# Statistics

There are three different types of t-tests. The one sample t-test is used the mean of a given sample with that of a known population or a known fixed value. The second type is the independent sample t-test which is used in the evaluation of the statistical differences that exist in the means of two groups. The paired samples t-test is used to compare two repeated means from the same test scores in different times or measures and also to compare samples that are paired (Johnson & Bhattacharyya, 2009).

**Calculate the test statistic of the two samples whose sample size, mean and variance are 6&11, 78&113, and 430&95 respectively.**Moving from the use of z-tests to use t-tests is done when there are large samples i.e. where n > 30. This is because if the z-test is used where n < 30, it gives results that are different to those obtained when the t-test is used. This is because in t-test samples, there are certain fluctuations that occur and these are not existent in z-tests. Where the samples do not follow a normal distribution, tests are moved from the z-tests to t-tests.

**Explain the differences between t-tests and z-tests giving reasons why t-tests find more use than z-tests.**In statistics, there are values that are present during the last calculation of the statistic and these values can vary (Sternstein, 1996). These values are the ones that are referred to as degrees of freedom. Generally, from the sample size, one calculates the degrees of freedom. Calculating degrees of freedom measures the information that is used up from the sample. This means that with each calculation of a statistic from a sample, a degree of freedom is used.

**A sample of 16 students is picked from an entire class and the sample mean is 55 kilograms and the standard deviation is 10. Calculate the mean of the population at a 95% confidence level.**Directional hypothesis testing is the testing that is used in a one-tailed test, that is when one sample mean is greater than the other. Non-directional hypothesis testing which is a two-tailed test is used where both negative and positive differences of the sample means are equally important in testing of the null hypothesis.

**State the directional and non-directional hypothesis of the following question. How is family financial stability related to student performance.**When comparing sample data to population data, different parameters are used. These include the mean, and the variance. Where both mean and variance are known, the z test is used. On the other hand, when only the mean is known, then the t test is used to make the comparison.

**Why is it statistically important to compare the sample and population data? In statistics, the comparison of two unique groups is common.**The data that is used to compare these groups is usually given as a summary of scores or means that characterize each group (Rubin, 2009). Statistical analysis is thus important so as to evaluate if the differences are real or they occur due to the variations in measurement.

**Give two statistical techniques that can be used for comparing two normally distributed groups. What are their differences and when does each test gain usage?**. Data that is obtained from one group at two different points can be statistically different. This is because of some external factors. For example, data can fluctuate due to defaults in the measuring instrument, inaccuracy in taking results, climatic conditions, as well as emotions when carrying out psychological test. Therefore, to be able to accept or nullify the hypothesis, this data needs to be compared.

**Which sample test method is used to compare group data from two different points? Why is this method preferred over the others?**

**References**

Johnson, R.A & Bhattacharyya, G.K. (2009). *Statistics: Principles and Methods*. 6^{th} Ed. John Wiley and Sons.

Rubin, A. (2009). *Statistics for Evidence-Based Practice and Evaluation*. 2^{nd} Ed. Cengage Learning.

Sternstein, M. (1996). *Statistics. *Barron’s Educational Series.

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