Two-tailed Test

Stuck in mathematics dire,
Knee-deep in statistical mire...

Whoa! I still don't seem to have gotten over those early days when I was being initiated to the intricacies of statistics - even today the terror of those graphs and hypotheses makes me rhyme! I remember the time when, during the first semester of my business management studies course, I and most of my classmates shared the intellectual agony of completing and submitting difficult statistics assignments within stingy deadlines at the tremendous opportunity cost of relinquishing our right to more entertaining pursuits. Anyway, before I embark upon a full-fledged walk down the avenues of nostalgia, let me retrace my steps towards statistics help and enable you to understand what a two-tailed test is all about.

What is a Two-Tailed Test?

Before letting out the details of two-tailed test, let's take a quick look at statistical inference. A statistical inference refers to the process of drawing conclusions from a given set of data that may have an influence of random variations such as sampling variations and observational errors. Two-tailed test is a statistical test which is used to arrive at a statistical inference. In a two-tailed test, the null hypothesis H₀ is rejected in case the value of the test statistic, which is a function of the sample, is either sufficiently small or sufficiently large.

This is where the one tailed test vs two-tailed test difference comes into picture as in case of the former, only one among the two rejection areas (either sufficiently small or sufficiently large) is preselected in accordance with the fact that the alternative hypothesis is chosen and rejection takes place only in case of the test fulfilling this said criterion. When you draw a bell curve, which is what a normal data distribution curve is known as, you get two data tails at the horizontal extremities, parallel to the X-axis. It is due to these data tails that this test gets its name.

When to Use This Test?

This test is used for statistical inference when the statistical hypothesis is rejected due to its value being either sufficiently large or sufficiently small. In other words, when the null hypothesis gets rejected due to its value falling under either tails of the sampling distribution curve, a two-tailed test is used to detect changes in the parameter.

Two-tailed Test Formula

The formula for drawing statistical inference using this tailed test goes as follows:-

ɀ = (Arithmetic Mean-μ)/(σ/√n)

Where,
ɀ = test statistic
μ = mean of population values
σ = standard deviation of differences between all pairs
n = total number of pairs

Therefore, using this formula, we can take a look at the following example to better understand this statistical concept.

Problem
During a recruitment drive, 150 job applicants were asked take a test to evaluate their Intelligence Quotient (IQ). From the data provided by the applicants, the recruitment personnel found out that the mean IQ is 120 and the standard deviation of the results is 11.3. They also estimated that the population mean would be less than 116. Use a 0.05 significance level and decide whether the recruiters are correct in their hypothesis.

Solution
H₀ = 116
H₁ = >116
Therefore, using significance level 0.05, critical value = 1.645
Since, ɀ = (Arithmetic Mean-μ)/(σ/√n),
Hence, ɀ = (120-116)/(11.3/√150) = 4.34

We can see, on plotting the bell curve, that the test statistic 4.34 will fall to the right of the critical value 1.645, falling inside the critical region. This rejects the null hypothesis H₀. Therefore, it can be concluded from the above two-tailed test that the alternative hypothesis H₁ has been accepted, citing the value of the population mean to be greater than 116. Hence, the statistical verdict contradicts the recruiters' estimate.