Answer to Question #253670 in Statistics and Probability for Goldie

Question

Getting acquainted with the essential terms and concepts on testing the difference between two dependent sample means, it is now time for you to explain thoroughly your answers to the following questions.

Explain the role of the following statistical terms in testing the difference of two dependent samples.

a. Hypotheses (Null and Alternative Hypotheses) 

b. Test of significance (One Tailed Test and Two Tailed Test) 

c. Level of significance 

d. Critical region 

e. Critical value 

f. Test value 

Answer 1

a.

two tailed test

Null Hypotheses: $\mu_1=\mu_2$ , the means are equal

Alternative Hypotheses: $\mu_1\neq\mu_2$ , the means are not equal

one tailed test

Null Hypotheses: $\mu_1-\mu_2<C$ , difference of means is less than some value

Alternative Hypotheses: $\mu_1\ge\mu_2$ ,difference of means is equal or more than some value

b.

A test of significance is a formal procedure for comparing observed data with a claim.

We compare test statistic with critical value; if test statistic is less than critical value, we accept null hypothesis. For two tailed test critical value is for half of level of significance for one tailed test.

c.

The level of significance (alpha) is the percentage of risk we are willing to take while rejecting the null hypothesis.

d.

A critical region, also known as the rejection region, is a set of values for the test statistic for which the null hypothesis is rejected.

e.

A critical value is the value of the test statistic which defines the upper and lower bounds of a confidence interval, or which defines the threshold of statistical significance in a statistical test.

f.

A test statistic is a standardized value that is calculated from sample data during a hypothesis test. The procedure that calculates the test statistic compares your data to what is expected under the null hypothesis.

For example, for z test, test statistic is:

$z=\frac{\mu_1-\mu_2}{\sqrt{\sigma_1^2/n_1+\sigma_2^2/n_2}}$