Example of ANOVA Calculation

One factor analysis of variance, also known as ANOVA, gives us a way to make multiple comparisons of several population means. Rather than doing this in a pairwise manner, we can look simultaneously at all of the means under consideration. To perform an ANOVA test, we need to compare two kinds of variation, the variation between the sample means, as well as the variation within each of our samples. We combine all of this variation into a single statistic, called the ​F statistic because it uses the F-distribution. We do this by dividing the variation between samples by the variation within each sample. The way to do this is typically handled by software, however, there is some value in seeing one such calculation worked out. It will be easy to get lost in what follows. Here is the list of steps that we will follow in the example below: Calculate the sample means for each of our samples as well as the mean for all of the sample data.Calculate the sum of squares of error. Here within each sample, we square the deviation of each data value from the sample mean. The sum of all of the squared deviations is the sum of squares of error, abbreviated SSE.Calculate the sum of squares of treatment. We square the deviation of each sample mean from the overall mean. The sum of all of these squared deviations is multiplied by one less than the number of samples we have. This number is the sum of squares of treatment, abbreviated SST.Calculate the degrees of freedom. The overall number of degrees of freedom is one less than the total number of data points in our sample, or n - 1. The number of degrees of freedom of treatment is one less than the number of samples used, or m - 1. The number of degrees of freedom of error is the total number of data points, minus the number of samples, or n - m.Calculate the mean square of error. T his is denoted MSE SSE/(n - m).Calculate the mean square of treatment. This is denoted MST SST/m - 1.Calculate the F statistic. This is the ratio of the two mean squares that we calculated. So F MST/MSE. Software does all of this quite easily, but it is good to know what is happening behind the scenes. In what follows we work out an example of ANOVA following the steps as listed above. Data and Sample Means Suppose we have four independent populations that satisfy the conditions for single factor ANOVA. We wish to test the null hypothesis H0: ÃŽ ¼1 ÃŽ ¼2 ÃŽ ¼3 ÃŽ ¼4. For purposes of this example, we will use a sample of size three from each of the populations being studied. The data from our samples is: Sample from population #1: 12, 9, 12. This has a sample mean of 11.Sample from population #2: 7, 10, 13. This has a sample mean of 10.Sample from population #3: 5, 8, 11. This has a sample mean of 8.Sample from population #4: 5, 8, 8. This has a sample mean of 7. The mean of all of the data is 9. Sum of Squares of Error We now calculate the sum of the squared deviations from each sample mean. This is called the sum of squares of error. For the sample from population #1: (12 – 11)2 (9– 11)2 (12 – 11)2 6For the sample from population #2: (7 – 10)2 (10– 10)2 (13 – 10)2 18For the sample from population #3: (5 – 8)2 (8 – 8)2 (11 – 8)2 18For the sample from population #4: (5 – 7)2 (8 – 7)2 (8 – 7)2 6. We then add all of these sum of squared deviations and obtain 6 18 18 6 48. Sum of Squares of Treatment Now we calculate the sum of squares of treatment. Here we look at the squared deviations of each sample mean from the overall mean, and multiply this number by one less than the number of populations: 3[(11 – 9)2 (10 – 9)2 (8 – 9)2 (7 – 9)2] 3[4 1 1 4] 30. Degrees of Freedom Before proceeding to the next step, we need the degrees of freedom. There are 12 data values and four samples. Thus the number of degrees of freedom of treatment is 4 – 1 3. The number of degrees of freedom of error is 12 – 4 8. Mean Squares We now divide our sum of squares by the appropriate number of degrees of freedom in order to obtain the mean squares. The mean square for treatment is 30 / 3 10.The mean square for error is 48 / 8 6. The F-statistic The final step of this is to divide the mean square for treatment by the mean square for error. This is the F-statistic from the data. Thus for our example F 10/6 5/3 1.667. Tables of values or software can be used to determine how likely it is to obtain a value of the F-statistic as extreme as this value by chance alone.

Piagets Four Stages of Cognitive Development - 1621 Words

Describe Piaget’s four stages of cognitive development. Based on this understanding describe how you would tailor a coaching session to suit the needs of a child who is 6 years old. â€Å"The influence of Piaget’s ideas in developmental psychology has been enormous. He changed how people viewed the child’s world and their methods of studying children. He was an inspiration to many who came after and took up his ideas. Piagets ideas have generated a huge amount of research which has increased our understanding of cognitive development.† (McLeod 2009). Piaget purposed that we move through stages of cognitive development. He noticed that children showed different characteristics throughout their childhood development. The four stages of†¦show more content†¦Whilst playing around, teddy actually messed up one row of sweets. Once he was safely back in a box the children were asked if there were the same number of sweets. The children were between four- and six-years-old, and more than half gave the correct answer.† (McLeod 2009). They claimed that Piagets tasks prevents a child from conserving at an younger age. The Formal Operational stage- This stage occurs to a child who is above the age of twelve and goes onto their adult life. The child no longer requires material objects to think logically, they start to imagine and also start to predict outcomes. Advanced tactical knowledge is developed at a later stage. Everything the child does, the actions and thoughts are organised systematically. The child begin to manipulate ideas using his/hers brain. In problem solving children at the formal operational stage use a more strategic approach where as in the earlier stages the children and infants used a trial and error method to solve problems. The â€Å"third Eye Problem† used by Piaget was a successful way to demonstrate the difference between the earlier stages and the formal operational stage. Children of the concrete operational stage were asked where they would place an extra eye if they were given one and most of them answered with â€Å"on my forehead† compared to when chil dren from the formal operational stage were questioned, they replied by suggesting,Show MoreRelatedJean Piagets Four Stages Of Cognitive Development1326 Words   |  6 PagesTheory Overview Jean Piaget’s theory of cognitive development proposes that the developmental process of a child is the result of their brains maturity, their nervous system, and environmental factors. He believes the foundation of a childs ability to learn is through discovery learning (Gordon Browne, 2016). Piaget suggests that a child’s logic of thinking is different from that of an adults. 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Should Internet Have A Tax Essay Example For Students

Should Internet Have A Tax? Essay In 1998, retailers sold nearly $8 billion goods and services to consumers over the Internet, or on-line, while business to business online commerce was valued at an estimated $17 billion. Business weekly magazine (June 22, 1998) predicted that Internet commerce would increase the U.S. gross domestic product (GPD) by between $10 billion and $20 billion annually by 2002. They argue that imposing new Internet taxes, at least during the next few years, would bog down the Internets growth and stunt a sector of the economy that is currently flourishing. For now industry leaders say it is important to build consumer confidence in the Internet by refraining from imposing taxes or other regulatory barriers that may deter people from shopping on line. Internet retailers must charge a sales tax only if the company has some kind of physical presence, such as a warehouse or an office, in the state where the customer is buying the item. Otherwise, companies do not have to add the sales tax to the purchase price. In 1997, Sen. Ron Wyden introduced the legislation that developed into the Internet Tax Freedom Act. The ITFA called for a moratorium of approximately six years on the taxation of Internet transactions, access, or communications. Wyden called the moratorium a time out period that would give the Internet the opportunity to continue to grow. The goal of the legislation was to give lawmakers and Internet industry time to figure out a national taxation policy. Many businesses, he says, would be scared away from the Internet if they were burdened with the responsibility of monitoring and enforcing a thicket of conflicting sales taxes imposed by various states and municipalities.Indeed the potential loss of tax revenue for states and municipalities is one of the biggest concerns. Unlike the federal government, which does not impose a sales tax, states are heavily dependent on sales taxes to raise revenues. Sales taxes comprise 49% of tax revenues collected by the states, while state income taxes comprise only 33%, according to the federal statistics. Critics of the ITFA say that if states municipalities are not permitted to collect taxes on Internet transactions, they could lose much needed revenue that helps pay for government services such as highway construction and public education. Others say that not collecting sales tax from the Internet is inherently discriminatory since businesses that do not engage in electronic commerce must still charge sales taxes. It is fundamentally unfair that Main Street retailers should be required to collect a sales tax while Internet and mail order vendors can sell the same goods and not be required to collect a sales tax, argues Utah Gov. Michael Leavitt. Once consumers and businesses become accustomed to tax-free shopping on the Internet, there may be a backlash against attempts to impose sales taxes when the moratorium ends. Government should impose a tax on all items sold over the Internet. The simplest and easiest to implement, would be a flat rate that applies to all states. I think that if there is no tax collected from these sales, each state will start to suffer from lose of revenue that helps pay for some of lifes necessities such as roads, education, and law enforcement. I think that Congress knew that there was a problem and by putting a moratorium, it would give them time to come up with some ideas. But by the time they come up with ideas it will be to late. If they do enforce a tax on Internet sales, people will either pay it or start going back to Main Street businesses, then we have a problem because Internet will no longer have the same sale volume. If they do not enforce a tax, I think that the Main Street businesses will either shut their doors or go to the Internet for their sales. In the latter event they will probably still shut their doors and move to an office building. Why have a huge store that they no longer need when they can run everything from a computer. .u41fab4e8818473a00013b66843bcdac3 , .u41fab4e8818473a00013b66843bcdac3 .postImageUrl , .u41fab4e8818473a00013b66843bcdac3 .centered-text-area { min-height: 80px; position: relative; } .u41fab4e8818473a00013b66843bcdac3 , .u41fab4e8818473a00013b66843bcdac3:hover , .u41fab4e8818473a00013b66843bcdac3:visited , .u41fab4e8818473a00013b66843bcdac3:active { border:0!important; } .u41fab4e8818473a00013b66843bcdac3 .clearfix:after { content: ""; display: table; clear: both; } .u41fab4e8818473a00013b66843bcdac3 { display: block; transition: background-color 250ms; webkit-transition: background-color 250ms; width: 100%; opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #95A5A6; } .u41fab4e8818473a00013b66843bcdac3:active , .u41fab4e8818473a00013b66843bcdac3:hover { opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #2C3E50; } .u41fab4e8818473a00013b66843bcdac3 .centered-text-area { width: 100%; position: relative ; } .u41fab4e8818473a00013b66843bcdac3 .ctaText { border-bottom: 0 solid #fff; color: #2980B9; font-size: 16px; font-weight: bold; margin: 0; padding: 0; text-decoration: underline; } .u41fab4e8818473a00013b66843bcdac3 .postTitle { color: #FFFFFF; font-size: 16px; font-weight: 600; margin: 0; padding: 0; width: 100%; } .u41fab4e8818473a00013b66843bcdac3 .ctaButton { background-color: #7F8C8D!important; color: #2980B9; border: none; border-radius: 3px; box-shadow: none; font-size: 14px; font-weight: bold; line-height: 26px; moz-border-radius: 3px; text-align: center; text-decoration: none; text-shadow: none; width: 80px; min-height: 80px; background: url(https://artscolumbia.org/wp-content/plugins/intelly-related-posts/assets/images/simple-arrow.png)no-repeat; position: absolute; right: 0; top: 0; } .u41fab4e8818473a00013b66843bcdac3:hover .ctaButton { background-color: #34495E!important; } .u41fab4e8818473a00013b66843bcdac3 .centered-text { display: table; height: 80px; padding-left : 18px; top: 0; } .u41fab4e8818473a00013b66843bcdac3 .u41fab4e8818473a00013b66843bcdac3-content { display: table-cell; margin: 0; padding: 0; padding-right: 108px; position: relative; vertical-align: middle; width: 100%; } .u41fab4e8818473a00013b66843bcdac3:after { content: ""; display: block; clear: both; } READ: Cyber-Crime is a Serious Problem Essay I think government should put a tax on Internet sales. BibliographyReferencesCoy, Peter. You Aint Seen Nothin Yet. Business Week (June 22, 1998) p. 130 Denver Post. Then Internet Tax Dodge. (March 2, 1998) p. B6Gleckman, Howard. The Tax Man Eyes the Net. Business Week (April 6, 1998) p. 131Gruenwald, Juliana. Moratorium on Internet Taxes Advanced by House Pannels. Congressional Quarterly Weekly (October 11, 1997) p. 2475Hof, Robert; McWilliams, Gary; Saveri, Gabrielle. The Click Here Economy. Business Week (June 22, 1998) p. 122Social Issues