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)
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• Apple and Android represent more than 75% of the smartphone market
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• 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