Two-Tailed Test for Hypothesis Testing in Statistics, Research and Lean Six Sigma

Hello everyone,
In this blog, we'll study a two-tailed test in the context of hypothesis testing. So far with respect to hypothesis testing, we have studied,

• The concept of hypothesis testing, 
• The test statistic, level of significance and level of confidence, 
• Concept of Type 1 and Type 2 errors, and 
• Risk of rejecting a null hypothesis in case it is true

Testing a Hypothesis

For testing a hypothesis we have three different situations

Case 1: H0: μ=μ0 and Ha: μ not equal μ0

When the null hypothesis stated as the population mean equals to the hypothetical population mean and alternative hypothesis stated as the population mean is not equal to the hypothetical population mean. It implies the population mean is either lower than or higher than the hypothetical population mean

Case2: H0: μ=μ0 and Ha: μ < μ0

When the null hypothesis stated as the population mean equals to the hypothetical population mean and alternative hypothesis stated as the population mean less than the hypothetical population mean

Case3: H0: μ=μ0 and Ha: μ>μ0

When the null hypothesis stated as the population mean equals to the hypothetical population mean and alternative hypothesis stated as the population mean greater than the hypothetical population mean

Now In case 1, we use two-tailed test to test the null hypothesis and in two other situations we use a one-tailed test such as left-tailed test and right-tailed test for case 2 and 3 respectively

Rejection Regions, Acceptance Region, and Critical Limits

In a two-tailed test, there are two rejections regions also known as critical regions, one on each tail of the curve. For a 5% significance level, the value of alpha (α) is 0.05. it defines the probability of the rejection area for the null hypothesis when it is true. And if we apply the two-tailed test, then it equally splits on both sides of the curve such as (α/2)=0.025

Now the acceptance region is 95% or 0.95, here in this region the null hypothesis will be accepted. The acceptance region is the area under the normal curve between the critical limits defined by Z=+/-1.96 at a 5% significance level. You may name these limits as lower critical limit and the upper critical limit for suitability

Now the area under the left half of the curve up to the lower critical limit Z= -1.96 is 0.475 and the area under the right half of the curve up to the upper critical limit  Z=1.96  is also 0.475 and both taken together equals to 0.95 or 95% area of the curve.

Now to find the area under the normal curve for Z=1.96, you may refer the table “Area under the standard normal distribution”  in any book of business statistics

Here 1.96 is the critical value, and if the value of test statistic like ‘Z’ test comes out less than  or equal to1.96  then the null hypothesis will remain in acceptance region and if the value of test statistic comes out above 1.96 at 5% significance level then the null hypothesis will rest in the rejection region.

Hence mathematically Acceptance Region is defined by;

A: |Z|<=1.96

And Rejection Region is defined by;

R: |Z|>1.96

Now to better understand the concept of the two-tailed test, let's take an example

Example Problem

A sample of 400 male students is found to have a mean height of 67.47 inches. And the mean height of a large population of male students is 67.39 inches and the standard deviation is 1.30 inches. Conclude that at a 5% significance level, whether the sample is taken from the population having a population mean 67.39inches.
Now to conclude that whether the sample is taken from this population at a 5% significance level, we first state the null and alternative hypothesis…..

Solution

So the null and alternative hypothesis for the given problem can be stated as
H0: μ=67.39 and Ha: μ not equal 67.39
Here alternative hypothesis is two-sided hence we use two-tailed test, also it is given that

• The sample mean (xbar)=67.47inches,  
• The hypothetical population mean (μ0)=67.39 inches, and 
• Population standard deviation (Std. Dev.p)=1.30 inches
• And Number of students in sample (n) = 400

Now let's process the z test by replacing the value of  x bar, mu not, sigma p, and N in the z statistic, assuming the population to be normal

Z= (xbar-μ0)/(Std. Dev.p /sqrt(n)

Z= (67.47-67.39)/(1.30/sqrt(400) = 0.080/0.065=1.231

The calculated value of ‘Z’ is 1.231 less than the critical value of Z=1.96 at a 5% significance level under a two-tailed condition hence it will lie in the acceptance region and therefore null hypothesis is accepted.
And we may conclude that the given sample with mean height of 67.47 Inches drawn from the population with a mean height of 67.39 inches and a standard deviation of 1.30 inches at a 5% significance level.

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So this was the concept of a two-tailed test with respect to hypothesis testing.

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Thanks for reading, see you in my next blog with a new topic.

Narender Sharma