Two-Tailed t-test

Personally, I hated to study statistics in those early days of my college. The endless calculations, the horrifying formulas and the tangential hypothetical concepts would terrify me in my every single semester exam. If you ask my status yet today, I haven't overcome that dread state. But as of now, I still have the mettle to look out on the brighter side of my reverence and provide you with the needful in this article. Basically, a t-test in the subject of statistics is defined as a statistical hypothesis test where in the test statistics complies if a null hypothesis is substantiated and that follows a Student's t distribution. This distribution generally follows when the test statistic also complies a normal distribution. A normal distribution is practically a 'bell-shaped curve.' All the data is stacked and calculated from far left to far right under that bell-shaped normal distribution figure. A two-tailed t-test has an inferring theoretical word called hypothesis. H₀ is called the null hypothesis. Which means that the 'mean' is equal to 'x'. Now in a statistical test, the value of the resultant test is either amply large or amply small. In the two-tailed test the null hypothesis is rejected for all those values which fall into either the tail of the sampling distribution. Similarly on the other hand, statistics also has a one-tailed t-test in contrast to the former. There in, the resultant test has only one rejection answer, large or small. Hence, in a one-tailed test the hypothesis is rejected if the test statistics fulfills any one particular criteria.

Some More on Two-Tailed t-test

In the books of two-tailed vs one-tailed t-test, both are actually determined by calculating the total area of 'a'. The part to be verified is that, whether a is placed in one tail or is equally subdivided into the two-tailed test. The concept of one-tailed t-test should be comprehended if you see the result proceeding in one direction. In contrast to that, the concept of a two-tailed test should be determined if you see the result estimating in either direction. The choice of these two definitely has an impact on the hypothesis statistical testing process in a host of ways. Significantly if we speak, there is a level of 0.05 figure, which is your 'alpha'. The two-tailed version allots this same alpha as half of the test statistic. That is one alpha is allotted in one direction and the other in the other direction. That means 0.25 is allotted in each tail of the normal distribution during the test statistic.

Two-Tailed Test Formula

ɀ = (Arithmetic Mean-μ)/(σ/√n)

where,
ɀ = test statistic
μ = mean of population values
σ = standard deviation (difference between all the pairs)
n= total number of pairs

This is the basic formula for drawing a statistical inference. With this take, let's take a look at an example.

Problem
Statistically noting the previous years data, if the pay phones at any airport are voided in every 14days, its seen that the coin collectors are 70% full on the average. The pay phone company tries to docket the collection visits at a figure of 70% full, because the money is doomed if the phones are full and not usable. Also, visiting the phones very frequently gets expensive on the other hand. It is obvious that the company has a record of the data during the collection, in case of increase or decrease of frequency. So with this, the problem declares that during the last visit, if suppose 5 pay phones were 50%, 40%, 70%, 75% and 45% full, respectively. What do you think, do the frequency of visits need to be changed? Or is this case a chance variation?

Solution
In the above problem the μ = 70% (this is in either direction since it's a two-tailed t-test). Now the accurate hypothesis and test procedures for the above problem are:

1. Hypothesis: H₀: μ = 70 versus H₁: μ ≠ 70
2. Test Statistic: ɀ = (Arithmetic Mean-70)/(σ/√n) = 56.0 - 70/ 15.57/√5 = -2.01
3. P-value: Presuming H₀ is true out here, the probability of chance variation that yields a t- statistic more extreme is -2.01 on either side of 0 on X-axis. (Note: Since H₁ direction is both low and high, the figure is .11)
4. Conclusion: Since P-value > 0.05 we do not reject H₀ here. Thus the sample doesn't provide quite a lot of evidence to show that the mean fill has shifted from 70%.

Hence the summary for μ is:
H₀: μ = c versus H₁: μ ≠ c
P-value: The total area greater than |t| and less than -|t| under the t-curve with n-1 degrees of freedom. If it is far enough from 0 on the X-axis on either side (the direction of H₁) the value of P would be small.

Conclusion: If P-value ≤ 0.05 we reject H₀ with a statistical significance.
If P-value ≤ 0.01 we reject H₀ with a high significance.
If P-value > 0.05 We do not reject H₀.

With the value of -2.01 we plotted the same on the bell curve and notice that the null hypothesis H₀ cannot be rejected. Hence the result cited is that the sample doesn't have enough evidence to show the mean shift from 70%.

Tables for One and Two-tailed t-tests

Most of the tables pose a critical t-value which only gives the values for either one or two-tailed tests. But definitely not both. But this table tells you in short how to use and remember the formulas for both the tests.

Table That You Have	Operation	To Obtain
One Tailed	Divide P by 2	Two-tailed Test for P/2
Two-tailed	Multiply P by 2	One Tailed Test for 2P

Well, if you are still holding a candle to the impossibilities of clearing stats, I advice don't!! Just open this article and remember a few simple things to entangle your doubts. All the Best!