35+ demographic is growing rapidly, now with over 30% of the entire Facebook user base. The core 18-24 year old segment is now growing the fastest at 74% year on year. Almost 72% of all US internet users are on now Facebook, while 70% of the entire user base is located outside of the US.

Over 700 Billion minutes a month are spent on Facebook, 20 million applications are installed per day and over 250 million people interact with Facebook from outside the official website on a monthly basis, across 2 million websites. Over 200 million people access Facebook via their mobile phone. 48% of young people said they now get their news through Facebook. Meanwhile, in just 20 minutes on Facebook over 1 million links are shared, 2 million friend requests are accepted and almost 3 million messages are sent.

 G.K. Elio

STATISTICAL FACTS 

40% of American adults believe in the theory of evolution.

A survey, published in The New York Times, states that 47% of dog owners in America sleep with their dogs.

According to the Gallup poll, in 2009, 77% of Americans claim Christianity as their religion (down from 91% in 1948).

The age with the highest number of children in the foster system is 16 (42,272 out of the 510,000 are 16-year olds).

The top 4 types of child maltreatment in 2007 were the following: 59.0 percent were neglected, 10.8 percent were physically abused, 7.6 percent were sexually abused, and 4.2 percent were psychologically abused.

According to Centers for Disease Control and Prevention, in 2007, there were 35,962 diagnosed with AIDS. This brings the total number of AIDS diagnoses in the US and the district of Colombia to 1,018,428.

National Institute on Alcohol Abuse and Alcoholism (NIAAA) estimated the cost for alcohol abuse in the United States in 1998 to be $184,636 millions (the cost includes healthcare and prevention treatments, loss of productivity and earnings, crime and crashes).




G.K. Elio

World Facts and Stats

Places to See Before They Disappear

Glaciers, Glacier National Park

United States and Canada

Glacier National Park contains some of the most beautiful, primitive wilderness in the Rocky Mountains. There are more than 200 glacier-fed lakes, high peaks, sheer precipices, large forests, waterfalls, much wildlife, and a great variety of wildflowers. However, temperature fluctuations have caused glacier growth and depletion. Ten thousand years ago, the area of Glacier National Park was covered by ice up to one mile below sea level. The latest warm period has caused the number of glaciers to decrease from 150 in 1850 to 26 today. If current global warming trends continue, there will be no glaciers left in Glacier National Park by 2030.

Venice, Italy

With as many as 40 floods per year between March and September, Venice is slowly sinking at an estimated rate of 2.5 inches every 10 ten years. Venice, a city of beauty and charm, was built as a collection of 118 separate islands, relying entirely on a canal system of about 150 canals, mostly very narrow, crossed by some 400 bridges. A severe flood in December 2008 brought renewed attention to Venice's vulnerable state and imminent fate as an underwater city.

Mexico City, Mexico

In the past 100 years, Mexico City has sunk more than 30 feet. The original city was built on the site of a former lake—the Aztecs built the city on a series of aquatic platforms, but when the Spanish conquered the city, they drained the lake, causing it to sink. As the city population ballooned and the demand for water increased in the 20^th century, the government began pumping much of the city's supply out of the underground aquifer that once fed the lake, causing the city to sink further. No practical plan has been made for the future to provide the 22 million inhabitants of Mexico City with the water they need without destroying the city.

G.K. Elio

STATISTICAL FACTS ABOUT CHILDREN

Siblings and Extended Families:
Over 2 million American children live with grandparents or other relatives because their parents cannot care for them. When relatives provide foster care (known as kinship care), siblings can often stay together. Kinship care also improves stability by keeping displaced children closer to their extended families, their neighborhoods, and their schools.

Total Population:
513,000 children were in the U.S. foster care system on September 30, 2005. Most children are placed temporarily in foster care due to parental abuse or neglect.

Gender:

Male 52%
Female 48%

Age:
Average Age: 10.0 years
6% < 1 year
26% 1-5 years
20% 6-10 years
28% 11-15 years
18% 16-18 years
2% >19 years


G.K. Elio

FACEBOOK STATISTICS:

1.57% of people talk to other people online than they do in real life.
2.48% of 18 to 34 year olds check there facebook right when they wake up.


D. Bertiz

Testing the Difference between Two Variances

Characteristics of F Distribution
1. The values of F cannot be negative, because the variances are always positive or zero.
2. The distribution is positively skewed
3.The mean value of F is approximately equal to 1
4. The F distribution is a family of curves bases on the degrees of freedom of the variance of the numerator and the degree of freedom of the variance of the denominator.

C.Ordoyo

COOL STATISTICAL FACTS

1. Only a fraction of the money that individuals put into the lottery is ever used to pay winnings. A large portion of it goes back to the government to spend however they please.

 2. You are actually 16 times more likely to get killed in a car accident on the way to purchase your lottery ticket than you are to actually win with the lottery ticket you buy there!

3. Statistically, you will be struck by lightning 5,000 times before you ever win the lottery during your life.

4. If you buy more lottery tickets, you're not that much more likely to win. Until you start purchasing thousands of tickets, your odds don't go up that much at all.

5. If you put $20 a month into buying lottery tickets from age 25 to 65, you'll likely come out with next to nothing. If you instead put that into a mutual fund, you would have $93,626.41, after taxes!

6. If every single person on the planet had purchase a power ball ticket during any one week, only 44 of them would win the lottery. At least it'd make some small child in a third world country happy for a while!

7. If you put $10,000 a week into playing the lottery, it would take you on average 2,809 years to win.

8. You are over 8000 times more likely to be murdered than playing the lottery. You're probably more likely to be murdered on the way to purchase you're lottery ticket than you are to actually win it!

9. You are 213 times more likely to die in your bathtub than you are to win the lottery and 146 times more likely to die of a fireworks accident than you are to win the lottery.

G.K. Elio

STATISTICAL FACTS

According to a study by Swiss researchers, people are 14% more likely to die on their birthday than on any other day of the year. 

Half of the population of Uganda is under 15 years of age.

100 people a year choke to death on ball-point pens.

A group of researchers counted the number of troubles in the world, and came up with a total of 2,653.

In 1997, U.S. News & World Report surveyed its readers as to whether they believed various well-known figures to be "very likely" or "somewhat likely" to get into heaven. More people (87%) picked themselves as being likely to get into heaven than anyone else, ahead of people such as Mother Teresa (second place, 79%) or Oprah Winfrey (third place, 66%).

J.Santillan

STATISTICAL FACTS


1.You might think that the most common street name in the United States would be "First", but according to statistics from the U.S. Census Bureau, the most common street name is "Second", with 10,866 occurrences. Strangely enough, "First" isn't in second place either; "Third" is in second place with 10,131 occurrences, while "First" is in third place with 9,898. 

2.The population of the Earth can be a source of many interesting statistics. In the year 8,000 B.C., there were only 5 million people on Earth. Four thousand years later, the population had only risen by 2 million people, to 7 million people. Nowadays, Earth's population rises by 2 million roughly every nine days.

3.In 1662, John Graunt, a London merchant, published the first set of actuarial tables in his book Observations on the Bills of Mortality. Graunt provides many interesting statistics regarding causes of deaths in London in 1632. Seven people are listed as being murdered, 10 people as having died from cancer, and no specific mention is made of heart ailments. On the other hand, 13 people are listed as having died from "planet", 38 from "king's evil", and 98 from "rising of the lights". Possibly the saddest statistic, however, is that out of 9,535 deaths that year, infants made up 2,268 of them, over 23%.

4.Worldwide, around 265 people are born every minute and 115 people die, for a net increase in population of 150 people every minute. 

5.The current population of Earth is over 6,970,000,000. Around 1900 there were only 1,600,000,000 people, meaning that Earth's population has more than quadrupled in slightly over 100 years' time. 

6.Only 1% of the population has a "genius" IQ, one of 140 or higher.

7.Over 88% of the world's population lives north of the Equator.

8.In 1985, NASA estimated the probability of a space shuttle accident to be 1 in 100,000. However, on the 25^th shuttle launch on January 28, 1986, Challenger exploded after take-off, killing all seven astronauts aboard; on February 1, 2003, the 113^rd mission, Columbia exploded on re-entry, again with the loss of all seven astronauts. Other groups had earlier estimated the probability of failure as being closer to 1 in 100, a probability that now seems more reasonable.

9.In a study of 3,000 people who made New Year's resolutions in 2007, only 12% stuck to them. The resolution with the greatest chance of success was "to enjoy life more".
In 1936, Literary Digest magazine polled 10 million people (more people than in any previous survey prior to a presidential election) using the telephone and its mailing list to try to predict the outcome of the United States presidential election. Their results indicated that Alf Landon would defeat Franklin Roosevelt by a margin of 370 electoral votes to 161; however, in the election, Landon was trounced by Roosevelt by a margin of 523 electoral votes to 8, at the time the largest landslide in a presidential election. The problem with the survey was that, during the Great Depression, telephones and magazine subscriptions were luxuries that not everyone could afford. Those who could afford such luxuries tended to vote Republican, but the voting public in general was more inclined to vote Democrat.

10.In 1915, statistics were compiled from 18 U.S. states of the number of deaths from three branches of outdoor sport. 16 people were killed in football, 59 in hunting, and 59 in baseball.
In 1938, a United States presidential commission concluded that the nation's population would never reach 140 million. The population exceeded that figure only eight years later.



J.Santillan

COOL FACTS

A car is stolen almost every minute in the United States.

More than 50% of the accidents happen during the weekend.

An average person speaks about 30,000+ words per day.

The average person spends about 2 years of their life talking over the phone.

Law of Right – 90% of people who enter a store turn to the right.

1% of the world population at any given time of a day is drunk.

Chances of you being murdered is one in twenty thousand.

Odds of being killed by Blizzard or Tornado is one in a million.

Fifty percent of teenager boys say they would rather be RICH than SMART.

200-400 babies are born worldwide every minute.


G.K. Elio

DID YOU KNOW?

Only 5% of people shot to the head survive.

Woodpeckers can peck at a rate of 20 times per second.

A queen bee can lay up to 1,500 eggs in a day.

Almost 10,000 birds a year die from smashing into windows.

Some lions mate over 50 times a day.
















G.K. Elio

STATISTICAL FUN FACTS

• Worldwide, around 265 people are born every minute and 115 people die, for a net increase in population of 150 people every minute.

• Only 1% of the population has a "genius" IQ, one of 140 or higher.

• Over 88% of the world's population lives north of the Equator.

• In 1915, statistics were compiled from 18 U.S. states of the number of deaths from three branches of outdoor sport. 16 people were killed in football, 59 in hunting, and 59 in baseball.

• According to a study by Swiss researchers, people are 14% more likely to die on their birthday than on any other day of the year.

• 100 people a year choke to death on ball-point pens.

G.K. Elio