T-test
A t-test is any statistical hypothesis test in which the test statistic follows a Student's t distribution if the null hypothesis is supported. It can be used to determine if two sets of data are significantly different from each other, and is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known. When the scaling term is unknown and is replaced by an estimate based on the data, the test statistic (under certain conditions) follows a Student's t distribution.Unpaired and paired two-sample t-test
Two-sample t-tests for a difference in mean involve independent samples, paired samples and overlapping samples. Paired t-tests are a form of blocking, and have greater power than unpaired tests when the paired units are similar with respect to "noise factors" that are independent of membership in the two groups being compared. In a different context, paired t-tests can be used to reduce the effects of confounding factors in an observational study.
(a) Independent samples
The independent samples t-test is used when two separate sets of independent and identically distributed samples are obtained, one from each of the two populations being compared. For example, suppose we are evaluating the effect of a medical treatment, and we enroll 100 subjects into our study, then randomize 50 subjects to the treatment group and 50 subjects to the control group. In this case, we have two independent samples and would use the unpaired form of the t-test. The randomization is not essential here—if we contacted 100 people by phone and obtained each person's age and gender, and then used a two-sample t-test to see whether the mean ages differ by gender, this would also be an independent samples t-test, even though the data are observational.
(b) Paired samples
Paired samples t-tests typically consist of a sample of matched pairs of similar units, or one group of units that has been tested twice (a "repeated measures" t-test).
A typical example of the repeated measures t-test would be where subjects are tested prior to a treatment, say for high blood pressure, and the same subjects are tested again after treatment with a blood-pressure lowering medication. By comparing the same patient's numbers before and after treatment, we are effectively using each patient as their own control. That way the correct rejection of the null hypothesis (here: of no difference made by the treatment) can become much more likely, with statistical power increasing simply because the random between-patient variation has now been eliminated. Note however that an increase of statistical power comes at a price: more tests are required, each subject having to be tested twice. Because half of the sample now depends on the other half, the paired version of Student's t-test has only 'n/2 - 1' degrees of freedom (with 'n' being the total number of observations). Pairs become individual test units, and the sample has to be doubled to achieve the same number of degrees of freedom.
A paired samples t-test based on a "matched-pairs sample" results from an unpaired sample that is subsequently used to form a paired sample, by using additional variables that were measured along with the variable of interest. The matching is carried out by identifying pairs of values consisting of one observation from each of the two samples, where the pair is similar in terms of other measured variables. This approach is sometimes used in observational studies to reduce or eliminate the effects of confounding factors.
Paired samples t-tests are often referred to as "dependent samples t-tests" (as are t-tests on overlapping samples).
(c) Overlapping samples
An overlapping samples t-test is used when there are paired samples with data missing in one or the other samples (e.g., due to selection of "Don't know" options in questionnaires or because respondents are randomly assigned to a subset question). These tests are widely used in commercial survey research (e.g., by polling companies) and are available in many standard crosstab software packages.
Year
|
t value
|
P value
|
Null hypothesis
|
2003-04
|
-10.4625
|
0.00
|
reject
|
2004-05
|
-10.1878
|
0.00
|
reject
|
2005-06
|
-8.47942
|
0.00
|
reject
|
2006-07
|
-6.92902
|
0.00
|
reject
|
2007-08
|
-1.71972
|
0.06
|
accept
|
2008-09
|
0.716896
|
0.24
|
accept
|
2009-10
|
-8.8544
|
0.00
|
reject
|
2010-11
|
-4.72108
|
0.00
|
reject
|
2011-12
|
-3.09829
|
0.01
|
reject
|
CORRELATION
Correlation is a measure of the degree of relatedness of variables. It can help a business researcher determine, for example, whether the stocks of two airlines rise and fall in any related manner. For a sample of pairs of data, correlation analysis can yield a numerical value that represents the degree of relatedness of the two stock prices over time.
Correlation is determined using sample coefficient of correlation, r, where r is a measure of the linear correlation of two variables.
The Correlation between 2 variables can be computed using the Product Pearson – Moment Correlation Coefficient which can be given by
In probability and statistics, mean and expected value are used synonymous to refer to one measure of the central tendency either of a probability distribution or of the variable characterized by that distribution. In the case of a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability P(x), and then adding all these products together, giving .
µ =Σ x P(x)
µ =Σ x P(x)
An analogous formula applies to the case of a continuous probability distribution. Not every probability distribution has a defined mean; see the Cauchy distribution for an example. Moreover, for some distributions the mean is infinite: for example, when the probability of the value is for n = 1, 2, 3, ....
For a data set, the terms arithmetic mean, mathematical expectation, and sometimes average are used synonymously to refer to a central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values. T bar". If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is termed the sample mean to distinguish it from the population mean. .
For a finite population, the population mean of a property is equal to the arithmetic mean of the given property while considering every member of the population. For example, the population mean height is equal to the sum of the heights of every individual divided by the total number of individuals. The sample mean may differ from the population mean, especially for small samples. The law of large numbers dictates that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean.
SUBMITTED BY : Palak Jain(2013185)
GROUP No : 7
Nidhi Sharma (2013169)
Nitesh Singh Patel(2013178)
Nitin Boratwar(2013179)
Palak Jain(2013185)
Pallavi Bizoara (2013186)
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