Question

What are the two, main statistics calculated in the study to test a hypothesis.

Answer 1

There are two main types of hypotheses we can test: one-tailed hypotheses and two-tailed hypotheses. Our critical region will be constructed differently in each case.

A one-tailed test is a statistical test in which the critical area of a distribution is one-sided so that it is either greater than or less than a certain value, but not both. If the sample being tested falls into the one-sided critical area, the alternative hypothesis will be accepted instead of the null hypothesis

Example:

Suppose we wanted to test whether or not girls, on average, score higher than 600 on the SAT verbal section. Our underlying theory is that girls do score higher than 600, which would give us the following null (denoted H0) and alternative (denoted H1) hypotheses:

H0 : µ ? 600

H1 : µ > 600

where µ is the average score for girls on the SAT verbal section. This is an example of what is called a one-tailed hypothesis. The name comes from the fact that evidence against the null hypothesis comes from only one tail of the distribution (namely, scores above 600). When constructing the critical region of size ?, one finds a critical value in the sampling distribution so that the area under the distribution in the interval (critical value, ?) is ?. We will explain how to find a critical value in later sections.

A two-tailed test, also known as a non directional hypothesis, is the standard test of significance to determine if there is a relationship between variables in either direction. Two-tailed tests do this by dividing the .05 in two and putting half on each side of the bell curve.

Example:

Suppose instead that we wanted to see if girls scored significantly different than the national average score on the verbal section of the SAT, and suppose that national average was 500. Our underlying theory is that girls do score significantly different than the national average, which would give us the following null and alternative hypotheses:

H0 : µ = 500

H1 : µ 6= 500

where again µ is the average score for girls on the SAT verbal section. This is an example of a twotailed hypothesis. The name comes from the fact that evidence against the null hypothesis can come from either tail of the sampling distribution (namely, scores significantly above AND significantly below 500 can offer evidence against the null hypothesis). When constructing the critical region of size ?, one finds two critical values (when assuming the null is true, we take one above the mean and one below the mean) so that the region under the sampling distribution over the interval (??, critical value 1) ? (critical value 2, ?) is ?. Often we choose symmetric regions so that the area in the left tail is ?/2 and the area in the right tail is ?/2; however, this is not required. There are advantages in choosing critical regions where each tail has equal probability. There will be several types of hypotheses we will encounter throughout our work, but almost all of them may be reduced to one of these two cases, so understanding each of these types will prove to be critical to understanding hypothesis testing.