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Statistical Analysis The idea behind parametric tests is to provide the researcher with a statistical inference about the population by conducting statistically significant tests (like t-test) on the sample drawn from the population. The parametric test called t-test is based on a student’s t statistic. This statistic assumes that variables are drawn from the normal population. The mean of the population in this statistic of t-test has been assumed to be known. The distribution, called t-distribution, has a similar shape to that of a normal distribution, i.e. a bell shaped appearance. Statistics Solutions is the country’s leader in statistical consulting and t-test analysis. Use the calendar below to schedule a free 30-minute consultation.  Discover How We Assist to Edit Your Dissertation ChaptersAligning theoretical framework, gathering articles, synthesizing gaps, articulating a clear methodology and data plan, and writing about the theoretical and practical implications of your research are part of our comprehensive dissertation editing services.
The parametric test called t-test is useful for testing those samples whose size is less than 30. The reason behind this is that if the size of the sample is more than 30, then the distribution of the t-test and the normal distribution will not be distinguishable. The parametric test is used for conducting statistically significant tests in the testing of hypotheses. There are basically three types of t-tests: one sample t-test, two independent sample t-test and paired t-test. In the case of a one sample t-test, if a researcher in the field of psychology is working on a study where he wants to make sure that at least 65% of students will pass the IQ test, he can use the t-test. So, one sample t-test will be used after the hypothesis has been formulated in this particular case. The parametric test is then calculated by selecting an appropriate formula of t-test. In this case, the appropriate formula will be a t-test for a single mean. A selection of the level of significance is conducted to check the t-test of the null hypothesis. Usually, the researcher takes 0.05 as the appropriate level of significance while conducting the t-test. The level of significance refers to the minimum probability that there will be a false rejection of the null hypothesis. Now, if the value calculated from the t-test is more than the tabulated value, then the null hypothesis gets rejected at a particular level of significance. Similarly, if the value calculated from the t-test is less than the tabulated value, then the null hypothesis gets accepted at a particular level of significance. In two independent sample t-tests, two samples that are not at all related to each other are tested. The main idea behind two independent sample t-tests is to draw out a statistical inference about the comparison of two independent samples of data. For example, in the field of psychology, if the researcher wants to compare the IQ level of students living in region A and region B, then a two independent sample t-test is useful. The region A and the region B are not at all related to each other, i.e. they are independent of each other. The procedure for conducting this t-test is the same, except that now the sample number is double instead of single. Also, in the case of t-test for single mean and two independent sample t-tests, there are different formulas for the degree of freedom. The degree of freedom is referred to as the restriction that a researcher puts forward while conducting parametric tests, like t-test in this case. The paired sample t-test refers to that type of sample in which the variables form paired categories. For example, if a researcher wants to compare male and female smokers, paired sample t-test comes into play if the variable is in following form: male chain smoker and female chain smoker, male occasional smoker and female occasional smoker, etc. Reference: Introduction to the theory of statistics: Mood A.M., Graybill F.A., Boes D.C Note: Some of the statements in the text below are disputed. For small sample size non-parametric tests like the Mann-Whitney U test or the Wilcoxon rank-sum test might rather be used than a t-test.
A one sample t-test has the following null hypothesis: where the Greek letter (mu) represents the population mean and c represents its assumed (hypothesized) value. In statistics it is usual to employ Greek letters for population parameters and Roman letters for sample statistics. The t-test is the small sample analog of the z test which is suitable for large samples. A small sample is generally regarded as one of size n<30. A t-test is necessary for small samples because their distributions are not normal. If the sample is large (n>=30) then statistical theory says that the sample mean is normally distributed and a z test for a single mean can be used. This is a result of a famous statistical theorem, the Central limit theorem. A t-test, however, can still be applied to larger samples and as the sample size n grows larger and larger, the results of a t-test and z-test become closer and closer. In the limit, with infinite degrees of freedom, the results of t and z tests become identical. In order to perform a t-test, one first has to calculate the "degrees of freedom." This quantity takes into account the sample size and the number of parameters that are being estimated. Here, the population parameter, mu is being estimated by the sample statistic x-bar, the mean of the sample data. For a t-test the degrees of freedom of the single mean is n-1. This is because only one population parameter (the population mean)is being estimated by a sample statistic (the sample mean). degrees of freedom (df)=n-1 For example, for a sample size n=15, the df=14. Example[edit | edit source]A college professor wants to compare her students' scores with the national average. She chooses a simple random sample (SRS) of 20 students, who score an average of 50.2 on a standardized test. Their scores have a standard deviation of 2.5. The national average on the test is a 60. She wants to know if her students scored significantly lower than the national average. Significance tests follow a procedure in several steps. Step 1[edit | edit source]First, state the problem in terms of a distribution and identify the parameters of interest. Mention the sample. We will assume that the scores (X) of the students in the professor's class are approximately normally distributed with unknown parameters μ and σ Step 2[edit | edit source]State the hypotheses in symbols and words.
The null hypothesis is that her students scored on par with the national average. The alternative hypothesis is that her students scored lower than the national average. Step 3[edit | edit source]Secondly, identify the test to be used. Since we have an SRS of small size and do not know the standard deviation of the population, we will use a one-sample t-test. The formula for the t-statistic T for a one-sample test is as follows: where is the sample mean and S is the sample standard deviation. A quite common mistake is to say that the formula for the t-test statistic is: This is not a statistic, because μ is unknown, which is the crucial point in such a problem. Most people even don't notice it. Another problem with this formula is the use of x and s. They are to be considered the sample statistics and not their values. The right general formula is: in which c is the hypothetical value for μ specified by the null hypothesis. (The standard deviation of the sample divided by the square root of the sample size is known as the "standard error" of the sample.) Step 4[edit | edit source]State the distribution of the test statistic under the null hypothesis. Under H0 the statistic T will follow a Student's distribution with 19 degrees of freedom: . Step 5[edit | edit source]Compute the observed value t of the test statistic T, by entering the values, as follows: Step 6[edit | edit source]Determine the so-called p-value of the value t of the test statistic T. We will reject the null hypothesis for too small values of T, so we compute the left p-value: p-valueThe Student's distribution gives at probabilities 0.95 and degrees of freedom 19. The p-value is approximated at 1.777e-13. Step 7[edit | edit source]Lastly, interpret the results in the context of the problem. The p-value indicates that the results almost certainly did not happen by chance and we have sufficient evidence to reject the null hypothesis. The professor's students did score significantly lower than the national average. See also[edit | edit source]What if sample size is less than 30 for tThe parametric test called t-test is useful for testing those samples whose size is less than 30. The reason behind this is that if the size of the sample is more than 30, then the distribution of the t-test and the normal distribution will not be distinguishable.
When n is smaller than 30 then the tWhen n is small (less than 30), how does the shape of the t distribution compare to the normal distribution? It is taller and narrower than the normal distribution. It is almost perfectly normal.
When n is small less than 30 how does the shape of the tFor practical purposes, the shape of the t-distribution is identical to the normal distribution when sample size is large. However, when sample sizes are small (below 30 subjects), the shape of the t-distribution is flatter than that of the normal distribution, and the t-distribution has greater area under the tails.
When the sample size is less than 30 then it is known as?For example, when we are comparing the means of two populations, if the sample size is less than 30, then we use the t-test.
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