CHI-SQUARE TEST (χ²):
Chi-square is a statistical test commonly used to compare observed data with data we would expect to obtain according to a specific hypothesis. For example, if, according to Mendel's laws, you expected 10 of 20 offspring from a cross to be male and the actual observed number was 8 males, then you might want to know about the "goodness to fit" between the observed and expected. Were the deviations (differences between observed and expected) the result of chance, or were they due to other factors. How much deviation can occur before you, the investigator, must conclude that something other than chance is at work, causing the observed to differ from the expected. The chi-square test is always testing what scientists call the null hypothesis, which states that there is no significant difference between the expected and observed result. The formula for calculating chi-square (χ²) is:

2= (o-e) ²/e
That is, chi-square is the sum of the squared difference between observed (o) and the expected (e) data (or the deviation, d), divided by the expected data in all possible categories.

INTERPRETATION OF CHI-SQUARE TEST
1. Determine degrees of freedom (DF). Degrees of freedom can be calculated as the number of categories in the problem minus 1. 2. Determine a relative standard to serve as the basis for accepting or rejecting the hypothesis. The relative standard commonly used in biological research is p > 0.05. The p value is the probability that the deviation of the observed from that expected is due to chance alone (no other forces acting). In this case, using p > 0.05, you would expect any deviation to be due to chance alone 5% of the time or less. 3. Refer to a chi-square distribution table Using the appropriate degrees of 'freedom, locate the value closest to your calculated chi-square in the table. Determine the closest probability(p) value associated with your chi-square and degrees of freedom. Step-by-Step Procedure for Testing Your...

...Chi-square requires that you use numerical values, not percentages or ratios.
Then calculate 2 using this formula, as shown in Table B.1. Note that we get a value of 2.668 for 2. But what does this number mean? Here's how to interpret the 2 value:
1. Determine degrees of freedom (df). Degrees of freedom can be calculated as the number of categories in the problem minus 1. In our example, there are two categories (green and yellow); therefore, there is I degree of freedom.
2. Determine a relative standard to serve as the basis for accepting or rejecting the hypothesis. The relative standard commonly used in biological research is p >0.05. The p value is the probability that the deviation of the observed from that expected is due to chance alone (no other forces acting). In this case, using p >0.05, you would expect any deviation to be due to chance alone 5% of the time or less.
3. Refer to a chi-square distribution table (Table B.2). Using the appropriate degrees of 'freedom, locate the value closest to your calculated chi-square in the table. Determine the closestp (probability) value associated with your chi-square and degrees of freedom. In this case (2=2.668), the p value is about 0.10, which means that there is a 10% probability that any deviation from expected results is due to chance only. Based on our standard p > 0.05, this is within the...

...CHI-SQUARE AND TESTS OF CONTINGENCY TABLES
Hypothesis tests may be performed on contingency tables in order to decide whether or not effects are present. Effects in a contingency table are defined as relationships between the row and column variables; that is, are the levels of the row variable diferentially distributed over levels of the column variables. Significance in this hypothesis test means that interpretation of the cell frequencies is warranted. Non-significance means that any differences in cell frequencies could be explained by chance.
Hypothesis tests on contingency tables are based on a statistic called Chi-square. In this chapter contingency tables will first be reviewed, followed by a discussion of the Chi-squared statistic. The sampling distribution of the Chi-squared statistic will then be presented, preceded by a discussion of the hypothesis test. A complete computational example will conclude the chapter.
REVIEW OF CONTINGENCY TABLES
Frequency tables of two variables presented simultaneously are called contingency tables. Contingency tables are constructed by listing all the levels of one variable as rows in a table and the levels of the other variables as columns, then finding the joint or cell frequency for each cell. The cell frequencies are then summed across both rows and columns. The sums are...

...Chi-SquareTestChi-square is a statistical test commonly used to compare observed data with data we would expect to obtain according to a specific hypothesis. For example, if, according to Mendel's laws, you expected 10 of 20 offspring from a cross to be male and the actual observed number was 8 males, then you might want to know about the "goodness to fit" between the observed and expected. Were the deviations (differences between observed and expected) the result of chance, or were they due to other factors. How much deviation can occur before you, the investigator, must conclude that something other than chance is at work, causing the observed to differ from the expected. The chi-squaretest is always testing what scientists call the null hypothesis, which states that there is no significant difference between the expected and observed result.
The formula for calculating chi-square ( [pic]2) is:
[pic]2= [pic](o-e)2/e
That is, chi-square is the sum of the squared difference between observed (o) and the expected (e) data (or the deviation, d), divided by the expected data in all possible categories.
For example, suppose that a cross between two pea plants yields a population of 880 plants, 639 with green seeds and 241 with yellow seeds. You are asked to propose the genotypes of the...

...THE CHI-SQUARE GOODNESS-OF-FIT TEST
The chi-square goodness-of-fit test is used to analyze probabilities of multinomial distribution trials along a single dimension. For example, if the variable being studied is economic class with three possible outcomes of lower income class, middle income class, and upper income class, the single dimension is economic class and the three possible outcomes are the three classes. On each trial, one and only one of the outcomes can occur. In other words, a family unit must be classified either as lower income class, middle income class, or upper income class and cannot be in more than one class. The chi-square goodness-of-fit test compares the theoretical, frequencies of categories from a population distribution to the observed, or actual, frequencies from a distribution to determine whether there is a difference between what was expected and what was observed. For example, airline industry officials might theorize that the ages of airline ticket purchasers are distributed in a particular way. To validate or reject this expected distribution, an actual sample of ticket purchaser ages can be gathered randomly, and the observed results can be compared to the expected results with the chi-square goodness-of-fit test. This test also can be used to determine...

...Chi-squaretests
1. INTRODUCTION
1.1 χ2 distribution and its properties
A chi-square (χ2) distribution is a set of density curves with each curve described by its degree of freedom (df). The distribution have the following properties:
- Area under the curve = 1
- All χ2 values are positive i.e. the curve begins from 0 (except for df=1) increases to a peak and decreases towards 0 as its asymptote
- The curve is skewed to the right, and as the degree of freedom increases, the distribution approaches that of a normal distribution
Fig. 1 Graph of χ2 distribution with differing degrees of freedom
Each χ2 value is computed by the formula:
χ2 = Σ (O-E)2
E
where O = observed counts from the sample Equation 1
and E= expected counts based on the hypothesized distribution
1.2 Types of χ2 tests and their purpose
For a single population, to determine if the observed distribution in the population conforms to a specific known distribution or a previously studied distribution, the χ2 test for goodness-of-fit can be used.
An example of this usage include: Mendel’s genetic model predicts that the phenotypic distribution of two phenotypes, each phenotype having a dominant and recessive allele, will follow the ratio of 9:3:3:1. A study done to confirm this makes use of χ2 test for goodness-of-fit to determine if the observed...

...A chi-squared test, also referred to as chi-squaretest or χw² test, is any statistical hypothesis test in which the sampling distribution of the test statistic is a chi-squared distribution when the null hypothesis is true. Also considered a chi-squared test is a test in which this is asymptotically true, meaning that the sampling distribution (if the null hypothesis is true) can be made to approximate a chi-squared distribution as closely as desired by making the sample size large enough.
Some examples of chi-squared tests where the chi-squared distribution is only approximately valid:
Pearson's chi-squared test, also known as the chi-squared goodness-of-fit test or chi-squared test for independence. When the chi-squared test is mentioned without any modifiers or without other precluding context, this test is usually meant (for an exact test used in place of χ², see Fisher's exact test).
Yates's correction for continuity, also known as Yates' chi-squared test.
Cochran–Mantel–Haenszel chi-squared test.
McNemar's...

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Predictability of Social Media Usage to Grade Averages
IB Math Studies
Spring 2013
Table of Contents:
Introduction/Purpose……………………………………………………………..p.3
Data Collection Method……………………………………………………….....p. 3 - 4
Data Analysis: Chi-Squared Statistic
Frequency Table…………………………………………………………p. 4 - 5
Contingency Table……………………………………………………….p. 5 – 6
Chi – Squared Statistic…………………………………………………...p. 7
Degrees of Freedom………………………………………………………p. 7
Critical Value……………………………………………………………..p. 7 – 8
Conclusion……………………………………………………………………....p. 9 - 10
Introduction/Purpose
Predictability is the degree to which a correct prediction or forecast of a system’s state can be made quantitatively. It is necessary to achieve maximum productivity or potential when completing a task. The ability to foresee an occurrence before others allows one to alter different aspects in time to garner the utmost productivity. The question that has been plaguing today’s generation is the use of social media sites, and their consequences on our lives. A restaurant only orders an amount of food adequate for its needs. If the restaurant orders more than it needs, then the extra food will spoil and the restaurant will lose some of its inventory. A trader that has inside information on a corporation may sell his stock minutes before the price drastically falls, saving himself a loss of hundreds of...

...parametric and Chi-square distribution
Brief Summary:
I worked for a logistic company. My major responsibility was in charge of the storage and transportation of parts of cars between two areas, which are about 1400 miles apart. One of my jobs is collecting the goods from suppliers and arranging the trucks to deliver them. There are five truck drivers, and each of them is assigned to deliver on each weekday throughout a whole year. Before the delivery, we will check the quality of the goods and Make sure that there are no damaged goods. When arriving at the destination, the staff will check the goods again and record the damaged goods that occurred in transit. At the end of every month, we will pay for the compensation according to the number of the defective goods. In order to reduce the number of the damaged goods during the delivery, I want to identify the reasons why they are damaged. In this study, I want to find out that whether some drivers are more prone to make the goods damaged during their delivery.
Variable to be measured:
Two variables are to be measured. The first variable is just the five truck drivers, and the second one is the quality of the goods after the delivery.
Determination of Population:
Population in this case is defined as the all goods delivered from Tianjin area to Guangzhou area.
Statistical method:
To analyze relationship between the two variables above which are both nominal in...