Question

I have to submit a question and a solution to a problem for an assignment. Please help!

Select a single concept or topic covered in this course (e.g. regression, t-testing, ANOVA, correlation, etc) and create a supplemental material explaining it and its use. Aspects of your explanation may include:

- When to use this statistical test.

- How this statistical test functions

- How this statistical test is computed

- What are the assumptions of this statistical test?

- Worked examples of this statistical test in action.

- Real world examples of when the statistical test might be appropriate.

Answer 1

We have to select a single concept or topic mentioned in the problem. Let us choose t-testing.

Area of use-

In case we are given a set of sample data about which we know that the population is normally distributed (either mentioned or we can test) but we do not know the value of population standard deviation (or population variance), we use t-testing. Few instances when we use this testing procedure are as follows.

• Test of population mean of a normally distributed data when population variance is unknown
• Test of equality of two population means of a normally distributed data when population variances are unknown but equal
• Paired t-test for two population.
• Test of significance of population regression coefficient
• Test of significance of population correlation coefficient

and lot more

Function of this statistical test-

In this statistical test, we first propose null and alternative hypotheses as per the demand of the problem. Then we calculate sample mean(s) and sample standard deviation(s) along with number of sample and other required parameters. We then use formulae obtained through different theoretical approaches and compute test statistic. At the given level of significance for the degrees of freedom (calculated from sample size) we obtain critical value using standard table or any software. We compare computed value of test statistic and critical value to reject or not to reject our null hypothesis. Another approach is to calculate p-value corresponding to test statistic. We reject null hypothesis in case of p-value being smaller than level of significance and do not reject otherwise.

Computation of statistical test-

At first, we calculate sample mean(s) and sample standard deviation(s), sample size and other such parameters as required. We then use formulae obtained through different theoretical approaches and compute test statistic.

Assumptions of this statistical test-

This test can be done in case of normally distributed data only. So, in many instances when it not given that the population corresponding to the given sample are normally distributed we try to test weather the given data comes from normally distributed data or merely assume that the population is normally distributed.

Example-

Suppose, we are given pairs of data for same population in different situations. Further we are interested in detection weather there is any significant difference in two situations or not. We then perform paired t-test. We compute difference in two situations for same sample unit, compute average difference and standard deviation of differences and calculate for test statistic as follows.

We have to test for null hypothesis

against the alternative hypothesis

Xi	Yi	Di=Xi-Yi
x1	y1	d1=x1-y1
x2	y2	d2=x2-y2
......	…..	…..
xn	yn	dn=xn-yn

Test statistic is given by

Degrees of freedom

Level of significance

We calculate p-value corresponding to the test statistic.

If p-value < we reject null hypothesis, otherwise we fail to reject.

Real world example-

Suppose, we are interested to know weather there is any impact of a special class upon students regarding a topic. For this we choose a sample of students, examine them through a test, provide special class and finally take an alike test after completion of the special class. Based on paired scores of each student (sample unit) we perform analysis (as mentioned theoretically in the previous part). Based on our p-value and required level of significance, we reach to a conclusion.