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,

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

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;

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

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…..

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

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.

Narender Sharma

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 meanNow 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 asH0: 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

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.

### Like Comment and Share

So this was the concept of a two-tailed test with respect to hypothesis testing.- If you have any questions or suggestion then please do write your feedback in the comment box below and hit the like button if you liked this blog and share it among your friends and colleagues.
- Subscribe ‘Shakehand with Life’ YouTube channel and press the bell to receive notifications of my latest videos
- Visit on shakehandwithlife.in to download the course notes and eBooks in PDF

Narender Sharma

## 3 comments:

It's very useful blog post with inforamtive and insightful content and i had good experience with this information.I have gone through CRS Info Solutions Home which really nice. Learn more details About Us of CRS info solutions. Here you can see the Courses CRS Info Solutions full list. Find Student Registration page and register now. Go through Blog post of crs info solutions. I just read these Reviews of crs really great. You can now Contact Us of crs info solutions. You enroll for Pega Training at crs info solutions.

Contour all your sufferings with the post COVID services offered by Aiwo covid rehab centre chennai. Aiwo has launched its rehabilitation services for post-COVID patients. Aiwo has worked with numerous patients who suffered from the COVID.Highly trained physios for Post-Acute Rehabilitation with MASHing and Stretching.

Post acute rehabilitation treatment will help functional recovery and will improve the quality of life. Know the effects of treatment in rehabilitation on the patients with covid-19 rehabilitation medicine center .

.

Post a Comment